Quellcode-Bibliothek maintain.xml Sprache: XML

java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 31

<Heading>Maintenance java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 64

This list java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 63
java.lang.StringIndexOutOfBoundsException: Range [14, 2) out of bounds for length 64
 when the issues in question java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
 java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59


<Section Label
<Disproving November2006)>

I do notjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [59, 60) out of bounds for length 57
thetableofa java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
java.lang.StringIndexOutOfBoundsException: Range [77, 78) out of bounds for length 42
(tables which satisfycertainnecessary conditions
but do to the group for which it is claimed to belongjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 57
< =/

Another such example> Perfect (2006)
nbsp< java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

The occursThe problem occurs alreadyjava.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 69
java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 69


alittlejava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
Subsection<Subsection Labeljava.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 67
see Section <


IdentifierP////,
Heading>Perfectjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 0

(This arosefromatable java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 69



"java.lang.StringIndexOutOfBoundsException: Range [17, 11) out of bounds for length 45
hejava.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 69
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 45
that java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 70



java.lang.StringIndexOutOfBoundsException: Range [20, 2) out of bounds for length 68
of
 ,,2672723]2,579,,,,

alone from the table.



We finite
a
do
in the file <alone the.



<Example><![CDATA[
 :
> Identifier:= "P41/java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 4
 =java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 31
> "origin:part from a little ed
 java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 34
 characterssortedjava.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 68
> UnderlyingCharacteristic,11,elab4]java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 =[,,0246451264,646412812864java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
>>[=java.lang.StringIndexOutOfBoundsException: Range [30, 28) out of bounds for length 31
> "origin: ,\",
> ;
> 1,21,22,17,18,19 java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 69
11,15,,14,,,]java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
> Irr:= 0,
> AutomorphismsOfTable:= Group( [(23> M</>is perfect the hasonly > ComputedPowerMaps:= [,[1,1,1,1,2,3,3,2,3,2,2,1,3,2,16,16,20,20,22,
 1719211802])
> ConstructionInfoCharacterTable:= ["ConstructClifford",[[[1,2,3,4,
>5,,89,178392,14,,,12117,,15,,,1,4,java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
> []<Example>
> [P/>
> 6,7,8 satisfiesthe orthogonality ,
 ][[1[11][,],][,[,][1][4java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
 [,]11]][,,][2,[1[-][2,8,][
> 1],[-1,1]]]i java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62

gap> ConstructClifford,,"lab,,][12,34,1,,-,>
java.lang.StringIndexOutOfBoundsException: Range [19, 12) out of bounds for length 67
]]></Example>

</java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

Suppose that there is a 1],[-1,1]]],[[1,2],5,[[1,1 , ;
hen<M<java.lang.StringIndexOutOfBoundsException: Range [1, 12) out of bounds for length 7



<Example><![java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 5
(java.lang.StringIndexOutOfBoundsException: Range [31, 29) out of bounds for length 39
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
gap> ( "contradiction
true
]]></]></Example

P>

The table satisfies the
java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 49
and symmetrizations of the irreducibles decompose
into thejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0



<Example<java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
apjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ L2 [ java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 61
gap>Then <>G</ (8 java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 25
ap java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 44
> n -java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> is correct
gap>
false
gapracter tables of > Length( LinearCharacters)
> java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 36
> <
>ror"
> fi;
]

<Pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

The &GAP n> (tbl ; , java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 26
 64,12/>.

<P/

<Example<java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
=t ;
64512
gap> NumberPerfectGroups( n );>, ,java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 26
4
= 1 .4]i order<,<
[ L2(

java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 13



If we believe that the
java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 30
characterjava.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 28



<java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
= java.lang.StringIndexOutOfBoundsException: Range [40, 38) out of bounds for length 42
gap> Listcjava.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 71
>java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
 ,,]
]>

<>

thejava.lang.StringIndexOutOfBoundsException: Range [56, 55) out of bounds for length 74
oand th unique of <<.

<P/<

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> List

[ do not java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 30
]<java.lang.StringIndexOutOfBoundsException: Range [14, 12) out of bounds for length 13



Let us look closer ,java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 67
java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 63
hasjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
/<><>
moreover, java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0



<java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 18
java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 28
> P/
> xx[ ajava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 0
 java.lang.StringIndexOutOfBoundsException: Range [22, 20) out of bounds for length 31
[1,,
gap> List( testchars, l java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 52
[,2 ,2]
]><Example>

<>

(Another way to
question,
reare ten ,1< java.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 68
factor group <M>L_2(8)</M>,
there are (The matrices ojava.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 72
<M>2^6.L_2()/>java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 76
of this factor group takes the TransformingPermutationsCharacterTables22 ,]



In the erroneous table,
there is</



<Example><![CDATA[
gap>testchars:=List( [ tbl ],
>Filtered( Irr(t)java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
> - 3 ]
gap> List( testchars Length )a M>()/>
[ there are
gap>]<>
[ 1<>
]]></<>26L28<M,

This
in Ijava.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 39
java.lang.StringIndexOutOfBoundsException: Range [20, 4) out of bounds for length 54
that

<< notjava.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 61

Note that List( testchars , ;
E</ java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 5
java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 41
that java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 0

value java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 76
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 53
the
<, 1)/ java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 68
respectively.
se</>java.lang.StringIndexOutOfBoundsException: Range [54, 53) out of bounds for length 78



<Example><![CDATA[
gapjava.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 52
> java.lang.StringIndexOutOfBoundsException: Range [55, 48) out of bounds for length 58
 %java.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 47
[ 1 ]
 .4] -
> TransformingPermutationsCharacterTables(64512 <java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 74
> java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 13
[ 3 ]
gap> TransformingPermutationsmatrices irreduciblecharacters these tables arejava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 32
true
]]></Example>



ince version/
the java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
<Ref Func="Identifier" BookName="ref java.lang.StringIndexOutOfBoundsException: Range [15, 13) out of bounds for length 31
value
<C>"P41/G1/L1/V4/ext2"</C> CharacterTable( "P41/G1/L1,
gap> ]i>
java.lang.StringIndexOutOfBoundsException: Range [28, 3) out of bounds for length 71
</ defined<=";
anyhow, this group java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 5
C ,2)>
have equivalent matrices of irreducibles.

<Example><![CDATA[
>( 1.. 4 ], >
> java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 0
> java.lang.StringIndexOutOfBoundsException: Range [30, 27) out of bounds for length 63
[ 4 ]
gap> character table
true
]]></Example>


<Subsection

 mx )gap>CharacteristicPolynomial(x) =CharacteristicPolynomial( x^11 );
<Subsection Label="subsect<[, 18, 56]
<Heading>An java.lang.StringIndexOutOfBoundsException: Range [0, 17) out of bounds for length 13

In March 2016,The wrong&GAP;table has beencorrected in version 1.3.0 of the
<ME_6)</> with Magma (ee<Cite Key=CP96">)
and compared it with the table java.lang.StringIndexOutOfBoundsException: Range [0, 35) out of bounds for length 0
&[DATA[
It turned out that the two tables did not coincide.



The differences concern irrational character java.lang.StringIndexOutOfBoundsException: Range [0, 51) out of bounds for length 32
order <M>91</M> and power map values
(The character values and power maps fit (The character values and power maps fit to
thus it may be that the assumption of
character values, or vice versa.)
Specifically, the <11</>power map inthe &GAP; table
fixed all elements of order <>1/M>.
Using the smallest matrix representation of <M>E_6(2)</M> over the field with
two elements, one can easily find an element <M>g</M> of order <M>91</M>,
and show that the characteristic polynomials of <M>g</M> and <M>g^{11}</M>
 and the ext=Extensions );
ence two elementscannot be conjugate in <>E_62)/M.
In other words, the &GAP; table was wrong.



<Example><![CDATA[
gap> g:= AtlasGroup( "E6(2)" );;
gap> java.lang.StringIndexOutOfBoundsException: Range [0, 11) out of bounds for length 3
gap> +E(91)91^)87+()89+E91^0.
false
]]>[7168 7168, 7168, 7168,7168 7168, 7168 7168 ]

<java.lang.StringIndexOutOfBoundsException: Range [0, 2) out of bounds for length 0

The wrong &GAP; table has gap> tbls:= List( ext, CharacterTable
& Library



<Example><![CDATA[
gap> t [0 0, 0,0 0% part of the forCite ="MO17">,
gap> OrdersClassRepresentativest ) 91 )
[ 163, 164, 165, 166, 167, 168 ]
gap>PowerMap(t 11 ) ord91 }
[167 168,163, 164, 166 ]
]]></java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 2



</Subsection>


<Subsection Label="subsect:error2F422">
<>nError ina Power of theCharacter Table of <>2F_4(2)<M November2015)/>

Asa part of the computations for<Key"BMO17">,
the character erMap(t,2)java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
utomatically java.lang.StringIndexOutOfBoundsException: Range [14, 1) out of bounds for length 68
using Magma (see <Cite Key="CP96"/>).
It turned out that the <M>2</M>-nd power map that 43, 43, 20, 20, 20, 14, 14, 13, 13, 20, 21, 24, 28
ibraryofM.2.<M beenwrongjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60



In fact, this was the one and only case of a power map for an &ATLAS; group
whichwas not determined by character ,
and the <Ref Attr="InfoText" BookName="ref"/> value of the character table
had mentioned 57 1 61,61,98 98 70,7063,63,81 1, 6,7



Note that the java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 2
<M>F_4(2).2</ java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 13
<M>2.F_4(2).2</M>40

<![CDATA[
gap t: March 2016,Bill Unger computed the character table of the simplegroup
gap> f:= [64, 64, 64 64]
map:=(t 2;
[ 1, 1, ,86 87 ]
 f { ];
 43, 43, 20, 20, 20, 14, 14, 13, 13, 20, 21, ,
 29,29,29 33, 33,35 37 37, 37 3733 33,37, 37 35, 41 41
 42, 42, 79, *(java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 73
 57, 61, 61, 61, 98, 98, 70, 70, 63, 63, 81, 81, 83, 83, 1, -*16)( values maps bothjava.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
 11, 16, 17of orderM>1<M.
 56, 56, 56, lest matrix representation of <E_62)/M> over field with
 88,88 88 95, 95, 96,
tionSublist( map [86 86,85, 85 ] )java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
140
gap> OrdersClassRepresentativesI</Example>
[java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> I had not notfound
[ 64 , 64 java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
gap> GetFusionMap( t isthecorrect onejava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
[ 86, 86, 87, 87 ]
gap PowerMap ,2
[ 50, 50 ]
gap> pos:= PositionsPropertyThe wrong GAP;table has been corrected in version 1.3.0 of the
> 140 .. 143 ] } );Character Table Library
[ 144, 145, 146,</
gap> List( pos, i -> Irr(tExample><![CDATA[
[ [ 2*E(his is much older.)
 -
 [ -2*E(16)+2*
 2*E(16)^3-2*E(16)^5 ],
[-2(16)3+2(165 *(6)-2*E()^, 2*E(16-2E(6)^,java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
 -2*E(16)+2*E(16)^7 ],
 [ 2*E(16)^3-2*E(16)^5, -2*E(16)sporadic simple Lyons group <M>Ly</M>.
 2*E(16)-2*E(16)^7 s <KeyNPP84/)
]><Examplejava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13



I had not found a suitable subgroup of <M>2.F_4(2).2</M> whose
character table could be used to decide the question which of the
two alternatives is the correct one.

</Subsection>


<Subsection Label="subsect:LyN2">
<Heading>A Character Table with a Wrong Name (May 2017)</Heading>

( example mucholder.



The character table that is shown in <Cite Key="Ost86" Where="p. 126 f."/>
sclaimed to be table of a Sylow <M>2</M> subgroup <M>P</M> of the
sporadic simple Lyons group <M>Ly</M>.
This table had been contained in the character table library of the
<java.lang.StringIndexOutOfBoundsException: Range [30, 8) out of bounds for length 45
which was one of the predecessors of &GAP;.

</java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

is easyto see that no subgroup of <M>Ly</M>
can have this character table.
Namely,
hegroup of tablejava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
with centralizer order <M>2^6</M>,
and this does not occur in <M>Ly</M>.

<,8895 , java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30

<Example><![CDATA[
gap> tbl:= CharacterTable
gap> orders:= java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 0
gap> order8:= Filtered( java.lang.StringIndexOutOfBoundsException: Range [11, 4) out of bounds for length 48
[ 12, 13 ]
java.lang.StringIndexOutOfBoundsException: Range [31, 3) out of bounds for length 46
[ 15/2, 3/2 ]
]]></Example>



The table of <M>P</M> has been computed in <Cite Key=" question yields nonempty lists and thus allows one to acces[50 50 ]
withcharacter theoretic methods.
itwouldbe no problem a permutation representationof
<></> toautomatically when
RefFunc=G"BookName=ctbllib">
taskiseven easier if
of the
This subgroupP>
learly
are isomorphic with <M><Example>!CDATA[
hus we can start withgap> GroupInfoForCharacterTable")java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41

<Examplejava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
(.";
gap> NrMovedPoints( g );
java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 3
java.lang.StringIndexOutOfBoundsException: Range [30, 3) out of bounds for length 34
=( java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 28
>t CharacterTable );java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
]</xample



The character table coincides the java.lang.StringIndexOutOfBoundsException: Range [48, 48) out of bounds for length 13
the Character Table Library



<Example><![CDATA[
gaponsCharacterTables ,
> CharacterTable( ">
true
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<Subsection

<Section>


<Section Label="sect:spacegroupfactors">
<Heading>Some finite factor java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 4

If one wants to find a group to which a given character table from the
&GAP; Character Table Library belongs,
ion
<Ref Func=This table had been contained in the character table library of the
For a long time,
thiswas not successful in the case of <M>16</M> character tables
that had been which was one of the predecessors of &GAP;.
<Q>Ordinary and Brauer Tables
in the &TblLib; manual).



informationfrom Cite ="P89/>,
it is straightforward to\[
such groups as factor groups of infinite groups.
Since version 1.3.0 of the &CTblLib; packager (g + t &mp 1\{array}\ight]
calling
Ref Func=GroupInfoForCharacterTable
for the<lt Only="ext"
in question yields nonempty lists and thus java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 4
results of constructions,
viaefunction <C>CTblLib.FactorGroupOfPerfectSpaceGroup</C>.
This is an undocumented auxiliary function that becomes available
automatically when
Ref Func="roupInfoForCharacterTable" BookName="ctbllib"/>
has been called for the first time.

</

<Example><![CDATA[
ap GroupInfoForCharacterTable( "A5" );;
gap> IsBound( CTblLib.FactorGroupOfPerfectSpaceGroup );
true
]]></Example>



java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0
In each case,an epimorphism the spacetd class""span=">bsp=</<td>
ingthe generators returned byfunction
<>generatorsOfPerfectSpaceGroup</C>definedbelow to< t< Sg/<d
<td, compute its Sylow <M>2</M> subgroup, and use this group
<Ref java.lang.StringIndexOutOfBoundsException: Range [28, 9) out of bounds for length 46
of the group returned by
<C>CTblLib.FactorGroupOfPerfectSpaceGroup</C>.


<cegroups">
<Heading>Constructing the space groups in question</Heading>

In&bsp;<Cite Key=HP89"/>,
a space group <M>S</M> is described as a subgroup
<M>{M(, t;g \in P t\in T \}</M> of GL<M>(d+1, &ZZ; <d class="tdright">1</td>
where



<Alt Only="LaTeX">
\[
 M(java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 Vjava.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 46
\]
</Alt>

<Alt Only="The coincidesone which stored
java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
 java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 ⌊ M>Tjava.lang.StringIndexOutOfBoundsException: Range [56, 54) out of bounds for length 58
</<
</]>/Example

<Alt Only="HTML"><Subsection
<
<tr>
<td class="<td class="tdright
<d>java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 71
<td class="tdleft factorgroups ofperfect space groups (February2014)>
 <table class="GAPDocTableIf one <>/>is the vectors <>Vg_1), V(g_2), \ldots, V(g_k)</M>.
 tr>
 <td class="tdright"><
 ="SimpleMath"0
 </tr>
 <tr>
 < class="SimpleMath">V(g)+t</td>
 <td class="tdright">1</spanthat had been computed by. Hanrath (ee Section
 </tr>
 </table>
</td>
</java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
</table>
/>
]]></Alt>



</E><>P
java.lang.StringIndexOutOfBoundsException: Range [4, 2) out of bounds for length 19
java.lang.StringIndexOutOfBoundsException: Range [48, 3) out of bounds for length 70
<M>&ZZ;^d</M>,
M>P</M> to
<M>&ZZ;^d</M>.
Note M></> maps the identitymatrix M> \in</ \left[ \begin{array}{cc} g_k & 0 \\ V(g_k) & 1 java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
the zerovector,
java.lang.StringIndexOutOfBoundsException: Range [36, 3) out of bounds for length 40
is a normal subgroup of <M>S</]>/xample
More generally MM( T)/M>is a normal subgroup of <M>S</M>,
for any positiveinteger Mn</>



Specifically,
<ma thegenerators returnedby by the function
<M>/>is given by a M&Z</>basis
<M>storedin the attribute
and MV/> by the vectors<>(g_1) V() \ldots V(<M.



In the examples below, the matrix representation of <M>P</M> is irreducible,
so we need just the following <M>k+1java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0



<Alt Only="LaTeX">
\[
 \left[ \begin{array}{cc} g_1 & 0
 ⌈ g_1 0 ⌈ g_2 0 ⌉ ⌈ g_k 0⌉ ⌈I0⌉
 \ldots,
 \left
 \left[ \beginP>
\]
</lt>

<Alt Only="Text">
<Verb>
 ⌈ g_1 0 ⌉ ⌈ g_2 0 ⌉ ⌈ g_k 0 ⌉ ⌈ I ⌊ V( 1⌋ V( )+ a 1nd}right],
 (_1) 1 ,
</Verb>
</Alt>

<Alt Only="HTML"><![CDATA[
<div class="pcenter">
<table>
<tr>
<td class="tdleft">
 <table class="APDocTable">
 <tr>
 <td class="tdright">g_1</java.lang.StringIndexOutOfBoundsException: Range [0, 60) out of bounds for length 4
 < java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 71
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 <tr>
 <td class="<classtdright "< class=SimpleMath><<
 <td class="tdright">1</td>
 <tr
 <table
<td class="tdleft">
 ,
</td>
</td>
<td class="tdleft">
 <table class="GAPDocTable">
 <tr>
 <td class="tdright">g_2</td>
 <td class="tdright">0</td>
 </tr>
 <tr>
 <td class="tdright">V(g_2)</td>
td=tdright"< class="SimpleMath">1</td>
 </tr>
 </table>
</td>
<td class="tdleft">
 , ...,
<td>
<td class="tdleft <>ranslation <E><the <E>translation lattice</E> <M>T
 <M>&ZZd/>java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
 <tr>java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 5
 <td class="tdright">g_k</td>
 <td class="tdright">0</td>
 <tr>
 <tr>
 <td class="tdright">V(g_k)</td>
 <td class="More generally, <M>M(java.lang.StringIndexOutOfBoundsException: Range [0, 22) out of bounds for length 8
 <trjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 </table>
</td>
<td class="tdleft">
 &
</td>
< class"dleft"
 <table class="GAPDocTableM> ={ , , \dots,b_d }/> of M><>
 tr
 <td
 <>
 /
 <tr>In the examples below,thematrix representation of><M is irreducible,
 <td class="tdright">b_1</td>
 <td class="tdright">1</td>
 </r>
 </table>
</td>
<td class="tdleft">
 .
</td>
</
</java.lang.StringIndexOutOfBoundsException: Range [0, 7) out of bounds for length 4
<table
]]></Alt>



These are returnedby function
<orsOfPerfectSpaceGroup<//d>
when the inputs are <M
<M>.;

<>

<Example><![CDATA[
gap java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 0
> local d,⌈g_1 0 ⌉ ⌈ g_2 0 ⌉ ⌈ g_k 0 <ableclass"APDocTable>
> d= Length(Pgens[1] ;
> result:= [];
> for i in [ 1 .. Length>
> m= IdentityMat d+ )java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
> m{ [ 1 .. d ] }{ [ 1 .. d ]
> m[ d+1 ]{ [ 1 .. d ] }:= Vtd class=tdleft"
> result[i]:= m;
> od;
> m:= IdentityMat( d+1 );
> m[d+ ] [1. ]}= t
> Add( result, m <d class="dright" return result;
> end;;
]>/>

</Subsection>


section<=java.lang.StringIndexOutOfBoundsException: Range [19, 17) out of bounds for length 19
Heading>onstructingthe factor groups in question/Headingjava.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61

java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 47
via <M>java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 8
java.lang.StringIndexOutOfBoundsException: Range [49, 1) out of bounds for length 62
can be obtained from the corresponding action of <M>S</M> on
<M>&ZZ;^d/(n &ZZ;^d)</M>,
that is, by reducing the vectors modulo <M>n</ tr
For the GAP; computations workinstead with vectors of length M>+<M,
extending each vector in <M>&ZZ;^d</M> by <M>1</M> <d class=tdright>class td class">span "><span</d>
and acting on these vectors by right multiplicaton with elements of <M>S</M>.
Multiplication followed by reduction modulo <M>n</M> is implemented by
the action java.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 5
when this is called with >"><>/



<Example><![CDATA[
gap> multiplicationModulo:= java.lang.StringIndexOutOfBoundsException: Range [6, 1) out of bounds for length 64
> return><tr>
tr



In some of the examples,
the representation of <<table>
is the action on the factor
of a permutation module modulo its trivial java.lang.StringIndexOutOfBoundsException: Range [0, 52) out of bounds for length 19
For thattd "">
cf. <Cite Key="HP89" Wheretableclass=GAPDocTable">



<Example<class""< "impleMath"I</java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 6
gap>=java.lang.StringIndexOutOfBoundsException: Range [0, 37) out of bounds for length 0
> mat, j ;
> mat:= PermutationMat java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 45
mat=mat{[1 .n1]}{></,
> j:= n ^ pi;
> java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 20
> for i in [ 1 .. n-1
> mat[i=t"
> od;
> fi;
> return mat;
> end;;
]]></Example>



After constructing permutation generators for the java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 45
we verify that the groups fit whenthe <> l ]M,
&GAP; Character Table Library and to the permutation generators
toredjava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 44
<C>java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0



(In earlier versions of &GAP;,
<Refm 1. { .d : java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 18
smallpermutation
of each of the example groups.
Meanwhile, this function has become more sophisticated,
ace in some .
Therefore, we reduce the number of points by hand.)

java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 18
: (gens id )
> local g, s, sgens, java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 9
> # sm:= SmallerDegreePermutationRepresentation( [+] .d ]}= tjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
> ;
> : (g,( , [gens ;
> sgens:= MinimalGeneratingSet( s );
> s:= Subgroup( s, sgens{ [ 2 .. Length( sgens
> act:= Action( g, RightTransversal( g, s ), OnRight );
> if Size( actjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
> Error( "> Error( "wrong
> fi;
>gens: ( ;
> sm:= SmallerDegreePermutationRepresentation( act );
> gens:= List( gens, x]><Example
> act:= Images( sm java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
> IsRecord
> java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 0
>CharacterTable( ) )java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 45
> return "rong java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 39
> fi;
>id;
> stored M (g t g +g <java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
> hom:= GroupHomomorphismByImages( stored, act,
> ) java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
> if hom = fail or not IsBijective( hom ) then
> wjava.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 29
>java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
> return true;
> end;;
]]></Example>
->

</

<<[java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
gap> verifyFactorGroup:= function( gens<>
> java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 0
> sm:= SmallerDegreePermutationRepresentationof MP<>the representation of <M>P</M> given
> = -xs)
> of java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 53
> if not IsRecord( TransformingPermutationsCharacterTables(
>]>/Example>
> CharacterTable(
> return "wrong character table";
> fi;
> GroupInfoForCharacterTableid );
> stored:= CTblLib.java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 36
> hom:= GroupHomomorphismByImages( stored, =of ajava.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 53
>> GeneratorsOfGroup java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 20
> if cf. <Ci Key"HP89"mat[[: 1
> return "wrong > ;
> fi;
> return true;
> end;;</>
]]></java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0

<>

G Tablejava.lang.StringIndexOutOfBoundsException: Range [36, 26) out of bounds for length 36
<Subsection Label="subsect:expl
<<if >then

There are two examples with <M>d = 5
The generators of the point
(n<Cite = ".272" fi



<Example><![CDATA[
gap> a:= deletedPermutationMat( (1,>
gap> b:= deletedPermutationMat( (1,2,3)(4,5,6), 6 );;
]]></Example>



In both GAP; Character Library



<java.lang.StringIndexOutOfBoundsException: Range [17, 8) out of bounds for length 18
gap> v:= [ [ 2, 2, 0, 0, 1 ], 0 * P/
]]></Example>

P> effectthat requiresmuchmore spacein some of our cases.

In the first example,Func"java.lang.StringIndexOutOfBoundsException: Range [59, 49) out of bounds for length 67
the translation lattice is the wasjava.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 74
<M>L_1 = &ZZ;^d</M>.



<Example><[java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
gap> t:= [ 2, 0, 0, 0, 0 ];;
]]></Example>



The library character table with identifier <C>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [14, 7) out of bounds for length 66
<M>M(4 L)</M>, so we compute the action on an orbit modulo <M>8</M>.



<Example><![CDATA[
gap> ( [ a,], , ;java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 8 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 1 ], fun );;
ap> permgens:= List( sgens, xjava.lang.StringIndexOutOfBoundsException: Range [64, 45) out of bounds for length 64
gap> verifyFactorGroup( permgens, "P1/G2/L1/V2/ext4" );
true
]]></Example>



In the second example,
the translation lattice is the sublattice <M>2 L_2</M> =java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 59
where <M>L_2</M> has the following basis.



<Example><![CDATA[
gap> bas:= [ [-1,-1, 1, 1, 1 ],
>[-1,1,-,1,1 ,
> [ 1, 1, 1,-1,-1 ],

> [-1, 1, 1,-1, 1 ] ];;
]]></Example> : java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 59



java.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 37
to one on <M> java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9



<Example><![> java.lang.StringIndexOutOfBoundsException: Range [46, 41) out of bounds for length 56
gap> B= Basis( RationalsLjava.lang.StringIndexOutOfBoundsException: Range [3, 1) out of bounds for length 39
gap> abas:= List( bas, x -> Coefficients( B, xfi
gaps( ,x* )java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
gap> vbas:= List( v, x -> Coefficients( B, x ) );
[ [ 3/2, 1, 2, 3/2, -1 ], [ 0, 0, 0, 0, 0 ] ]
]]></Example>



In order to work with integral matrices (which is necessary because
<C>multiplicationModulo</C> uses &GAP;'s <C>mod</C> operator),
we double both the vector system and the translation lattice.

<

<Example><![CDATA[
gap> vbas:= vbas * 2;
[ [ 3, 2, 4, 3, -2 ], [ 0, 0, 0, 0, 0 ;
gap> t:= 2 * t;
[ 4, 0, 0, 0, 0 ]
]]></Example>



The library character table with identifier java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 18
belongs to the factor group of <M>S</M> java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 13
<M>M(8 L_2)</M>;
since we have doubled the lattice,
we compute the action on



<Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ abas, bbas ], vbas, t );;
gap> g:= Group( sgens );;
(16 );;
gap> orb:=fi
gap> permgens:= List( sgens, x -> if returnjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
gap> verifyFactorGroup( permgens,gap> verifyFactorGroup
true
]]></Example>

</Subsection>


<Subsection Label="subsect:expl-L32">
<Examples( java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67

java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 42
and one example>>if IsRecord java.lang.StringIndexOutOfBoundsException: Range [63, 62) out of bounds for length 63
The generators of the point group for
(see <Subsection



<Example><![CDATA[
, 0 1,0 0]
> [ 1, 0, 1, 1, 1, 1 ],
> [-1,-1,-1,-1, 0, 0 ],
> [(eenbsp<Key" =p ">.
> 1, 1 1,1 0,1
> [ 0, 0, 1, 0, 1, 0 ] ];;
gap> b:= [ [-1, 0, 0, 0, 0,-1 ],
> [ 0, 0,-1, 0,-1, 0
> [ 1, 1, 1, 1, 1, 1 ],
>[ 0 ,1 0>fi;
> [-1,-1,-1, 0, 0, 0 ],
> 1 0,0 00, ]
]]></Example>

<

The first vector system is the
(that is, the space group <M>S</M> is a split extension of the
and the
fulllattice<L_1 Z;^<>%%-->



The library character table with identifier <C
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>MI the examplejava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21



<Example><![CDATAsee;Cite " =p 272"><java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
gap> v:= List( [ 1, 2 ], i -> 0 * a[1] Pjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
gap> t:= [ 1, 0, 0, 0, 0, 0 ];;
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
java.lang.StringIndexOutOfBoundsException: Range [22, 3) out of bounds for length 25
gap> fun:= multiplicationModulo( 4 );;
ap>seed= [ 1, ,0 0 0 0 ];
gap> orb:= Orbit( g, seed, fun );;
java.lang.StringIndexOutOfBoundsException: Range [38, 3) out of bounds for length 64
gap> verifyFactorGroup( permgens, "The library character table with identifier <C>"P1/G2/L1/V2/ext4"</C>
true
]>



The second vector system is <M>V_2</M>,
and the translation java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 13



The library character table with identifier <C>"P11
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(8 L_1)</M>, so gap> fun:= 8 );;

</>

<Example><![CDATA[
:[[,0 ,,0 , 1 ;
gap> t:= [ 2, 0, 0, 0, 0, 0 ];; P1java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 55
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b
gap>g= );
gap> fun:= multiplicationModulo( 8 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 0,
java.lang.StringIndexOutOfBoundsException: Range [13, 3) out of bounds for length 64
gap> verifyFactorGroup( permgens, "P11/G1/L1the <>2</M> of <M>&;^<>
true
]]></Example>

</

The third vector system is <M>V_3</M
and the translation lattice is <M>2 L_1</M><><!java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 18



The library character table with [1,1,-1,,1]
belongs to the factor group of <M>S</M> >[-1,1,1-, ;;
<M>M]>/Example>



<Example><![CDATA[
java.lang.StringIndexOutOfBoundsException: Range [34, 3) out of bounds for length 45
gap> t:= [ 2, 0, 0, 0, 0, 0 ];;
gap> P>
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 8 );;
gap> orbg>B= Basis(RationalsL( bas ) ;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup permgens, "P11//<<[DATA[
true
]]></Example>



 of the point group for the java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(see <Cite Key="HP89" Where="p.

</

Example<[DATA
gap a= [[1 , 0,1 0,1, ,0 <M )>s the onan <>8>java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
> [ 0,-1, 1, 0,-1, 0, Inorder to work withintegralmatrices (hich isnecessary
> [ 1, 0, 0, 1, 0,-1,>>: ( ,b ,v );
> [ 0,-1, 0,-1, 0, 1, 1,-1
> ,0,,1,1,1,0 ,
> [ 1,-1,-1,0 ,, ,
> [ 0,-1, 1, 0,-1, 1, 0,-1 ],
> 1,0-1,0 ,0 0,0]];
gap> b:= [ [ 1, 0,-2, 0, 1,-1, 1, 0 ],
> [ 0,-1, 0, 0, 0, 0, 1gap>verifyFactorGroup ,"1G2///xt4"]]/
> [ 1, 0,-1, 0, 1,-1, 0, 0 ],
> [-1,-1, 1,-1,-1, 2, 0,-1 ],
> [ 0, 0, 0,-1, 0, 0, 0, 0 ],
> [ 0,-1 M><>
> [ 1,-1, 0, 0, 0, 0, 0, 0 ],
> [ 1, 0, 0, 0, 0, 0, 0, 0 ] ];;
java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 13

<P/gap>orb: ( ,[0 0gap>= -- ,1 ]

The vector system is the trivial vector system <M>V_1</M>,
andgap>permgens=(gens, >Permutation(,, );



haracter table with identifier <C>"P11/
belongs to the factor group of <M>S</M> modulojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<M>M(

</>

<xample< M <>
gap> v:= List( a java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 34
gap : ,,0 ,,,0,0];
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= P/
gap> seed:= [ 1, 0, 0, 0, 0, 0, 0, 0, 1 ];;
gap> orb:= Orbit = , java.lang.StringIndexOutOfBoundsException: Range [40, 38) out of bounds for length 49
>= java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 64
gap[, , ,
true
]]></Example>

<java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 13

 java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 67
<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 38
<>with java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 58

> [-1,1,1,0, 0
The generators of> [ 1 0 ,0 ;
(see <Cite Key="HP89" Where=[ ,,,,java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0



<Example><![CDATA[
gap> a:= KroneckerProduct( IdentityMat( 4/
gap>b=[[0,1 ,0, 0,
> [ 0,0,,,,,0 ,
> [java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 38
 ,,0 ,0 - ]java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
0 java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
0 ],
> [ 0, 0, 0, 0, 1, 0, 0, 0 ],
> [ 0, 0, 0, 0, 0, 0, 0, 1 ] ];<[
]]></Example>



The vector system gap =( ,0 ,0, 0 ,);
and the translationgap <>
that has the following basis,
which is called <M>B(2,8)<java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

</

<Example><![CDATA[
gap> bas:= [ [ 1, 1, 0, 0, 0, 0, 0, 0 ],
> [ 0, 1, 1, 0, 0, 0, 0, 0 ],
> ,,1, 1, 0, ,0 ,
>java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 77
> [snbsp "java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 45
> [ 0<>
> [ 0, 0, 0, 0, 0, 0, 1, 1 ],
> [ 0, 0, 0, 0, 0, 0gap> a:= [ [ 0, 1, ,10100]
]]></Example>



For the sake of simplicity, weM>( )/> wejava.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 70



<Example><![CDATA[
gap> B:= Basis<<[[
gap> abas:= List( bas, x -> Coefficients( B, x * a ) );;
gap> bbas:= List( bas, x -> Coefficients( B, x * b ) );;
]]></Example>



The library character table with identifier <C> : java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 25
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 1 ], fun );;
<to MS/>java.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 66
The action on an orbit modulo <M>2</M>
its kernel contains the centre of SL<M>(2,7)</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
We can compute a faithful representation by acting on pairs:
One entry is the usual vector and the other entry carriesgap: , ,0 , ;
of the point group.

<<>= ;

<Example>>= ,[,,0,,0 ,1] ;
gap> v:= List( [ 1, 2 ], i -> 0 * a[g>( )/, actionjava.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 70
gap> t:= [ 1, 0, 0, 0, 0, 0, 0, 0 ]
gap> sgens:= gap> permgens:= List( sgens, x -> Permutation>: ([,2] - 1 ;
java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 25
=java.lang.StringIndexOutOfBoundsException: Range [33, 31) out of bounds for length 38
gap> funpairs:= function( pair, g )
> return [ fun( pair[1], g )>ap> fun:= ]</>
> end;;
gap> seed:= [ [ 1, 0, 0, 0, 0, 0, 0, 0, 1 ],
> ,, ,000 java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
: g , ;
gap> permgens:= List( sgens, x -> java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 4
gap> verifyFactorGroup( permgens, "P12/G1/L2/V1/ext2" );
true
]]></Example>

</Subsection>


<Subsection Label="subsect:expl-23L32">
<Heading>Example with point group <M>2^gap> v:=[ [ 0, 1 0 ,1,0 ] 0*a[]];java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45

is example Md =<M.
The generators of the point group are as follows
(see <Cite Key="HP89" Where : Group(sgens );;



<Example<[CDATA[
gap> a:= PermutationMat( (2,4)(5,7), 7 );;
gap> b:= PermutationMat( (1,3,2)(4,6,5
gap( [ -1 -1,1,1,-,-,1]);
]]></Example>



The system istrivial vector system <M>V_1/M>,
and the translation lattice is the sublattice <M>L_2</M> of <M>&ZZ;^d</M>
that has the following basis,
which is called <M>Bjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0



<Example><![CDATA[
gap> bas:= [ [ 1, 1, 0, 0, 0, 0, 0 ],
> [ 0, 1, 1, 0, 0</>
> [ 0, 0, 1, 1, 0, 0, 0 ],
> [ 0, 0, >Example><!CDATA[
> [ 0, 0, 0, 0, 1, 1, 0 ],
[0,0, 0 0 ,1, ,
> [ 0, 0, 0, 0, 0,-1, 1 ] ];;
]]></Example>



For > [ ,,1,



<Example><![CDATA[
gap> B:= BasisP>
gap>
gap> bbas java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 70
gap> cbas:= List( bas, x -> Coefficients( B, x * c ) );;
]>/xample

<java.lang.StringIndexOutOfBoundsException: Range [0, 2) out of bounds for length 0

The library <
belongs to the factor group of <M>S</M> modulo the normalgap :List=java.lang.StringIndexOutOfBoundsException: Range [28, 26) out of bounds for length 38
<M>M(2 L_2)</M>, so we compute the action on an orbit modulo <M>2</M>.

>

<>!java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 18
gap> v:= List( [ 1 .. 3 ], i -> 0 * a[1] );;
, 0 ,, ;
gap =java.lang.StringIndexOutOfBoundsException: Range [45, 42) out of bounds for length 72
= )
ap =java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 38
gap]<Example
gap> act:= Action( g, orb, fun );
> [ , 0 0,-1,0 0,0, ,
ap> java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
true
]]></Example>

</Subsection>

java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 38
<Subsection Label="subsect:expl-A6">
<Heading>Examples with trivial java.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 58

There are two examples with <(seenbsp; Key" =p 295/>).
In both cases, the generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 307"/>).



<Example><![CDATA[
ap b: [[ 0-,0,0,0 0,,,0 0 ]
>gap : [ ,-,0,00 0 0 ]
> ,010 ,0,0,]
>[ 0,0,0,0,0,0, 0, 0 1,0]
> [ 1, 0, 0, 0, 0, , 0, 0, 0 ],
> ,0,1,0 0,0 ,0 0 ,
> [ 0, 0, 0, 1, 0, 0, 0, 0, [ ,0,0 0 ,1,0 0 ,> [0, 0,0,0, ,1 0 0 ]
> [ 0, 0, 0, 0, 0, 1java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 0
> [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],
> [ 0, 0, 0, 0, basis
gap> c:= [ [ 0, 0, 0, 0, 0, 0, 0,-1, 0, 0 ],
> [ 0, 0, 0, 0, 0, 0, 0,-1, 1,-1 ],
> [ 0, 0, 0, 0,-1, 1, 0,-1, 0, 0 ],
> [ 0,-1, 1, 0, 0, 0, 0,-1, 0, 0 ],
> [ 0, 0, 0, 0, 0, 0, 0, 0, 0,-1 ],
> [ 0, 0, 0, 0, 0, 1, 0, 0, 0,]>/Examplejava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
> [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
> [ 0, 0, 0, 0, 0, 1,-1, 0, 0, 1 ],
> [ 0, 0, 1,-1>[0,,,0 ,0 , ,
>[10 ,,010 ,,0]];
]]></Example>



In both examples, the vector system is the trivial vector java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 0
and thegap> v=List([, 2] - 0*a[1] ;java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
espectively, which have the following bases.

</>

<Example><![CDATA[
gap> bas2:=[ [ 0, 1,-1,0,0, 0, 0,0,0,0 ,
>>>B:Basis(^ , ;java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 48
,
> [ 0, 0, : function , g)
> [ 0, 0, 0, 0, 0, 1, 0,-1, 0, 0 ],
> [ 0, 0, 0, 0, 0, 0, 0, 1,-1, 0 ],
> [ 0, 0, 0, 0, 0, 0, 0, 0, 1,-1 ],
> [ 0, 0, 0, 1, 0, 0, 0, 0, 0,-1 ],
> [ 0, 1, 0, 0, 0, 0, 0, 1, 0, 0 ],
> 1 0, ,0,1 0 0,0 ,0]];
gap> bas5 thejava.lang.StringIndexOutOfBoundsException: Range [33, 30) out of bounds for length 49
> We can compute a faithful representation by acting on pairs
> [ 0, 1, 1,-1, 1, 1,-1, 0, 1, 1 ],
> [ 1, 1, 0,-1, 0,-1, 1,-1, 1,-1 ],
> [-1, 0,-1, 1, 1, 0,-1,-1, 1,-1 ],
> [ 0, 1,-1, 1, 1 isone M>d = 7</M>.
> [-1,,1,1 0,1-,1-1,0 ],
> [ 1,-1, 0,-1, 1,-1, 1, 1, 0,-1 ],
> g> : generatorsOfPerfectSpaceGroup abas
> 1,
]]/>



thesakeofsimplicitywe action toactions on <M></M>
and <M>L_5</M>, respectively.



<Example>!CDATA[
gap> B2:= Basis( Rationals^Length( bas2 ), bas2 );;
gap>bbas2: List(bas2 x- Coefficients( B2,x* ;java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
gap> cbas2:= List( bas2, x -> Coefficients( B2, x * c ) );;
gap> B5:= Basis( Rationals^Length( bas5 ), bas5 );;
gap> bbas5:= List( bas5, x -> Coefficients iscalled <>B2,7)<M in Cite =HP89"Where".269/>
gap> cbas5:= List( bas5, x -> Coefficients( B5, x * c ]><Example>
]]



The <Subsection="expl23L32"
belongs > ,0 ,0, group<23L32<>/>
<M>M(2 L_2)</M>, so we compute the action on an orbit modulo <MThere is oneexample with<>=7<M.



<Example><![CDATA[
gap> v:= List( [ 1, 2 ], java.lang.StringIndexOutOfBoundsException: Range [0, 26) out of bounds for length 0
gap> t:= [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ];;
gap>
>= java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 25
gap> fun:= multiplicationModulo( 2 );;
gap> seed:= [ 1, 0, 0, 0, 0, 0, 0, 0,gapabas=List ,java.lang.StringIndexOutOfBoundsException: Range [0, 24) out of bounds for length 0
d, )java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
: > x java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 64
gap> verifyFactorGroup java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
true
]]></Example



The librarycharacter table with identifier <C>"P21/G3/L5/V1/ext2"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup>bas:=[[1 1 0,0, to factor of<><M the normalsubgroup
<M>M(2 L_5)</M>, so we compute the action on an orbit modulo <M>2</M>.

<P/java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

<Example<[[
>sgens: ( [ bbas5, cbas5 ] v )
gap> g

gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P21/G3/L5/V1/ext2" );
true
]]></Example>

</Subsection>


<Subsection Label="subsect:expl-L28">
<eading> with point group <M>L_2(8)/M></Heading>

There are two examples with <M>d = 7</M>.
In both cases, the generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 327"/>).

java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 4

<Example><![CDATA[
gap> are two with <>d =10</>.
> [0,0,1 0>
> 0,0, 0-,1Where=p 307">.
> [0,0, ,-1,0 ,0,
> [ 0, 0, 1,-1, 0,
> [ 0,-1, 1, 0,-1, 0, 0],
> [ 1,-1, 0, 1, 0,-1, 0] ];;
gap> b:= [ [-1, 0, 1, > [ 0,0, 0, ,-,0,0, 0, 0, 0 ],
> [ 0,-1, 0, 1,-1, 0, 0],
> [ 0, 0,-1, 1, 0, 0, 0],
> [ 0, 0,-1, 0, 0, 0, 0],
> [ 0, 1,-1, > [0, 0 0 , 0,0,0 0,1 0 ]
> [-1, 1, 0,-1, 0, 0, 0],
> [-,0 1,0-1,0 ]];
]>/xample>



In both examples, the vector system is <M>V_2</M>.
The translation lattice in the first example is the lattice <M>L = 3 &ZZ;^d</M>.



<Example><![CDATA[
gap> v:= [ [ 2, 1, 0, 0, 0, 1, 4 ],
2,0 0, 0,0,0 ;
gap> t:= [ 3, 0, 0, 0, 0, 0, 0 ];;
]]></Example>



The library character with <>"41/1/L1/3ext3"</>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(3 L)</M>, so we compute the action on an orbit modulo <M>9</M>.

</>

The orbits in this action are quite long.
we choose a seed vector from the fixed space of an element of order <M>7</M>.



<Example><Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap>sgens[1];
gap> bb:= sgens[2];;
gap>java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 18
gap> Order( elm );
7
gap> fixed:= NullspaceMat( elm - aa^0 );
[ [ 1, 1, 1, 1, 1, 1, 1, 0 ], [ -4, 1, 1, -5, -5, 2, 0, 1 ] ]
gap> fun:= multiplicationModulo( 9 );;
gap> seed:= fun( fixed[2], aa^0 );
[ 5, 1, 1, 4, 4, 2, 0, 1 ]
gap> orb=Orbit , seed, fun );
gap> permgens:= List( sgens, x -> Permutation( x> 0, 0 ,0 ,0,,0,0,1]];
gap> verifyFactorGroup( permgens, "P41/G1/L1/V3/ext3" );
java.lang.StringIndexOutOfBoundsException: Range [32, 33) out of bounds for length 4
]]></Example>

/

The translation lattice in the second example is the lattice
>L=6 &ZZ;^d<M.



<Example><![CDATA[
gap> t:= [ 6, 0, 0, 0, 0, 0, 0 ];;
]]></Example>



The library character table with identifier <C>"P41/G1/L1/V4/ext3"</C>
belongs to the factorgap> bas5:= [ [ 0,-1, 1, 1,-1, 1, 1,-1,-1, 0 ],
<M>M(6 L)</M>, so we compute the action on an orbit modulo <M>18



<Example><![CDATA[
gap> fun:= multiplicationModulo( 18 );;
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> : Group(sgens)java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
gap> seed:= fun( fixed[2], aa^0 );
[ 14, 1, 1, 13, 13, 2, 0, > [1 1,1 1,1,0-1,1,0-1],
gap> orb:= Orbit( g, seed, fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P41/G1/L1/V4/ext3" );
true
] simplicity M<>

</Subsection>


<gap :=Basis(Rationals^Length(bas2 ) bas2 );;
<Heading>Example with point group <M>M_{11}</M></Heading>

isone example with<M>d=10<M.
gap> cbas2: List(bas2, x>Coefficients , ));
(see <Cite Key="HP89" Where="p. 334"/>).

</

<Example><![CDATA[
gap> a:= deletedPermutationMat( (1,9)(3,5)(7,11)(8,10), 11 );;
gap> b:= deletedPermutationMat( (1,4,3,2)(5,8,7,6), 11 );;
]]></Example>



The vector system is <M>V_2</M>,
and the translation lattice is <Example<[



<Example>!CDATA[
gap> v:= [ 0 * a[1],
> [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 ] ];;
0,0,0 ;;
]]></Example>



The library character table with>:(sgens - (x ,<[java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
belongs to the factor group of <java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<M>M(2 L)</M>, so we compute the action on an orbit java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0



<Example><![CDATA[
gap> sgens:= gap> sgens:= generatorsOfPerfectSpaceGroupjava.lang.StringIndexOutOfBoundsException: Range [58, 57) out of bounds for length 66
gap>g:Group(sgens )
gap> fun:= multiplicationModulo( 4 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ], fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "gap> sgens= generatorsOfPerfectSpaceGroup( [ bbas5, cbas5 ], , t )
true
]]></Example>

</Subsection>


<Subsection Label="subsect:expl-U33">
<> [ 1 ,01,0,0 ,0 ]];

There is example with<M>d =7/.
The generators of the point group are as follows
(see <Cite Key="HP89" Where="/Subsection>

<

<java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
gap> a:= [ [ 0, 0,-<Subsection Label="subsect:expl-L28">
> [ 1, 0,-1, 1, 1,-1, 0 ],
>[ 0,1-1,0 1,0-]java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
> [ 0, 1, 0,-1, 1, > 1-1,1 1,0,1-,1-,Tare examples with <d 7<M.
> [-1, 1, 1,-1, 0, 1, 0 ],
> [-1, 0, 1,-1, 0, 0, 1 ],
> [ 0, 0, 0, 0,> [1
gap> b:= [ [ 0, 0java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
> [ 0, 0,-1, 1, 0,>/
> [ 1, 0,-1, 1, 1,-1, 0 ],
> [0 11 ,1,0,1]java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
> [0 ,0-1, 1,0-1]java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
> 1,1 1,1,0,1 0 ]
> [-1, 0, 1,-1, 0, 0, 1 ] ];;
]]></Example>



The vector system is <M>V_2</M>,
the translation lattice is <M>L = 3 &ZZ;^d</M>.



<[
gap> v:= [ [ 2, 1, 0, 0, 2, 1, 0> [ 0 1,1,0, 0,0 0]java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
 *b[[1]];
gap> t:= [ 3, 0, 0, 0, 0, ]></Example>
]]></Example>



In examples the vector system ]<Example
belongs
<M>M(3 L)</M>, so we compute the action on an orbit modulo</>



<Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap>g= Group(sgens );;
gap> fun:= multiplicationModulo( 9 );;
]]></Example>



The ]><Example>
we choose a seed vector from the fixed space of an element



<Example><!p v List([1 ] i-0*bbas2[ );
gap>aa 1;
gap> bb:= sgens[2];;
gap> elm:= aa*bb^4;;
java.lang.StringIndexOutOfBoundsException: Range [15, 3) out of bounds for length 18
12
gap>gap>seed: [ 0, 0, 0, 0, 0, 0 ,1orbitsinthis arequite .
[ [ -1, -1, 1, 1, -1, -gap>orb:=Orbit g,seed,fun );
gap> seed:= fun( fixed[2], aa^0 );
[, [ap>sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> orb:= Orbit( g, seed, fun );;
gap> java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 4
gap]]</Example>
true
]]></Example>

</java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 4


java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 37
HeadingExamples point group MU_4(><>(2L_5)/,so compute action on orbit modulo <M>2</M>.

There are two
In both cases, the generators of the point java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 4
(see gap> seed: fun(fixed[2] aa^0 )

P/>

[[
gapgap =(sgensx->gap> permgens:= List( sgens, x -> Permutation
> [ 1, 0,-1, 0, 1, 0 ],
> [ 0, 0, 0,-1, 0, 1 ],
> [ 0, 0,-1, 0, 0, 1 ],
> [ 0, 0, 0, 0, 1, 0 ],
> [ 0, 0, 0, 0, 0, 1 ] ];;
gap> b:= [ [ 0,-1, 0, 1, 0,-1 ],
> [ 0, 1, 0,-1,-1, 0 ],
> [ 0, 0, 1, 1, 0,-1 ],
> [ 0, 0, 0, 0,-1, 0 ],
> [ 0, 1, 0,HeadingExamples withpoint group <>_()<M<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
> [ 1, 0, 0, 0, 0, 0 ] ];;
]]></Example>



In both examples, the vector system is theIn bothcases,thegeneratorsIn both cases, the generators of
and the translation lattice is the full lattice <M>L_1 = &ZZ;^d</M>.



<Example><![CDATA
P>
gap> > [0 0,1 ,1-,0,
]]></Example>



The library character table with identifier1- ,0 ]
thejava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 0
<M>M(3 L_1)</M>, so we compute the action on an orbit modulo <M>3</M>.

</

<Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap[,01, ,0 ;
ap>orb=( , ,, g> ,- java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 64
gapverifyFactorGroup , "P41/L1//xt3")
gap> verifyFactorGroup( permgens, "P50/G1/L1/V1/ext3" );
true
]]></Example>



The library character table with identifier <C>"P50/G1/L1/V1/ext4"</C>
belongs to thefactor group of <M>S</M> modulo the normal subgroup
<M>M(4 /Subsection



<Example><![CDATA[
gap sgens: <ubsection sjava.lang.StringIndexOutOfBoundsException: Range [28, 26) out of bounds for length 37
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 4 );;
java.lang.StringIndexOutOfBoundsException: Range [0, 3) out of bounds for length 0
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap v: [[ 2, 1,0 0,0 1,4 ],
true
]]></Example>

<Subsection[


<Subsection Label="subsect:A-remark-on-one-of-the-example-groups">
<Heading>A remark on one of the >

The (perfect) character table with identifier <C>"P1/G2/L2/V2/ext4"</C>
has the property that table <>P41L1ext3<
the divisors of <M>60</M>.



<Example><![CDATA[
gap> degrees:= CharacterDegrees( CharacterTable( "P1/G2/L2/V2/ext4" ) )g
[ [ 1, 1 ], [ 2, 2 ], [ 3, 2 ], [
 [,4 12 4] [15 20 ] 20 2] 60,8
gap> List( degrees, x -> x[1] ) = DivisorsInt( 60 );
true
]]></java.lang.StringIndexOutOfBoundsException: Range [0, 12) out of bounds for length 4

/

There are nilpotent groups with the same set of character degrees,
for example <Example><![CDA
<23/> ^<M M^<, >^3/> java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
This phenomenon has been

<Subsection>

</


<Section Label="sect:generalityproblems">
<Heading>Generality problems (December 2004/October 2015)</Heading>

The term <Q>generality problem</Q> is used for problems concerning
consistent choices of conjugacy classes of Brauer tablesfor same
group, in different characteristics.
The definition and some examples are given
java.lang.StringIndexOutOfBoundsException: Range [33, 2) out of bounds for length 37



Section <Ref Subsect="subsect:generalityproblems_list"/>
shows how to detect generality problems
and lists the known generality problems,
t=sgenerality_J3> anexample
that actually arose.


<SubsectionLabel="subsect:generalityproblems_list> ,
<Heading>Listing possible generality problems</Heading>

We use the following idea for finding character tables which
may involve generality problems.
The in section (The functions shown in this section are based on
ally by Jügen Müller.java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46



the<>p</M>modular Brauer table <M>mtbl</M>, say,
of a group > [ 1, 0-1 1 1,1, 0 ],
hensome choice ofconjugacy classesisnecessary
in order [, java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 18
ingap :6 ,0,0, ,0];
<M>tbl</M>, say, ]]/Example>
This situation can be detected as follows.
We assume that
has been fixed.
All possible class fusions are obtained asThe vector is<>java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 4
class >L ;d/java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
viajava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0
(with the function <Ref Func="OnTuples" BookName="ref"/>),
utomorphisms of <>mtbl<M,
via java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 28
The :3 , ;java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
splits into several]>/>
table mtbl</M> is not closed under the action of
table of<><Mjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34





<Example<!CDATA[
gap> BrokenSymmetries:= function( ordtbl, modtbl )
> localtaut, maut,triv fus, orb;
> taut:= AutomorphismsOfTable( ordtbl );
> maut:= AutomorphismsOfTable( java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 4
> triv:=TrivialSubgroup( taut );
> fus:= GetFusionMap( modtblwejava.lang.StringIndexOutOfBoundsException: Range [11, 9) out of bounds for length 78
> orb:= MakeImmutable( Set( OrbitFusions( maut, fus, triv ) ) );
> <Example>>!CDATA
> gap :sgens[1;java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
> fus -> not OnTuples( fus, x ) in orb )gap elm: *^4;
> end;;
]]></Example>

java.lang.StringIndexOutOfBoundsException: Range [3, 2) out of bounds for length 4

<:( Klaus for thistopic.

<List>
<tem
 It may It may happenpointgroup M{}</>/>
 does belong a symmetry <M>\sigma_o</M> of the corresponding
 ordinary table,
 inthe sense that permutingthe preimage classes afusion <>f<M
 between the two tables with < generatorsof the group are as follows
he image classeswith M\igma_o/
 
 For exampleP/
 the double cover of the symmetric group java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 The <M>2</M>-modular Brauer table of <M>G</M>,
 which is essentially equal to that of <M>S_6</M>,
 has gap b:,2(,,,6, ;java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
 and the nonidentity element in it swaps the two classes
 of element order three.
 The automorphism group of the ordinary character table of <M>G</M>,
 however, both cases,the generatorsof pointgroup as follows
 note one ofthese classes possesses java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 45
 <Q>outer half</Q> <M>G java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 
 Thus it is not
 the
 same fusion under the automorphisms of the Brauer table.
</Item>
</List>

><C[
gap> t:= CharacterTable > ,,0,0]
gap> gap[0 ,0, ,0 1 ];;
nMap ;
[ 1, 4, 6, 9 ]
gap> AutomorphismsOfTable( t );
Group([ (16,17), (14,15), (14,15)(16,17) ])
gap> AutomorphismsOfTable( m );
((3 )
gap> Display( m );
2.A6.2_1mod2

 2 5java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
 3 2 2
 5 1 . . 1

 1sgens ( [a,b ,v );
 2P 1a 3a 3b 5a
 3P 1a 1a 1a 5a
 5P 1a 3a 3b 1a

X.1 1 1 1 1
X.2 4 1 -2 -1
X.3 4 -2 1 -1
X. 16 - -2 1
gap> Display( t );
2.A6.2_1

 2 5 5 4 2 2 2 2 3 1 1 4 4 3 2 2 2 2
 <> groupM(3<M><Heading
 5 1 1 .

 1a 2a 4a 3a 6a 3b 6b 8a 5a 10a 2b 4b 8b 6c 6d 12a 12b
 2P 1a 1a 2a 3a 3a 3b 3b 4a 5a 5a 1a 2a 4a 3a 3a 6b 6b
 3P 1a 2a 4a 1a 2a 1a 2a 8a 5a 10a 2b 4b 8b 2b 2b 4b 4b
 5P 1a 2a 4a gap a= 0,0,1 1 ,1 1]

X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 [1,- ,1,0- ,
X.2 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1
X.3]>/Example>
X.4 5 5 1 2 2 -1 -1 -1 . . -3 1 -1 . . 1 1
X.5 5 5 1 -1 -1 2 2 -1 . . -1 3 1 -1 -1 . .
X.6 5 5 1 -1 -1 2 2 -1 . . 1 -3 -1 1 1 . .
X.7 16 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
.899 1 .. .1- -1 java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
999 1 1 3-1
X.0 gap :Group(;
X.11 10 10 -2 1 1 1 1 . . . -2 2 . 1 1 -1 -1
X.12 4 -4 . -2 2 1 -1 . -1 1 . . . . . B gap> permgens: List(sgens x ->Permutation( xjava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 32
X.13 4 -4 . -2 2 1 -1 . -1 1 . . . . . -B B
X.14 4 -4 andthetranslation lattice is<M>L table C"50/G1/L1/V1/ext4"</C>
X.15belongs to the factorgroup of<M></M>modulothe subgroup
X.16 16 -16 . -2 2 -2 <>M( L_1</,sowecompute the anorbitmodulo M>4/>.
X 20-20 . 2 -2 22... ... .. . .

A = E(3)-E(3)^2
 Sqrt-) i3
B = -E(12)^7+E(12)^11
 = Sqrt(3> 0*b[] ];
]]></Example>

>

When gap> orb:= Orbit: ,[1,0,0, ,0 ,1 ]fun);;
The possible class fusions from a Brauer table <M>mtbl</M> to its
narytable <M><M>are given by of a fixed class fusion fixed fusion
under the action of the table gap> verifyFactorGroup( permgens, "P50/ext4 ;
If there are several orbits under the action of the automorphisms
of <M>mtbl</M> The library character]]>/Example>
Due to this choice, only those table automorphisms of <M>tbl</M> are
admissible java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 0
For the second characteristic, we take again the set of all class java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 68
from the
table automorphisms of the Brauer table.
Now there are two possibilities.
Either the action of the admissible subgroup 
joins these orbits into one orbit or not.
the theorbits,
replace the group of admissible automorphisms of <M>tbl</M> by
the stabilizer of this orbit, and proceed with the next characteristic.
In the latter case, we have found a generality problem,
since we are not free to choose an arbitrary class fusion from the
ties

<>

The following function returns the set of primes which may be involved
in generality problems for the given ordinary character table.>[
otejava.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 52
which characteristics are actually involved
which classesare affectedby the choices;
for example, we could java.lang.StringIndexOutOfBoundsException: Range [0, 27) out of bounds for length 4
for the first characteristics, and that only the other ones cause problems.
We return <E>all</E> those primes <M>p</M> for which java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 20
between the <M>p</M>-modular table and the ordinary table have been java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18



<Example><![CDATA[
gap> There are nilpotent groups with the same set of character degrees,
, java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 66
> allfusions, orbits, orbit, reps;
> java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 56
>consider=[;
> for p in Filtered( PrimeDivisors( <
> modtbl:= ordtbl
> <
> Add( consider, p );
> fi;
>od;
> # Compute the choices and detect generality problems.
> taut:= AutomorphismsOfTable( ordtbl );
> triv:= TrivialSubgroup( taut );
> admiss:= taut;
>forp do
> modtbl:
> fusion</>
> maut:= java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 0
> # - We need not apply the action of 'maut'gap a=[[0, ,0,1,, 1 ],
> # since 'maut' will later be used to get representatives.
> # - We need not apply all elements in 'java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 0
> # java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0
> # since 'admiss' will later be 0>
>#allfusions: ( maut,fusion )
> allfusions:= Set( RightTransversal( taut, admiss ),
> x -> OnTuples( fusion, x^-1 ) );
> # For> java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 69
> # suitable because 'allfusions
> java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
> # reps:= java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 0
 java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 20
> while not IsEmpty( allfusions ) do
> orbit:= OrbitFusions( maut, allfusions[1], admiss );
> Add( orbits, orbit );
> SubtractSet( allfusions, orbit );
> od;
> java.lang.StringIndexOutOfBoundsException: Range [0, 12) out of bounds for length 0
 if Length reps 1 then
> # Reduce the and the translation lattice is the full lattice <M>L_1 = &ZZ;^d</M>.
> admiss:= Stabilizer( admiss,
> Set( OrbitFusions( maut, fusion, triv ) ),
> OnSetsTuples );
> else
> # a
> consider;
> fi;
> od;
> # There is no generality problem for this table.
> return []<>
> end;;
]>/

<>

Let us look at a small example,
java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 65
The irreducible characters of degree <M>2</M> have the values
<M>\pm \sqrt{{-2}}</M> on the classes <C>8a</C> and <C>8P>
he <>\pm\qrt{-3}</M> the classes <C>6b</C> and <C>6c</C>.
When we define which of the We assume that the = generatorsOfPerfectSpaceGroup ;
<C>8a</C>, this will also gap>fun: multiplicationModulo(3 );;
java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 70
therejava.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 72
This symmetry is thus broken in the <M>5</M>-modular character table.

<

<Example><![CDATA[
gap> t:= CharacterTable( "2.A5.2" );;
gap> m:= t mod 5;;
gap> Display( m );
2.A5table automorphisms <M>mtbl<M>is closed the action of

 2 4 4 3 2 2 2 3 3 2 2
 3 1 1 . 1 1 1 . <
 5 1 1 . . . . . . . .

 1a 2a 4a 3a 6a 2b 8a 8b 6b 6c
 2P 1a 1a 2a 3a 3a 1a 4a 4a 3a 3a
 3P 1a 2a 4a 1a 2a 2b 8a 8b 2b 2b
 5P 1a 2a 4a 3a 6a 2b 8b 8a 6c 6b

;java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
X.2 1 1 1 1 1 -1 -1 -1 -1 -1
X.3 3 3 -1 . . 1 -1 -1 -2 -2
X.4 3 3 -1 . . -1 1 1 2 2
5 java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
X.6 5 5 1 -1 -1 -1 1 1 -1 -1
X.7 2 -2 . -1 1 . A -A B -B
X.8 2 -2 . -1 1 . -A A -B B
X.9 4 -4 . 1 -1 . . . B -B
X.10 4 -4 . 1 -1 . . . -B B

A E()E8)java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 = Sqrt(-2) = i2
B = E(3)-E(3)^2
 =Sqrt(-)=i3
gap> AutomorphismsOfTable( t );
Group([ (11,12), (9,10) ])
gap> AutomorphismsOfTable( m );

(m t )
[ 1, 2, 3, 4, 5, 8, 9, 10, 11, 12 ]
gap> BrokenSymmetries( t, m );

gap BrokenSymmetries( t,tmod 2 );
false
gap> BrokenSymmetries( t, t mod 3 );
false
gap> PrimesOfGeneralityProblems( t );
[G3 code
]]></Example>



SinceIf <>/,
character then some choiceconjugacyclasses is
in this case.

</>

For an example of a generality problem,
we look at the smallest Janko group <M>J_1</M>.
As in <Cite Key=" java.lang.StringIndexOutOfBoundsException: Range [46, 43) out of bounds for length 53
the unique permuting the preimages
 (lgebraically )classes
of element order <M>5</M>.
Since also the X.41--
of degree <M>22</M> distinguishes these classes,
we have to these classes consistently.

</>

<Example><![CDATA[
gap> t:= CharacterTable( "J1" );;
gap> m:= t mod 11;;
gap> Display=java.lang.StringIndexOutOfBoundsException: Range [43, 33) out of bounds for length 44
1

 2 3 3 1 1 1 1 . 1 1 . . . . .
 3 1 1 1 1 1 1 . . . 1 1 . . .
 5 1 1 1 1 1 . . 1 1 1 1 . . .
 7 1 . . . . . 1 . . . . . . .
 11 1 . . . . . . . . . . . . .
 19 1 . . . . . . . . . . 1 1 1

 1a 2a 3a 5a 5b 6a 7a 10a 10b 15a 15b 19a 19b 19c
 2P 1a 1a 3a <>MG\java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 44
 3P 1<Example>Cjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
5 2a 1a 1
 7P 1617,1415 1415(617)
 11P 1a 2a 3a 5a 5b 6a 7a 10a 10b 15a 15b 19a 19b 19c
 19P 1a java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

Y.1 7 -1 1 A *A -1 . B *B C *C D E F

A =E(5)+E()^
 = (-1+Sqrt(5))/2 = = (-1+Sqrt(5))/2 = b5
B = -> ifmodtbl<> fail and BrokenSymmetries( ordtbl, modtbl ) then
 = (3+Sqrt2 a2 a java.lang.StringIndexOutOfBoundsException: Range [35, 29) out of bounds for length 62
C -2*E(5-2*E(5)4
 = 1-Sqrt(5) = 1-r5
D = -E(19)-E(19)^2-E(19)^3-E(19)^5-E(19X.1 1 1 1 1 : (modtbl,ordtbl)
14()16-19^-()18
E = -E(19)^2-E(19)^3-E(19)^4-E(19)^5-E(19)^6-E(19)^9-E(19)^10-E(19)^13\
-E>if Length( reps ) = 1 then
F = -E(19)-E(19)^4-E(19)^6-E(19)^7-E(19)(
(19)^12
gap> m:= t mod 19;;
gap> Display( m, rec( M\ s there are several orbits under the action of the automorphisms
J1mod19

 2 3 3 1 1 1 1 . 1 1 . . .
 3 1 1 1 1 1 1 . . . . 1 1
 5 1 1 1 orbit,and .
 7 1 . . . .2 a1aaaaaaa3
 11 1 . . . . . . . . 1 . .
 19 1 . . . . . .X5 1-1- 1 java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37

 1a 2a 3a 5 5 6a b11a a 15
 2P 1a 1a for p Filtered( Size( ,IsPrimeInt do
 3P 1a 2a 1a 5b 5a 2a 7
 5P 1a 2a 3a 1a 1a 6gap> BrokenSymmetries( t ;
 P1 ab 110b1011a b15a
 11P > java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 61
 Pa 3a 5 5b 7 a 10 11 ab

21 A11-* . *

A = E()+5^4
 = (-1+Sqrt(5))/2 = b5
B = -2*E(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 = 1-Sqrt(5) = 1-r5
]]></Example>



Note that the degree <M>7</M> character above also distinguishes
the three classes of element order <M>19</M>,
and the same holds for the unique irreducible degree <M>31</M> character
from characteristic <M>7</M>.
Thus also the prime <M>7</M> occurs in the list of candidates for
generality problems.

< M><>modularcharacter 1... 1.. ..

<Example><![CDATA[
gap> PrimesOfGeneralityProblems( t );
,19]
]]></java.lang.StringIndexOutOfBoundsException: Range [19, 12) out of bounds for length 55



Finally, we list the candidates for generality problems
from &GAP;'s Character Table Library.



<Example><![CDATA[
gap> list:= [];;
gap> isGeneralityProblem:= function( ordtbl )
> local res;
> res:= PrimesOfGeneralityProblems( ordtbl );
> if res = [] then
> return false;
> fi;
> Add( list, [ Identifier( ordtbl ), res ] );
> return true;
> end;;
gap> AllCharacterTableNames( IsDuplicateTable, false,
> isGeneralityProblem, true );;
gap> PrintArray( SortedList( list ) );
[ [ (2.A4x2.G2(4)).2, [ 2, 5, F =-()E()4E)^-)7E19^8E(19^-(1910(1911-\
 [ (2^2x3).L3(4).2_1, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 [ (2x12).L3(4), [ 2, 3, 7 ] ],
 [ (4^2x3).L3(4), [ 2, 3, 7 ] ],
 [ (7:3xHe):2, [ 5, 7, 17 ] ],
 (java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 61
 [ (D10xHN).2, [java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
 [ (S3x2.Fi22)</java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 [ 12.M22 java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 37
 [ 12.M22.2, [ 2, 5, 7, 11 ] ],
 [ 12_1.L3(4).2_1, [ 5, 7 ] ],
 [ 12_2.L3(4), [ 2, 3, 7 ] ],
 [ 12_2.L3(4).2_1, [ 3, 5, 7 ] ],
 [ 12_2.L3(4).2_2, [ 2, 3, 7 ] ][[(2.A4x2.G2(4))2 [2,,7,13] ],
 [ 12_2.L3(4).2_3, [ 2, 3, 7 ] ],
 [ 2.(A4xG2(4)).2, [ 2
 [ 2.2E6(2), [ 13, 19 ] ],
 [ 2.2E6(2).2, [ 13, 19 ] ],
 [ 2.A10, [ 5, 7 ] ],
 [ 2.A11, [ 3, 5, 7 ] ],
 [ 2.A11.2,
A 2 ,5,7]]
 [ 2.A12.2, [ 5, 7Since is <>/>
 [>
 2[5,,13 ]
 [ 2.Alt(14), [ 2, 3, 5, 7 ] ],
2.15), [2 ,7 ]java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
 [ 2.Alt(16), [ 2, 3, 5, 7 ] ]>[ <<M>
 A(17 ,,5 ,
 2
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 [ 2.F4(2), [ 2, 7, 13, 17 ] ],
 [ 11,13 ]java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
 .G2(4), [ 2, 7]]java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
 [ 2.G2(4).2, [ 5, 7, 13 ]<>
 [ 2.HS, [ 3, 5, 7, 11 ] ],
 [ 2.HS.2, [ 3, 11 ] ],

 [ 2.Ru, [ 5, 7, 13, 29 ] ],
 [ 2.Suz, [ 2, 5, 11 ] ],
 [ 2.Suz.2, [ 3, 7, 13 ] ],
 [ 2.Sym(15), [ 3, 5, 7 ] ],
 ), [ 3, 5, 7 ]],
 [ 2.Sym(17), [ 3, 5, 7 ] ],
 [ 2.Sym(18), [ 5, 7 ] ],
 [ 2.Sz(8), [ 2, 5, 13 ] ],
 [ 2^2.2E6(2), [ 13, 19 ] ],
 [ 2^2.2E6(2).2, [ 13, 19 ] ],
 [ 2^2.Fi22.2, [ 3, 11, 13 ] ],
 [ 2^2.L3(4).2^2, [ 5, 7 ] ],
 [ 2^2.L3(4).2_1, [ 5, 7 ] ],
 [ 2^2.Sz(8), [ 2, 5, 13 thepoint group forthe fourth example are as follows
 [ 2x2.F4(2), [ 2, 7, 13, 17 ] ],
 [ 2x3.Fi22, [ 2, 3, 5 ] ],
 [ 2x6.Fi22, [ 2, 3, 5 ] ],
 [ 2x6.M22, [ 2, 5, 11 ] ],
 [ 2xFi22.2, [ 11, 13 ] ],
 [ 2xFi23, [ 3, 17, 23 ] ],
 [ 3.Fi22, [ 2, 3, 5 ] ],
 [ 3.Fi22.2, [ 2, 5, 11, 13 ] ],
[ 3.J3, [ 2 17, 19] ],
 [ 3.J3.2, [ 2, 5, 17, 19 ] ],
 [ 3.L3(4).2_3, [ 2, 3, 7 ] ],
 [ 3.L3(4).3.2_3, [ 2, 3, 7
 [ 3.L3(7).2, [ 3, 7, 19 ] ],
 [ 3.L3(7).S3, [ 3, 7, 19 ] ],
 [ 3.McL, [ 2, 5, 11 ] ],
 [ 3.McL.2, [ 2, 3, 5, 11 ] ],
 [ 3.ON, [ 3, 7, 11, 19, 31 ] ],
 [ 3.ON.2, [ 3, 5, 7, 11, 19, 31 ] ],
 [ 3.Suz.2, [ 2, 3, 13 ] ],
 [ 3x2.F4(2), [ 2, 7, 13, 17 ] ],
 x2.i22.2, 11, 13] ],
 /ubsection>>
 [ 3xFi23, [ 3, 17, 23 ] ],
 [ 3xJ1, [ 7, 11, 19 ] ],
 [ 3xL3(7).2, [ 3, 7, 19 ] ],
 [ 4.HS.2, [ 5, 7, 11 ] ],
 [ 4.M22, [ 5, 7 ] ],
 [ 4_1.L3(4).2_1, [ 5, 7 ] ],
 [ 4_2.L3(4).2_1, [ 3, 5, 7 ] ],
 [ 6.Fi22, [ 2, 3, 5 ] ],
 [ 6.Fi22.2, [ 2, 5, 11, 13> [0,0,0,0,1 0,0,0 ,
 [ 6.L3(4).2_1, [ 5, 7 ] ],
 [ 6.M22, [he vector system is the trivial vector system <M>V_1</M>,
 [ 6.O7(3), [3,5, 13 ]]java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
 [ 6.O7(3).2, [ 3, 5, 13 ] ],
 [ 6.Suz, [ 2, 5, 11 ] ],
 [ 6.Suz.2, [ 2, 3, 5, 7, 13 ] ],
 [ 6x2.F4(2), [ 2, 7, 13, 17 ] ],
 [ A12, 2,3 ]],
 [ A14, [ 2, 5, 7 ] ],
 [ A17, [ 2, 7 ] ],
 [ A18, [ 2, 3, 5, 7 ] ],
 [ B, [ 13, 17, 23, 31 ] ],
 [ F3+, [ 17, 23, 29 ] ],
 [ +.2, [17 23 29 ],
,13 ]]java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
 [ Fi23, [ 3, 17, 23 ] ],
 [ HN, [ 2, 3, 11, 19 ] ],
 [ HN.2, [ 5, 7, 11, 19 ] ],
 [ He, [ 5, 17 ] ],
 [ He.2, [ 5, 7, 17 ] ],
 Isoclinic(12..),[ ,5, 7 ,
 [ Isoclinic(12_1.L3(4).2_1), [57],
 [ Isoclinic(12_2.L3(4).2_1), [ 3, 5, 7 ] ], <[
java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 61
 [ Isoclinic(2.A11.2), [ 5, 7, 11 ] ],
 [ Isoclinic(2.A12.2), [ 5, 7, 11 ] ],
 [ Isoclinic(2.A13.2), [ 5, 7, 13 ] ],
 2.2,[ , ,
 the <M>.42</ <>2B
 [ Isoclinic(2.HS.2), [ 3, 11 ] ],
 [ Isoclinic(2.HSx2), [ 3, 5, 7, 11 ] ],
 [ Isoclinic(2.L3(4).2_1), [ 5, 7 ] ],
 [ Isoclinic(2.Suz.2), [ 3, 7, 13 ] ],
 (41L3(4)._) [5 ]]java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
 [ Isoclinic(4_2.L3(4).2_1), [ 3, 5, 7 ] ],
 [ Isoclinic(6.Fi22.2), [ 2, 5, 11, 13 ] ],
 [ Isoclinic(6.L3(4).2_1), [ 5, 7 ] ],
 [ Isoclinic(6.O7(3).), ,5 13]]
 [ Isoclinic(6.Suz.2), [ 2, 3, 5, 7, 13 ] ]8/C> definewhich is C>6b</>
 [ J1, [ 7, >:/450255/>/>
 [7 ,19 ,
 [ J3, [ 2, 17, 19 ] ],
 [ J3.2, [ 2, 5, 17, 19 ] ],
 [ L3(4).2_3, [ 3, 7 ] ],
 [ L3(4).3.2_3, [ 2, 3, 7 ] ],
 [ L3(7).2, [ 3, 7, 19 ] ],
 [ L3(7).S3, [ 3, 7, 19 ] ],
 L3=3
 22 ,731] ,
 [ ( 78(,10
 [ 1, 2, 3, 4, 5, 8, 9, 10, 11
 [
 [ ON.2, [ 3, 5, 7, 11, 19, 31 ] ],
 [ Ru, [ 5, 7, 13, 29 ] ],
 [ S3xFi22.2, [ 11, 13 ] ],
 [ Suz.2, [ 3, 13 ] ] ]
]]></Example>

</



Note that but thissubgroup M3\ {}/><M3Fi_24}'/M>
and cannot toa subgroupof <M>3./M>.
(For example, the prime <M>2</M> was added to the entries for
extensions of <M>F_4(2)</M> when the <M>2</M>-modular table of <M>F_4(2)</M>
became available.)

</Subsection>


<Subsection Label="subsect:generality_J3">
<Heading>A generality problem concerning the group <M>J_3</M> (April 2015)</Heading>

<Alt Only="HTML">
<![CDATA[
<a name="generality_problem_J3">In March 2015,</a>
]]>
</Alt>
<lt=HTML">
In tolist all candidatesfor the class fusion,
</Alt>
Klaus Lux reported an inconsistency in the character data of &GAP;:



The sporadic simple Janko group <M>J_3</M> has a unique <M>19</M>-modular
irreducible Brauer character of degree <M>110</M>.
In the character table that is printed in the
&ATLAS; of Brauer characters <Cite Key="JLPW95" Where="p. 219"/>,
the Brauer character value on the class <C>17A</C> is <M>b_{17}</M>.
The &ATLAS; of Group Representations <Cite Key="AGRv3"/> provides
a straight line program for computing class representatives of <M>J_3</M>.
If we compute the Brauer character value in question,
we do not get <M>b_{17}</M> but its algebraic conjugate, <M>-1-b_{17}</M>.





<Example><![CDATA[
gap> t:= CharacterTable( "J3" );;
gap>m= t mod 19;
gap>
gap> Length( cand );
1
gap> slp:= AtlasProgram( "J3", "classes" );;
17:Position(slpo 17")
18
gap> info:= OneAtlasGeneratingSetInfo( "J3", Characteristic, 19,
> Dimension, 110 );;
gens: AtlasGeneratorsinfo);
gap> reps:= ResultOfStraightLineProgram( slp.program,
> gens.generators );;
gap> Quadratic( BrauerCharacterValue( reps[ 17a ] ) );
rec ATLAS : "1b17", a : -,b : -,d: 2
 display := "(-1-Sqrt(17))/2", root := 17 )
]]></Example>

<Pgap> for

How shall we resolve this inconsistency,
by replacing the straight line program
or by swapping the classes <C>17A</C> and <C>17B</C> in the character table?
Before we decide this, we look at related information.



=>java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 44
<Alt Not="LaTeX">The following table</Alt>
lists the <M>p</M>-modular irreducible characters of <M>J_3</M>,
according to <Cite Key="JLPW95"/>,
that can be used to define which of the two classes of element order <M>17</M>
shall be called <C>17A</C>;
a <M>+</M> sign in the last column of the table indicates that the
representation is available in the &ATLAS; of Group Representations.





<Table Label="valuesJ3" Align="|r|r|r|r|c|">
<java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 75
<HorLine/>
<Row>
<Item><M>p</M></Item>
<Item><M>\varphi(1)</M></Item>
<Item><M>\varphi(</M><C>17A</C><M>)</M></Item>
<Item><M>\varphi(</M><C>17B</C><M>)</M></Item>
<Item>&ATLAS;?</Item>
</Row>
<HorLine/>
<HorLine/>
<Row>
<Item><M>2</M></Item>
<Item><M>78</M></Item>
<Item><-{17>/>
<Item><M>2+b_{17}</M></Item>
<Item><M>+</M></Item>
</Row>
<Row>
<Item><M>2</M></Item>
<Item><M>80</M></Item>
<Item><M>3-b_{17}</M></Item>
<Item><M>4+b_{17}</M></Item>
<Item><M>+</M></Item>
</Row>
<Row>
<Item><M>2</M></Item>
<Item><M>244</M></Item>
<Item><M>b_{17}-2</M></Item>
<Item><M>-3-b_{17}</M></Item>
Item>M><M></>
</Row>
<Row>
<Item><M>2</M></Item>
<Item><M>966</M></Item>
<Item><M>r_{17}-3</M></Item>
<Item><M>-3-r_{17}</M></Item>
<Item><M>+<theunique M11<M- Brauercharacter ofdegree<java.lang.StringIndexOutOfBoundsException: Range [0, 70) out of bounds for length 27
</Row>
<Row>
<Item><M>19</M></Item>
<Item><M>110</M></Item>
<Item><M>b_{17}</M></Item>
<Item><M>-1-b_{17}</M></Item>
<Item><><M<Item>
</Row>
<Row>
<Item><M>19</M></Item>
<Item><M>214</M></Item>
<Item><M>1-b_{17}</M></Item>
<Item><M>2+b_{17}</M></Item>
<Item><M>+</M></Item>
</Row>
<Row>
<Item><M>19</M></Item>
<Item><M>706</M></Item>
<Item><M>-b_{17}</M></Item>
<Item><M>1+b_{17}</M></Item>
<Item><M>+</M></Item>
</Row>
<Row>
<Item><M>19</M></Item>
<Item><M>1214</M></Item>
<Item><M>-1+b_{17}</M></Item>
<Item><M>-2-b_{17}</M></Item>
<Item><M>-</M></Item>
</Row>
<HorLine/>
</Table>



Note that the irreducible Brauer characters in characteristic <M>3</M> and
<M>5</M> that distinguish the two classes <C>17A</C> and <C>17B</C> occur
in pairs of Galois conjugate characters.



The following computations show that the given straight line program
is compatible with the four characters in characteristic <M>2</M>
but is not compatible with the three available characters in characteristic
<M>19</M>.



<Example><![CDATA[
gap> table:= [];;
gap> for pair in [ [ 2, [ 78, 80, 244, 966 ] ],
> [ 19, [ 110, 214, 706 ] ] ] do
>p [];
> for d in pair[2] do
> info:= OneAtlasGeneratingSetInfo( "J3", Characteristic, p,
> Dimension, d );
> gens:= AtlasGenerators( info );
> reps:= ResultOfStraightLineProgram( slp.program,
> gens.generators );
> val:= BrauerCharacterValue( reps[ 17a ] );
> Add( table, [ p, d, Quadratic( val ).ATLAS,
> Quadratic( StarCyc( val ) ).ATLAS ] );
> od;
> od;
gap> PrintArray( table );
[ [ 2, 78, 1-b17, 2+b17 ],
 [ 2, 80, 3-b17,
 [ 2, 244, -2+b17, -3-<Subsection>
 [ 2, 966, -3+r17, -3-r17 ],
10 -, ,
 [ 19, 214, 2+b17, 1-b17 ],
 ] ]
]]></Example>



We see that the 32,734 <C.
the <M>19</M>-modular character table of <M>J_3</M>P
in <Cite Key="JLPW95"/>.
In particular, changing the straight line program would not help to
resolve the problem.



How shall we proceed in order to fix the problem?
We can decide to keep the <M>19</M>-modular table of <M>J_3</M>,
and to swap the two classes of element order <M>17</M> in the
<M>2</M>-modular table;
then also the straight line program has to be changed,
ndthe classesof elementorders <M>17/M>and<51<M
in the <M>2</M>-modular character table of the triple cover <M>3.J_3</M>
of <M>J_3</M> have to be adjusted.
Alternatively, we can keep the <M>2</M>-modular table of <M>J_3</M>
and the straight line program,
and adjust the conjugacy classes of element orders divisible
by <M>17</M> in the <M>19</M>-modular character tables of <M>J_3</M>,
<M>3.J_3</M>, <M>J_3.2</M>, and <M>3.J_3.2</M>.



We decide to change the <M>19</M>-modular character tables.
Note that these character tables —or equivalently, the corresponding
Brauer trees— have been described in <Cite Key="HL89"/>,
where lead to the shownBrauertrees.
Questions about the consistency with Brauer tables in other characteristic
had not been an issue in this book.
(Only the consistency of the Brauer trees among the <M>19</M>-blocks of
<M>3.J_3</M> is mentioned.)
In fact, the book mentions that the <M>19</M>-modular Brauer trees for
<M>J_3</M> had been computed already by W. Feit.
The inconsistency of Brauer character tables in different characteristic
has apparently been overlooked when the data for <Cite Key="JLPW95"/>
have been put together, and had not been detected until now.



<E>Remarks:</E>

<List>
<Item>
 Such a change of a Brauer table can in general affect the class fusions
 from and to this table.
 Note that Brauer tables may impose conditions on the choice of the fusion
 among possible fusions that are equivalent w. r. t. the
 table automorphisms of the ordinary table.
 In this particular case, injava.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
 see the sections <Ref Subsect="subsect:L2(16).4_in_J3.2"/> and
 Section <Ref Subsect="subsect:generality"/>.
 
</Item>
<Item>
 The change of the character tables affects the decomposition matrices.
 Thus the PDF files containing the <M>19</M>-modular decomposition
 matrices had to be updated, see
 <URL>http://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/dec/tex/J3/index.html</URL>.
</Item>
<Item>
 Jürgen Müller has checked that the conjugacy classes of all Brauer tables of
 <M>J_3</M>, <M>3.J_3</M>, <M>J_3.2</M>, <M>3.J_3.2</M> are consistent
 after the fix described above.
</Item>
</List>

</Subsection>


<Subsection Label="subsect:generality_HN">
<Heading>A generality problem concerning the group <M>HN</M> (August 2022)</Heading>

The classes <C>20A</C>, <C>20B</C> of the Harada-Norton group <M>HN</M>
in the <M>11</M>- and <M>19</M>-modular character tables
are determined by unique Brauer characters that have different values
on these classes.
Once we have <E>defined</E> these classes in one characteristic,
the two Brauer characters tell us how to <E>choose</E> them consistently
in the other characteristic.
Thus the[[, 432 734 ]]]
w.r.t. this property or not.



(Note that this question can be answered independently of all other
questions of this kind for <M>HN</M>,
because the permutation that swaps exactly the classes <C>20A</C> and
<C>20B</C> is a table automorphism
<M>HN</M>.)



We start with the ordinary character table of <M>HN</M>.
There are exactly two ordinary irreducible characters
that take different values on the classes <C>20A</C>, <C>20B</C>,
these are <M>\chi_{51}</M> and <M>\chi_{52}</M>.



<Example><![CDATA[
gap> t:= CharacterTable( "HN" );;
gap> pos20:= Positions( OrdersClassRepresentatives( t ), 20 );
[ 39, 40, 41, 42, 43 ]
gap> diff:= Filtered( Irr( t ), x -> x[39] <> x[40] );;
gap> List( diff, x -> Position( Irr( t ), x ) );
[ 51, 52 ]
]]></Example>



These values are irrational and lie in the field that is generated by
the square root of <M>5</M>.



<Example><![CDATA[
gap> List( diff, x -> List( x{ [ 1, 39, 40 ] },
> CTblLib.StringOfAtlasIrrationality ) );
[ [ "5103000", "2r5+1", "-2r5+1" ], [ "5103000", "-2r5+1", "2r5+1" ] ]
]]></Example>



In each of the characteristics <M>p \in \{ 11, 19 \}</M>,
the <M>p</M>-modular reductions of <M>\chi_{51}</M> and <M>\chi_{52}</M>
decompose differently into irreducibles.
Note that the Galois automorphism of the ordinary character table
that maps <M>\sqrt{5}</M> to <M>-\sqrt{5}</M> does not live in the
<M>11</M>- and <M>19</M>-modular Brauer tables.



For <M>p = 11</M>, the reduction of <M>\chi_{51}</M> is
<M>\varphi_{40} + \varphi_{48}</M>,
with <M>\varphi_{40}(1) = 1\,575\,176</M> and
<M>\varphi_{48}(1) = 3\,527\,824</M>,
and the reduction of <M>\chi_{52}</M> is
<M>\varphi_{39} + \varphi_{49}</M>,
with <M>\varphi_{39}(1) = 1\,361\,919</M> and
<M>\varphi_{49}(1) = 3\,741\,081</M>.



<Example><![CDATA[
gap> t11:= t mod 11;;
gap> rest11:= RestrictedClassFunctions( diff, t11 );;
gap> dec11:= Decomposition( Irr( t11 ), rest11, "nonnegative" );;
gap> List( dec11, Set );
[ [ 0, 1 ], [ 0, 1 ] ]
gap> List( dec11, x -> java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 18
[ [ 40, 48 ], [ 39, 49 ] ]
gap> List( Irr( t11 ){ [ 40, 48 ] }, x gap (ComplementClassesRepresentatives(G P ) )
[ 1575176, 3527824 ]
gap> List( Irr( t11 ){ [ 39, 49 ] }, x -> x[1] );
[ 1361919, 3741081 ]
]]></Example>



For <M>p = 19</M>, the reduction of <M>\chi_{51}</M> is

<>\arphi_{42} \varphi_{45}}/M,
with <M>\varphi_{42}(1) = 2\,125\,925</M> and
<M>\varphi_{45}(1) = 2\,977\,075</M>,
and the reduction of <M>\chi_{52}</M> is
<M\arphi_{33} + \varphi_{48}</M>,
with <M>\varphi_{33}(1) = 1\,197\,330</M> and
<M>\varphi_{48}(1) = 3\,905\,670</M>.



<Example><![CDATA[
gap> t19:= t mod 19;;
gap> rest19:= RestrictedClassFunctions( diff, t19 );;
gap> dec19:= Decomposition( Irr( t19 ), rest19, "nonnegative" );;
<xample>!CDATA[
[ [ 0, 1 ], [ 0, 1 ] ]
gap> List( dec19, x -> Positions( x, 1 ) );
[ [ 42, 45 ], [ 33, 48 ] ]
},x ->x[1]);
[ 2125925, 2977075 ]
gap> Listjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
[ 1197330, 3905670 ]
]]></Example>

The Frobenius-Schur indicators of all involved <M>p</M>-modular constituents
are <M>+</M>.
This implies that <M>\chi_{51}</M> reduces orthogonally stably modulo <M>11</M>
but not orthogonally stably modulo <M>19</M>,
whereas <M>\chi_{52}</M> reduces orthogonally stably modulo <M>19</M>
but not orthogonally stably modulo <M>11</M>.



<Example><![CDATA[
ap> Indicator( t11,2) [39,40,48,49] }
[ 1, 1, 1, 1 ]
gap> Indicator( t19, 2 ){ [ 33, 42, 45, 48 ] };
[ 1, 1, 1, 1 ]
]]</Example>



In version up to 1.3.4 of the character table library,
this condition was not satisfied:
The reduction of <M>\chi_{51}</M> modulo both <M>11</M> and <M>19</M> was
orthogonally stable,
and the reduction of <M>\chi_{52}</M> modulo both <M>11</M> and <M>19</M> was
not orthogonally stable.
However, this cannot happen, due to theoretical results about
orthogonal discriminants of the involved characters.
Thus we have found a way to decide the consistency of the classes
<C>20A</C> and <C>20B</C> in characteristics <M>11</M> and <M>19</M>:
Either the <M>11</M>-modular character table or the <M>19</M>-modular
character table of <M>HN</M> had to be changed for version 1.3.5,
by swapping the classes <C>20A</C> and <C>20B</C>.



We decided to change the <M>11</M>-modular table,
because there are no other generality problems for <M>HN</M> involving the
<M>19</M>-modular table, and hence we are sure that this table will not
need to be changed because of new solutions to generality problems.



For <M>HN.2</M>, the situation is similar.
two classes of elementM40<M that haveto be
swapped if <C>20A</C>under the natural epimorphismfrom<>2B</M to <M>/,
Thuswehave to change the <M>11</M>-modular table of <M>HN.2</M> accordingly.



Changes of this kind may affect also derived character tables in the library.
In this case, the <M>11</M>-modular table with identifier <C>"(D10xHN).2"</C>
was changed as well.
Note that this table is not stored explicitly 2 3 1 1 1 . . .
it gets constructed from ordinary java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 13
<Ref Func="ConstructIndexTwoSubdirectProduct" BookName="ctbllib"/>.

<> PossibleClassFusionstblV/, CharacterTable(" )));

</Section>


<Section Label="sect:derivebrauercharacters">
<Heading>Brauer Tables that can be derived from Known Tables<<>

In a few situations, one can derive
the <M>p</M>-modular Brauer character table of a group
oretic information.



For quite some time, a method is available in &GAP; that computes
the Brauer characters of <M>p</M>-solvable groups
(see <Ref Subsect="BrauerTable" BookName="ref"/> and
<Ref Subsect="IsPSolvableCharacterTable" BookName="ref"/>).



The following sections list other situations where Brauer tables
can be computed by &GAP;.


<Subsection Label="sect:brauercharactersbyconstruction">
<Heading>Brauer Tables via Construction Information</Heading>

If theto the library
java.lang.StringIndexOutOfBoundsException: Range [50, 3) out of bounds for length 57
then &GAP; may be tothemaximalsubgroupM(2 timesO_8+3).S_4/>ofM</>
/>modular of
<Q>ingredients</Q>.
This happens currently in the following cases.



<List>
<Item>
 <M>t</M> has been constructed with
 <Ref Oper="CharacterTableDirectProduct" BookName="ref"/>,
 and &GAP; can compute the <M>p</M>-modular Brauer tables
 of the direct factors.
</Item GetFusionMap(lib tbl2B));
<Item>
 <M>t</M> has
 <Ref Oper="CharacterTableIsoclinic" BookName="ref"/>,
 and &GAP; can compute the <M>p</M>-modular Brauer table
 of the table that is stored in <M>t</M> as the value of the attribute
 <Ref Oper="SourceOfIsoclinicTable" BookName="ref"/>.
</Item>
<Item>
 <M>t</M> has the attribute
 <Ref Attr="ConstructionInfoCharacterTable<M>ldots</M> andthat the compositions of fusions from <M>V</M> to <M>M</M>
 the first entry of this list <M>l</M>, say, is one of the strings
 <C>"ConstructGS3"</C> (see
 <Ref Subsect="sect:Character Tables of Groups of the Structure G.S_3"/>),
 <C>"ConstructIndexTwoSubdirectProduct"</C>
 (see <Ref Subsect="subsect:theorsubdir"/>),
 <C>"ConstructMGA"</C> (see <Ref Subsect="subsect:theorMGA"/>),
 <C>"ConstructPermuted"</C>,
 <C>"ConstructV4G"</C> (see <Ref Subsect="subsect:theorV4G"/>),
 and &GAP; can construct the <M>p</M>-modular Brauer table(s)
 of the relevant ordinary character table(s),
 which are library tables whose names occur in <M>lgap>java.lang.StringIndexOutOfBoundsException: Range [36, 34) out of bounds for length 49
</Item>
</List>

</Subsection>


<Subsection Label="sect:liftablebrauercharacters">
<Heading>Liftable Brauer Characters (May 2017)</Heading>

Let <M>B</M> be a <M>p</M>-block of cyclic defect for the finite group
<M>G</M>.
It can be read off from the set Irr<M>(B)</M>
of ordinary irreducible characters of <M>B</M>
whether all irreducible Brauer characters in <M>B</M> are restrictions
of ordinary characters to the <M>p</M>-regular classes of <M>G</M>,
as follows.



If <M>B</M> has only one irreducible Brauer character then all ordinary
characters in <M>B</M> restrict to this Brauer character.
So let us assume that <M>B</M> contains
at least
and consider the set <M>S</M>, say, of restrictions of Irr<M>(B)</M>
to the <M>p</M>-regular classes of <M>G</M>.



The block <M>B</M> contains exactly <M>|<>0<C> a java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 69
characters,
and the decomposition of the characters in <M>S</M> into these Brauer
characters is described by
an <M>|S|</M> by <M>|S| - 1</M> matrix <M>M</M>, say,
whose entries are zero and one, such that exactly two nonzero entries
occur in each column.
(See for example <Cite Key="java.lang.StringIndexOutOfBoundsException: Range [0, 32) out of bounds for length 0
which refers to <Cite Key="Dad66"Example><![CDATA[



If all irreducible Brauer characters of <M>B</M> occur in <gap> pos20:= Positions( OrdersClassRepresentatives( t ), 20
then the matrix <M><M contains M|S| 1/>rows
exactly one nonzero entry,
the consistsonlyof M<>.
e element M<M equal to
the sum of all other elements in <M>S</M>.
Conversely, if the element of largest degree in <M>S</M> is equal to
the sum of all other elements in <M>S</M> then
the matrix <M>M</M> has the structure as stated above,
hence all irreducible Brauer characters of <M>B</M> occur in <M>S</M>.



Alternatively,
one could state that all irreducible Brauer characters of <M>B</M>
are restricted ordinary characters if and only if the Brauer tree
of <M>B</M> is a <E>star</E> (see <Cite Key="HL89" Where="p. 2"/>.
If <M>B</M> contains at least two irreducible Brauer characters
then this happens if and only if one of the types <M>\times</M> or
<M>\circ</M> occurs for exactly one node in the Brauer graph of <M>B</M>,
see <Cite Key="HL89" Where="Lemma 2.1.13"/>,
and the distribution to types is determined by Irr<M>(B)</M>.



The default method for <Ref Oper="BrauerTableOp" BookName="ref"/>
that is contained in the &GAP; library has been extended in version 4.11
such that it checks whether the Sylow <M>p</M>-subgroups of the given group
<M>G</M> are cyclic and, if yes,
whether all <M>p</M>-blocks of <M>G</M> have the property discussed above.
(This feature arose from a discussion with Klaus Lux.)



Examples where this method is successful for all blocks
are the <M>p</M>-modular character tables of the groups PSL<M>(2, q)</M>,
where <M>p</M> is odd and does not divide <M>q</M>.



<Example><![CDATA[
gap> t:= CharacterTable( PSL( 2, 11 ) );;
gap> modt:= t mod 5;;
gap> modt <> fail;
true
gap> InfoText( modt );
"computed using that all Brauer characters lift to char. zero"
]]></Example>



Another such example is the <M>5</M>-modular table
of the Mathieu group <M>M_{11}</M>.



<Example><![CDATA[
gap> lib:= CharacterTable( "M11" );;
gap> fromgroup:= CharacterTable( MathieuGroup( 11 ) );;
gap> DecompositionMatrix( lib mod 5 );
[ [ 1, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
 [ 0, 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
 [ 0, 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0, 0 ],
 [ 0, 0, 0, 0, 0, 0, 1, 0, 0 ], [ 1, 0, 0, 0, 1, 1, 1, 0, 0 ],
 [ 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 1 ] ]
gap> fromgroup mod 5 <> fail;
true
]]></Example>



There are cases where all Brauer characters of a block lift
to characteristic zero but the defect group of the block is not cyclic,
thus the method cannot be used.
An example is the <M>2</M>-modular table of the Mathieu group <M>M_{11}</M>.



<Example><![CDATA[
gap> DecompositionMatrix( lib mod 2 );
[ [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ],
 [ 0, 1, 0, 0, 0 ], [ 1, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ],
 [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ], [ 1, 0, 0, 0, 1 ],
 [ 1, 1, 0, 0, 1 ] ]
gap> fromgroup mod 2;
fail
]]></Example>

</Subsection>

</Section>


<Section Label="sect:monstersubgroups">
<Heading>Information about certain subgroups of the Monster group</Heading>


<Subsection Label="sect:no2U42inMonster">
<Heading>The Monster group does not contain subgroups of the type <M>2.U_4(2)</M> (August 2023)</Heading>

In the context of a question about decomposition numbers of the
sporadic simple Monster group <M>&M;</M>,
Benjamin Sambale was interested in possible embeddings of certain groups
<M>G</M> into <M>&M;</M> such that the decomposition matrices of <M>G</M>
are known.
For a given <M>G</M>, the first steps were to compute the possible class
fusions of <M>G</M> in <M>&M;</M> and then to check whether the corresponding
embeddings would be interesting.



Apparently, calling <Ref Func="PossibleClassFusions" BookName="ref"/>
with its default parameters often runs very long and requires a lot of space
when <M>G</M> is a small group such as <M>2.U_4(2)</M>.
We We candifferentlyjava.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 40
<C>decompose:= false</C>.
This has the effect that one criterion is omitted that checks the
decomposability of restricted characters of <M>&M;</M> as an integral
linear combination of characters of the subgroup.
As a rule of thumb, if the number of classes of the subgroup is small
compared to the number of classes of the group
and if the result consists of many candidates then it might be faster
to omit the decomposability criterion.



<Example><![CDATA[
gap> s:= CharacterTable( "2.U4(2)" );;
gap> m:= CharacterTable( "M" );;
gap> sfusm:= PossibleClassFusions( s, m, rec( decompose:= false ) );;
gap> Length( sfusm );
2332
]<





Looking at the (many) candidates,
we see that all map the central involution of
<M>2.U_4(2)</M> to the class <C>2B</C> of <M>&M;</M>,
thus any subgroup of the type <M>2.U_4(2)</M> lies inside the <C>2B</C>
normalizer in <M>&M;</M>.
We compute the possible class fusions into this subgroup.



<Example><![CDATA[
gap> Set( List( sfusm, x -> x[2] ) );
[ 3 ]
gap> t:= CharacterTable( "MN2B" );
CharacterTable( "2^1+24.Co1" )
=( ,(decompose: ) );
gap> Length( sfust );
0
]]></Example>





Thus we have shown that <M>&M;</M> does not contain subgroups of the type
<M>2.U_4(2)</M>.

</Subsection>


<Subsection Label="sect:ML34inMonster">
<Heading>Perfect central extensions of <M>L_3(4)</M> (August 2023)</Heading>

There was <URL><LinkText>the question in MathOverflow</LinkText>
<Link>https://mathoverflow.net/questions/450255</Link></URL Frobenius-chur of involved<>p/>modularconstituents
This impliesthat <>chi_{1<M>reducesorthogonally modulo<><M>
 sporadicsimple Monster group <>M</.



First we get the list of perfect central extensions of <M>G</M> (asuming that
their character tables are contained in the character table library).

<Example><![CDATA[
gap> simp:= CharacterTable( "L3(4)" );;
gap> extnames:= AllCharacterTableNames( Identifier,
> x -> EndsWith(x, "L3(4)" ) );;
gap> ext:= List( extnames, CharacterTable );;
gap> ext:= Filtered( ext, x -> Length( ClassPositionsOfCentre( x ) ) =
> Size( x ) / Size( simp ) );;
gap> SortBy( ext, Size );
gap> names:= List( ext, Identifier );
[ "L3(4)", "2.L3(4)", "3.L3(4)", "2^2.L3(4)", "4_1.L3(4)",
 "4_2.L3(4)", "6.L3(4)", "(2x4).L3(4)", "(2^2x3).L3(4)",
 "12_1.L3(4)", "12_2.L3(4)", "4^2.L3(4)", "(2x12).L3(4)",
 "(4^2x3).L3(4)" ]
]]></Example>



fact ></>is Enot<> a <>&<Mjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
is shown in <Cite Key="HW08"/> (at the end of this paper).





And the following embeddings of central extensions of <M>G</M> in <M>&M;</M>
can be established using known subgroups of <M>&M;</M>.

<List>
<Item>
 <M>2.G < 2.U_4(3) < 2^2.U_6(2) < Fi_{23} < 3.Fi_{24}' < &MTare two classes of element order <M>40</M> that have to be
</Item>
<Item>
 ^2G&;He &t 3.{24} < M<M.
</Item>
<Item>
 <M>6.G < 2.G_2(4) < 6.Suz < 3^{1+12}_+.2java.lang.StringIndexOutOfBoundsException: Range [0, 54) out of bounds for length 0
</Item>
</List>

Note that <M>G</M> is a subgroup of <M>U_4Note that this is not storedexplicitly inthe ,
<>3.</ is a subgroupof <>_4)/> butnotof M.G_2()/>
and <M>G_2(4)</M> is a subgroup of <M>Suz</M> but not of <M>2.Suz</M>.
The positive statements follow from <Cite Key="CCN85" Where="pp. 52, 97, 131"/>
and the negative ones from the following computations.



<Example><![CDATA[
gap> Length( PossibleClassFusions( CharacterTable( "L3(4)" ),
> CharacterTable( "2.U4(3)" ) ) );
0
gap> Length( PossibleClassFusions( CharacterTable( "3.L3(4)" ),
> CharacterTable( "2.G2(4)" ) ) );
0
gap> Length( PossibleClassFusions( CharacterTable( "G2(4)" ),
> CharacterTable( "2.Suz" ) ) );
0
]]></Example>

</

The group <
If <M>3.G</M> is a subgroup of <M>&M;</M> then it is contained in
a <C>3A</C> centralizer (of the structure <M>3.Fi_{24}'</M>),
a <C>3B</C> centralizer (of the structure <M>3^{1+12}_+.2Suz</M>) or
a <C>3C</C> centralizer (of the structure <M>3 \times Th</M>).
Clearly the case <M>3C</M> cannot occur,
and <C>3B<=sbrauercharactersbyconstruction
and the <M>3B</M> normalizer <M>3^{1+12}_+.2Suz.2</M> is possible.



<Example><![CDATA[
gap> t:= CharacterTable( "MN3B" );
This thejava.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 46
gap> Length( PossibleClassFusions( CharacterTable( "3.L3(4)" ), t ) );
0
]]></Example>



If <M>3.G</M> is contained in the <C>3A</C> centralizer
then this embedding induces one of <M>G</M> into some maximal subgroup of
<M>Fi_{24}'</M>.
Using the known character tables of these maximal subgroups
in GAP's character table library,
one shows that only <M>Fi_{23}</M> admits a class fusion,
but this subgroup lifts to <M>3 \times Fi_{23}</M> in <M>3.Fi_{24}'</M>
and thus cannot lead to a subgroup of type <M>3.G</M>..



<Example><![CDATA[
gap> mx:= List( Maxes( CharacterTable( "Fi24'" ) ), CharacterTable );;
gap> s:= CharacterTable( "L3(4)" );;
gap> Filtered( mx, x -> Length( PossibleClassFusions( s, x ) ) > 0 );
[ CharacterTable( "Fi23" ) ]
]]></Example>



The other candidates <M>m.G</M> contain at least one central involution.
If <M>m.G</M> is a subgroup of <M><Item>
a <C>2A</C> centralizer (of the structure <M>2.B</M>) or
a <C>2B</C> centralizer (of the structure <C>2^{1+24}_+.Co_1</C>).
gainwe use<""BookName"/
to list all candidates for the class fusion,
but here we <ef Subsect=sect:CharacterTables ofGroups of the Structure G.S_3"/>),
centralizer as an image of one central involution in <M>m.G</M>.



<Example><![CDATA[
gap> done:= [ "L3(4)", "2.L3(4)", "3.L3(4)", "2^2.L3(4)", "6.L3(4)" ];;
gap> names:= Filtered( names, x -> not x in done );
["4_1L3(4)" 42L34" (x4).4" (^23)4"
 "12_1.L3(4)", "12_2.L3(4)", "4^2.L3(4)", "(2x12).L3(4)",
 "(4^2x3).L3(4)" ]
gap> invcent:=<List>
[ CharacterTable( "2.B" ), CharacterTable( "2^1+24.Co1" ) ]
gap> ForAll( invcent, x -> ClassPositionsOfCentre( x ) = [ 1, 2 ] );
true
gap> cand:= [];;
gap> ords:= "dummy";; # Avoid a message about an unbound variable ...
gap> for name in names do
> s:= CharacterTable( name );
> ords:= OrdersClassRepresentatives( s );
> invpos:= Filtered( ClassPositionsOfCentre( s ), i -> ords[i] = 2 );
java.lang.StringIndexOutOfBoundsException: Range [25, 26) out of bounds for length 25
> for t in invcent do
> init:= InitFusion( s, t );
> if init = fail then
> continue;
> fi;
> init[i]:= 2;
> fus:= PossibleClassFusions( s, t, rec( fusionmap:= init,
> decompose:= false ) );
> if fus <> [] then
> Add( cand, [ s, t, i, fus ] );
> fi;
> od;
> od;
> od;
gap> List( cand, x -> x{ [ 1 .. 3 ] } );
[ [ CharacterTable( "4_1.L3(4)" ), CharacterTable( "2^1+24.Co1" ), 3 ]
 , [ CharacterTable( "(2x4).L3(4)" ), CharacterTable( "2.B" ), 2 ],
 [ CharacterTable( "(2x4).L3(4)" ), CharacterTable( "2.B" ), 3 ] ]
]]></Example>





ote have< "=ref"
with the option <C>decompose:= false</C>, in order to save space and time.
See Section <Ref Subsect="sect:no2U42inMonster"/> for more details.)



Concerning the candidate <M>(2 \times 4).G</M>,
we see that only fusions are possible for which the central involution
in question is mapped and consider the set <M></M>, say, of restrictions of Irr<M>(B)</M>
Since we get candidates only for two out of the three java.lang.StringIndexOutOfBoundsException: Range [0, 61) out of bounds for length 44
we see that <M>(2 \times 4).G</M> does not embed into <M>&M;</M>.



Thus it turns out that exactly one group <M>m.G</M> cannot be excluded
this way.
Namely, these character-theoretical criteria leave the possibility
that <M>4_1.G</M> may occur as a subgroup of <M>2^{1+24}_+.Co_1</M>



Moreover, we see that if this happens then the centre <M>C</M>
of <M>4_1.G</M> lies inside the normal subgroup <M>N = 2^{1+24}_+</M>.
The centralizer of <M>C</M> in <M>N</M> has order <M>2^{24}</M>,
and the centralizer of <M>C</M> in <M>2^{1+24}_+.Co_1occur column
<M <KeyD/.
We see that <M>4_1.G</M>,
if it exists as a subgroup of <M>2^{1+java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
must lie inside the subgroup <>[2^24}./M.



<Example><![CDATA[
gap> s:= cand[1][1];
CharacterTable( "4_1.L3(4)" )
gap> t:= cand[1][2];
CharacterTable( "2^1+24.Co1" )
gap> fus:= cand[1][4];
[ [ 1, 5, 2, 5, 9, 8, 23, 27, 24, 27, 49, 49, 50, 50, 70, 74, 71, 74,
 70, 74, 71, 74, 114, 119, 115, 119, 114, 119, 115, 119 ] ]
gap> ClassPositionsOfCentre( s );
[ 1, 2, 3, 4 ]
gap> 5 in ClassPositionsOfPCore( t, 2 );
true
gap> siz:= SizesCentralizers( t )[5] / 2^24;
495766656000
gap> mx:= Filtered( List( Maxes( CharacterTable( "Co1" ) ),
> CharacterTable ),
> x -> Size( x ) mod siz = 0 );
[ CharacterTable( "Co3" ) ]
gap> Size( mx[1] ) = siz;
true
]]></Example>





I do not see a character-theoretic argument that could disprove
the existence of such an <M>4_1.G</M> type subgroup.

</Subsection>


<Subsection Label="sect:2BM12">
<Heading>The character table of <M>(2 \times O_8^+(3)).S_4 \leq 2.B</M> (October 2023)</Heading>

Consider a maximal subgroup <M M<>
in the sporadic simple Monster group.
The character table of <M>H</M> has</>
We can view <M>H</M> as <M>O_8^+(3).(3^2:2S_4) = O_8^+(3).F</M>.
The character table of <M>H</M> determines that of <M>F</M>,
and this table determines the isomorphism type of <M>F</M> as
<C>SmallGroup( 432, 734 )</C>.



<Example><![CDATA[
and distribution to isdeterminedbyIrrM(B)/.
CharacterTable( "(3^2:2xO8+(3)).S4" )
gap> N:= ClassPositionsOfSolvableResiduum( tblH );;
gap> tblF:= tblH / N;;
gap> Size( tblF );
432
gap> known:= NamesOfEquivalentLibraryCharacterTables( tblF );
[ "3^2.2.S4", "M12M7" ]
gap> Filtered( GroupInfoForCharacterTable( known[1] ),
 - Sjava.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 42
[ [ "SmallGroup", [ 432, 734 ] ] ]
]]></Example>

/

(Note that the precomputed
<Ref Attr="GroupInfoForCharacterTable" BookName="CTblLib"/>
information about &GAPgap> t: CharacterTable( PSL 2 11 );
one isomorphism type of groups fits to the character table of <M>F</M>.)



We compute that <M>O_3(F) \cong 3^2</M> has complements in <M>F</M>,
thus <M>H</M> has a subgroup <M>V
which is a complement of <M>O_3(H)</M> in <M>H</M>,
thus <M>V</M> is isomorphic with <M>H / O_3(H)</M>.



<Example><![CDATA[
gap> G:= SmallGroup( 432, 734 );;
gap> P:= PCore( G, 3 );;
gap> Length( [ 0, 0, 0, 0, 1, 0, 0, 0, 0 ], [ ,0 0,,0,,0]
1
>/Example>



We can derive the character table of <M>V</M> from that of <M>H</M>,
and compute that the group structure of 
<M>(2 \times O_8^+(3)).S_4</M>.
For we the orders of the unique subgroup
of order <M>2 |O_8^+(3)|</M>
If this normal subgroup would not be isomorphic with <M>2 \times O_8^+(3)</M>
then it would have one of the structures <M>java.lang.StringIndexOutOfBoundsException: Range [59, 46) out of bounds for length 59
<M>O_8^+(3).2_2</M>,
but then it would contain elements of the orders <M>24</M>.



<Example><![CDATA[
:= java.lang.StringIndexOutOfBoundsException: Range [42, 40) out of bounds for length 53
=sect>
9904359628800
gap> classes:= SizesConjugacyClasses( tblV );;
gap> 2N:= Filtered( ClassPositionsOfNormalSubgroups( tblV ),
 l - Sum classes l } ord);
;
1
gap> Set( OrdersClassRepresentatives( tblV ){ 2N[1] } );
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 18, 20, 26, 30 ]
Set ( CharacterTable O8+(3")))java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 18, 20 ]
gap> Set( OrdersClassRepresentatives( CharacterTable( "O8+(3).2_1" ) ) );
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 18, 20, 24, 26, 28,
 parameters runsvery requiresalot space
gap> Set( OrdersClassRepresentatives( CharacterTable( "O8+(3).2_2" ) ) );
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 18, 20, 24, 26, 28,
 30, 36 ]
]]></Example>



The class fusion of <M>V</M> into the andthejava.lang.StringIndexOutOfBoundsException: Range [26, 17) out of bounds for length 69
centralizes a <C>2A</C> element in the Monster,
Example[[



<Example><![CDATA[
gap> tblM:= CharacterTable( "M" );;
:=( tblV );
gap> Length( VfusM );
4
gap> ZV:= ClassPositionsOfCentre( tblV );
[ 1, 2 ]
gap> Set( List( VfusM, l -> l{ ZV } ) );
[ [ 1, 2 ] ]
]]></Example>



From the list of maximal subgroups of <M>B</M>,
we see that either <M>V</M> is contained in the preimage of <M>Fi_{23}</M>
under the natural epimorphism from <M>2.B</M> to <M>B</M>,
or <M>V</M> is equal to the preimage of <M>O_8^+(3).S_4</M>.



<Example><![CDATA[
gap> tblB:= CharacterTable( "B" );;
gap> mxB:= List( Maxes( tblB ), CharacterTable );;
gap>CharacterTable 2^+."java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
[ CharacterTable(
]]></Example>



The former possibility is excluded from the fact that the factor of <M>V</M>
a class fusion into<Fi_23}M>java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64



<Example><![CDATA[
gap> Length( PossibleClassFusions( tblV / ZV, CharacterTable( "Fi23" ) ) );
0
]]></Example>



We conclude that <M>V</M> is a maximal subgroup of <M>2.B</M>.



Thus we have used the character table ofwhich perfect simple <G=(4<>
the character table of a maximal subgroup of <M>2.B</M>,
with very little effort.



This table is meanwhile available in the table library,
with the identifier <C>"(2xO8+(3)).S4"</C>.



<Example><![CDATA[
gap> lib:= CharacterTable( "(2xO8+(3)).S4" );;
gap> TransformingPermutationsCharacterTables( tblV, lib ) <> fail;
true
gap> Irr( lib ) = Irr( tblV );
true
]]></Example>



In order to add the table to the library,
we have to provide also the class fusions from <M>V</M> to <M>2.B</M>,
to the maximal subgroup <M>(2 \times O_8^+(3)).S_4</M> of <M>B</M>,
and to the maximal subgroup <M>(3^2:2 \times O_8^+(3)).S_4</M> of <M>M</M>,
such that the compositions of fusions from <M>V</M> to <M>B</M>
java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48



<Example><![CDATA[
gap> tblU:= CharacterTable( "O8+(3).S4" );;
>tbl2B:CharacterTable( "2.B" );
gap> CompositionMaps( GetFusionMap( tblU, tblB ),
> GetFusionMap( lib, tblU ) ) =
> GetFusionMap ,tblB )java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
> GetFusionMap( lib, tbl2B ) );
true
]]></Example>



<M>\ldots</M> and that the compositions of fusions from <M>V</M> to <M>M</M>
via <M>(3^2:2 \times O_8^+(3)).S_4</M> and <M>2.B</M> are compatible.



<Example><![CDATA[
gap> tblH:= CharacterTable( "(3^2:2xO8+(3)).S4" );;
gap> CompositionMaps( GetFusionMap( tblH, tblM ),
> GetFusionMap( lib, tblH ) ) =
> CompositionMaps( GetFusionMap( tbl2B, tblM ),
> GetFusionMap
true
]]></Example>

</Subsection>
</Section
</Chapter>

Messung V0.5 in Prozent

¤ Dauer der Verarbeitung: 0.291 Sekunden ¤

Wurzel

Suchen

PVS Prover

Isabelle Prover

NIST Cobol Testsuite

Cephes Mathematical Library

Vienna Development Method

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung und die Messung sind noch experimentell.