Quellcode-Bibliothek maintain.xml Sprache: XML

<Chapter Label="chap:maintain">

<Section=">

<Heading>Maintenance Issues for the &GAP; Character Table Library</Heading>

This chapter collects examples of computations that arose
maintainingthe G; .
The sections been when the issues in arosemaintainingthe&;Character Libraryjava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
<


<java.lang.StringIndexOutOfBoundsException: Range [70, 2) out of bounds for length 70
<Heading

I notknowanecessary and sufficient criterionbelongs,
whethergivenmatrixtogetherwitha list maps
describes the java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [31, 8) out of bounds for length 42
(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<He> PseudoCharacter Table (ovember 2006</Heading>
.

Another such example is describedin
Section <java.lang.StringIndexOutOfBoundsException: Range [0, 17) out of bounds for length 0

The tables in the &GAP; Character Table Library withidentifier

there are table candidates for which these tests are not good enough. are attachedn;However,

there are table candidates for a finitegroup
 of java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 0

Another question would be whether a given editing,
belongs to the group for which it following input equals the data formerly
see


< Label<
<dingA Perfect Pseudo Character Table (November 2006)</Heading>

(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

< Identifier= "1114ext2"

Up to version 1.1.3 of the &GAP; Character
the table with identifier <C>P41/G1/L1/V4/xt2"/> was .
s already inthe microfiches
that are attached to <Cite >hetablewithidentifier <>P41/L1/"<C>was The already the

< 64,1281281818141414,4,,,,1818181818]

using221818,,6,7272323]1234,6,89101123,,java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
of a group,
apart from a little editing,
Currently we not know how to prove this inconsistency
alonefrom tablejava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21

uction java.lang.StringIndexOutOfBoundsException: Range [56, 50) out of bounds for length 57

the58>structure ^.()\"
in "(>" java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 68



<Example><![CDATA[
>6>java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 70
>java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 37
java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 31
 originHanrath library,\"
>>)java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
>"> ConstructClifford ConstructionInfoCharacterTable )
> : 0
> java.lang.StringIndexOutOfBoundsException: Range [0, 21) out of bounds for length 13
> 64,,,18181414141414141818181818 ,,,,,,,,,,4,262528,27,],
>java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
> 22,818,,2627,,23,23],[,2,34,,6,,8,,10,112,3, (171921)18,,2) )
>1,,,,,,,6151516165,,[,2345,,,,,
> 11,12,13,14,15,16,4,1,4,1,4,1,26,25,28,27,23,24]],
> : ,
> java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 4
> (17,19,21)(18,20,22)]
> ConstructionInfoCharacterTable> ][,]-][13[,][,,11][,,6]
> 5,6,7,8,[,,,,, 11]1]][1,]7,][[1,,],,]8,,,
> ],["23"][[,,34,
 11,][",4,25],[12344,4,>
> 6,7,],[elab"1017],[[,2],[,4],[1,][-11
>,1]][13,53,[,][-11],[13,64,
> [11][,]][12,72
> [9,5](tbl,n)))java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
> ;
gap> ConstructCliffordg> Decompositionjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gapgap>ifForAny ([1 .NrConjugacyClasses)] gap>Length(LinearCharacters tbl));
]java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 5



that <>,this.
M/L2 2E1java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25



<java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
gap( tbl ) )
1
gap> IsPerfectCharacterTable( tbl );
true
]]></Example>



The
<Ex> E( "contradiction";
andgap>]/Example>
into



<Example><![CDATA[
ap>IsInternallyConsistent( tbl );
true
gap> irr:= Irr( tbl );;
gap>test:= Concatenation List([2..7
> -Symmetrizations tbl, irr,n)));
gap]<Example
P>
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
> ForAny( Tuples( [ 1 .. NrConjugacyClasses( tbl |=64\,12<M
> t java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
> int- TransformingPermutationsgap>n: Size(bl)java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
>fail fail,fail fail ]
> fi;
]]><]>/>

<P/

&AP4
four gap>grps: List([ 1 . 4 ] ictgroup of <64512/>

</

<<Example>
gap,the <M7<M occurs exactlyjava.lang.StringIndexOutOfBoundsException: Range [0, 50) out of bounds for length 0
64512
( n);
4
gap> Example>[[
[ gap> :List(tbls,
 - Filtered (t)>
]]></ x

<P[1,1 1

e believethat java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 75
is correct then ,2,2 ]
haracter tables of these four

<

<question,
there ten[fail,,ail fail,fail ]
java.lang.StringIndexOutOfBoundsException: Range [6, 3) out of bounds for length 16
> x- matrices irreducible ofthe four groups
[ faild notat the unique class in which an irreducibledegreeof factor group takesthe value M>/> M><<>)
]]>/>



In> t
do not fit to the given



<Example><![ >]>/>
gap> List - 1 =63andSet 1 0
ormingPermutations( Irr(t ) (tbl)))java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67

]]></Example>

<P/

Letus look at the tables <> and M3/>
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
 one irreducible character
the java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
moreover,thevalue <>7/ occurs in testchars: (tbls,

<P/java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<Example><![CDATA[
>testcharsjava.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 31
>t> 1java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
> ]/
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 14
the ten>java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 68
, java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
]java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 4

/

(java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0
question ,
there are ten preimage classes of the involution class x- [3
ctorgroup<L_2<M
java.lang.StringIndexOutOfBoundsException: Range [0, 9) out of bounds for length 4
<2^._()/>
and java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
of of this



In thein f =dentifierref"
there is only onejava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5



<![<Example><![CDATA table is notcorrect.
gap> java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 4
> t->Filtered((t )java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
> x ->
gap> List(testchars, Length )java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
[ 1 ]
gap> List( testchars,in thehave equivalent ofirreducibles
[ 1 ]
]]></Example>



This>TransformingPermutationsCharacterTables
in fiche <M>2</M>, row <M>4</M>, CharacterTable("P41G1L1/V4/ext2 the table, these library tables are
withthename<>L1<>^7lt>() V4 <C>
and it turns out that this table is not correct.



java.lang.StringIndexOutOfBoundsException: Range [13, 4) out of bounds for length 78
and there were <E>three</E> tables of groups of that order
in the java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
that contain <C>origin: Hanrath library</C> in their
<>TransformingPermutationsCharacterTables( tbls[i],
value.
Besides the incorrect table> CharacterTablejava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
the charactergap>Filtered([1 . 4 ],i-
<C>PerfectGroup 64512, 1 )< it might thata perfectgroupof <M64\512/>
respectively.
(The matricesof irreducible charactersof these tables equivalent(java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 32

<

<Example><%<xample<![CDATAjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
p Filtered([ 1 . 4 ] i ->
> TransformingPermutationsCharacterTablesjava.lang.StringIndexOutOfBoundsException: Range [5, 6) out of bounds for length 5
> / [4]
[ 1 ]
>iltered([1. 4] i->
> The choice of this group was somewhat arbitrary since the vector system
> CharacterTable(P41G1L1/V2>V4<> tobenot in;;Cite Key=P89>
[ 3 ]
gap> TransformingPermutations( Irr( tbls[1] ), Irr( tbls[3] ) )<>PerfectGroup(64512 2 )/C,
java.lang.StringIndexOutOfBoundsException: Range [0, 4) out of bounds for length 0
]]>gap>Filtered([ 1 i-java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31



Since version 1.2 of the CharacterTable( "P41/G1/L1/V4/ext2" ) ) <> fail );
the withthe
<ef java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 4
value
<C>"java.lang.StringIndexOutOfBoundsException: Range [0, 7) out of bounds for length 4
<>PerfectGroup64512, 4 )/C>
The of this group was somewhat arbitrary since the vector system
<V4/>seemsto be in;<Key=HP89/>
anyhow, this group and the remaining perfect group,
<C>PerfectGroup( 64512, 2 )</C>,
have equivalent matrices of irreducibles.

<Example><![CDATA[
gap> Filtered%Then we consider the factor group <M>F</M> of <M>G</M> modulo its centre.
> TransformingPermutationsCharacterTables<Example>![DATA
> CharacterTable( "P41/G1/% gap> cen:= ClassPositionsOfCentre( tb

gap> TransformingPermutations( Irr( tbls[2] %gap facttbl: tbl /en;
true
]]></Example>


<Subsection Label="subsectg> (t ) 140.143 ;
<Heading>An Error in the Character

2016 java.lang.StringIndexOutOfBoundsException: Range [69, 68) out of bounds for length 74
<M>E_6(2)</M> with Magma (see> java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
and compared it with the table that was contained in the g>( { ,87 java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
&; Table since 2000.
It turned out that the x -> x[1] <> x[2] and29,29 ,33 ,,,37 ,, ,,, ,35 ,,



 irrationalcharacter values of
order <M>91</M> and power map values -*(16^32*()5 ]
T and toeachotherin ;
thus it may be that the 2*E(16)^3-2*E(16)^5 ]
character[-*E16-*)7
Specifically, the <M>11</M>th power map in the &GAP; table
fixed all elements <M>1<>
<>()the
two , can find,,88,95 96 ]
and show , ,,85 ;
differ.
Hence these two elements 140
Inn]></java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 13



<Example 64,64, 64,64]
gap> g:= AtlasGroup( "E6 table used todecide the which of the
gap> repeat x:= PseudoRandom( g ); until
ap ( x ( x^ ;
false
]]></Example>



The &;java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 64
GAP; .

</

<java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 18
gap> t:= CharacterTable( "E6(2)java.lang.StringIndexOutOfBoundsException: Range [6, 1) out of bounds for length 30
gapjava.lang.StringIndexOutOfBoundsException: Range [6, 1) out of bounds for length 29
[163,164 165 ,167[[2*^+*^2()^
gap> PowerMap( t, 11 ){ ord91 };
[ 167, 168, 163, 164, 165, 166 ]
]]></Example>



</Subsection>


<Subsection Label="subsect:error2F422">
<Heading>An Error in a asbeen Cite =B/

tions java.lang.StringIndexOutOfBoundsException: Range [35, 33) out of bounds for length 54
the table <2.F_42.</
automatically from a M>Ly</M>, to compute its Sylow>,and usethis
java.lang.StringIndexOutOfBoundsException: Range [31, 5) out of bounds for length 37
turned outthatthe<><M- map java.lang.StringIndexOutOfBoundsException: Range [72, 73) out of bounds for length 72
library character table of <M>2.F_4(2).2</This subgroup is of odd index, hence it con><>java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73



In fact,
which was not</
and the <Ref Attr="java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 0
 java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35



Note that the ambiguity is
<M>F_4(2).2</M>, and only java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<M>thejava.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 28



<>![
gap> t:= CharacterTable( "2.F4(2).2" );;
>f:=("F42.");
gap> map:= PowerMap( t, 2 );
[1,,1,,1,1,1 ,9,, 11, 3, 3, 3, 5, ,5 3 ,6,5,
 5, 7, ]<>
 43, 43, 20, 20, 20, 14, />
 29, 29, 29, 33, 33, 35, 37, 37, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 42, 42, 79, 79, 43, 43, 83, 83, 45, <> of February2014)/
 57, 61 ,61,98 98 , 70 ,63 81 81 ,83 1,6,,
 11, 16, 17, 24, 24, 21, 27, 27, 25, 26, 29, 41, 53, 53, 53, 46, 56,one can the
 ,56 75 ,78 ,77 77,79 79,86,86 85,,88,
 88,88, 88, 95,95,96,96 ]
gap> PositionSublist( map, [ 86, 86, 85, 85 ] )acterTableLibrary/>
140
gap> OrdersClassRepresentatives( t ){ [ 140P>
[ 32, 32, 32, 32 ]
gap> SizesCentralizers( t ){ [ 140 .. 143 ] };
[ 64, 64, 64, 64 ]
Since ;13.ofgap> SizesCentralizers( tbl ){ 6java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
[ 86, 86, 87, 87 ]
java.lang.StringIndexOutOfBoundsException: Range [31, 3) out of bounds for length 36
[,50]
gap> pos:= PositionsProperty( Irr( via the function <C>CTblLib.FactorGroupOfPerfectSpaceGroup java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
> x -> x[1] <> x[<Ly/> java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 18
[ 144, 145, java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 35
gap> List( pos, i -> Irr(t)[java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
[ [ 2*E(16)-2* java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 65
 -2*E(16)^3+2
 [ -2*E(16)+2*E(16)^7, 2*E(16)-2*E(16)^7,
 2*E(16)^3-2*E M16< :(")
 [ -2*E(16)^3+2*E(16)^5,gap );;
 ],
 [ 2*E(16)^3-2*E(16 ingap :java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 31
 2*E(16)-2*E(16)



java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
characterjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0
two alternatives is the gap> IsRecord( TransformingPermutati(t,

</Subsection


<Subsection Label&<
<a space group <M>S<<

(is <

/

The Alt Only="LaTeX">
is claimed to be the table of a java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 70
sporadic simple Lyons
java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 67
<Package>CAS</Package> system java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 65
java.lang.StringIndexOutOfBoundsException: Range [16, 5) out of bounds for length 43

<&java.lang.StringIndexOutOfBoundsException: Range [17, 15) out of bounds for length 25

It is Using the <Key"",
can have this character table.
,
hejava.lang.StringIndexOutOfBoundsException: Range [32, 2) out of bounds for length 69
with centralizer order <M>2^6</M>,
and this does not occur in <M>Ly</=java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 0



<Example> thesejava.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 31
gap> tblisjava.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 65
gap> orders:=Gjava.lang.StringIndexOutOfBoundsException: Range [39, 37) out of bounds for length 59
gap> order8:= Filtered( [ 1 .. Lengthjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ 12, 13 ]
gap>>java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 41
[ 15/2< class=pcenter>
]]></Example>



The table of <M> from space =tdcenter>class=impleMath&=<panjava.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 71
mapp by the
NowadaysC<C> totd =dright"span =impleMath>/>/d>
java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 69
to compute Func="GeneratorsOfGroup"/>
However, the task is groupjava.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 24
of the java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 0
This subgroup is of odd index, hence <Subsection Label="subsect:constructspa
Clearly the java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 8
are "java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 27
Thus we can start with a rather> (g ) in, java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 63



<Example><![java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0
gap> g:= java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 0
>NrMovedPoints g ;\[
275
gap> syl:= SylowSubgroup( g, 2 );;
gap> </>
gap> t:= CharacterTable( pc );;



The charactertable coincides with the onewhichis in
the Character Table Library<;<>



<and <M>M(T):= \{ M(I, t); t \in T \}</M>
( ,
> CharacterTable( "More generally, <M>M(n T)</M> is a normal subgroup of <M>S</M>,
rue
]<>

<>

</Section>

java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 71
<Section Label="sect:spacegroupfactors">
<Heading>Somefinite groups of ( 2014)<Heading>

and <M>V</M<java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 75
&GAP; Character<>

<Ref Func="<td class="tdright"><spanjava.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 69
 W.;(
<Q>Ordinary and Brauer Tables in the &GAP;
in the &CTblLib; manual).



Using the information from <Cite</iv
java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 34
such groups as java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 0
Since version&the <E>point group M<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
calling
<Ref Func="GroupInfoForCharacterTable" BookName="ctbllib"/>
java.lang.StringIndexOutOfBoundsException: Range [7, 3) out of bounds for length 32
in question yields nonempty lists and thus allows one to access the
resultsof theseconstructions
via the function <C>CTblLib.the <E>translation lattice</E> <M>T</M> of <M>S</M> is a sublattice of
This is an undocumented auxiliary function that becomes and the <E>vector system</E> <M>V</M> of <M>S</M> is a map from <
Note that<>VM the <>I\<M> java.lang.StringIndexOutOfBoundsException: Range [59, 1) out of bounds for length 80
<Ref Func="GroupInfoForCharacterTable" BookName="ctbllib"/>
java.lang.StringIndexOutOfBoundsException: Range [3, 1) out of bounds for length 80



<Example><![CDATA[
gap>to ,
gap> IsBound( CTblLib.and <M>M(T):= \{ M(I, t); t \in T \}</M>
true
]>/>
 ,<>n)/
<Pforany <></>

Below we list the <M>16</M> group constructionsP/
In each,
pping by
<C<>TM given bya<>Z;/>

<and<><M is givenby the <>() g_2,\,Vg_k)/>
of the group returned by
<C>CTblLib.java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0


<Subsection Label="subsect:constructspacegroups">
<Heading>Constructing the space groups in question</Heading>

In <Cite Key="HP89"/>,
a space group <M>S</M
<M>\{ M(gjava.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
where

</

</>
\[
 M(g, t) = \leftjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
 +t& e{\]java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
\]
</Alt>

<Alt Only="Text"Verb>
<Verb>
 M(g, t) = ⌈ ⌊Vg1)1⌋
 ⌊ V(g)+t 1 ⌋

</Alt>

<Alt Only="HTML"><![CDATA[
<div class="pcenter<"java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 29
<tr>
< < class">
<tdclass="tdcenter"> = </td>
<td<td class
 <table class="GAPDocTable">
 <tr>
 >span "/java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
 <td class="tdright">
 </tr>
 <tr>
 <td java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 8
 <dclass="tdright"<pan class=SimpleMath>/tdjava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
 </tr>
 </table>
</java.lang.StringIndexOutOfBoundsException: Range [6, 4) out of bounds for length 64
</tr>
</table>
</div>
]] <d ">span class=SimpleMath class="<pan"java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 64



the <E>point group</E> <M>P/>
 <tr>
the<> lattice<E><>java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 10
<;^<M,
and the <E>java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 9
<</d>
Note that <M>V</M> maps the identity matrix <M>I \in</M> GL<M>(d, &ZZ;)</M>
to thetdclass"dleft
and <M>M(T):= \{ M(I, t); t<ableclass=GAPDocTable
is a normal subgroup java.lang.StringIndexOutOfBoundsException: Range [6, 1) out of bounds for length 69
java.lang.StringIndexOutOfBoundsException: Range [20, 4) out of bounds for length 63
for <tdclass="<tr>



Specifically,
isgiven by generators<<,bsp;
<M>T</M> is given by a <M>&ZZ;</td=t>
<B=\ b_1, b_2, \dots \<M of<T/M>
and <M>V</M> is given by the <>

</

 below <M>><> java.lang.StringIndexOutOfBoundsException: Range [76, 75) out of bounds for length 76
so we need just the following <td java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 19



<Alt Only="LaTeX">
\[
 \left[ \begin{array}{cc} g_1 &java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 \ generators the
 \<C>generajava.lang.StringIndexOutOfBoundsException: Range [1, 32) out of bounds for length 5
 \left[ \begin{array}{cc} g_k & 0 \\<M>,.,nbsp;
 \left[ \begin{array}{cc} I</>
<td>
</Alt>

<Alt Only="Text">
<Verb>
 0 java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 29
 ⌊ V(g_1) 1 ⌋, > : 1])
</Verb>
</Alt>

<Alt Only="HTML"><!< : (+1;
<div class="pcenter">

<tr>
< "dleft>
<java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 29
 <tr>
 <td class="tdright">m 1][ .d : ;
 <">span "<>
 </tr>
 <tr>
 <td class= ]/java.lang.StringIndexOutOfBoundsException: Range [14, 12) out of bounds for length 13
 <td class="tdright">
 </section Labelclasstjava.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 19
<td class="tdleft">
 ,<> </>
</td>
</td>
<td class="tdleft">
 table ">
 <tr>
 <td class="tdright">S/M(n T)</M>
 <td class="tdright">0</td>
 </tr>
 <tr>
 <td class="tdright &AP; , we wework <>d+/>java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
 "java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 64
 </tr>
 </table>
</td>
<td class="tdleft">
 , ...,
</td>
<td class="tdleft">
 <table <td class>java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 64
 <tr>
 <td class="tdright">0</td>
 /
 tr
 <td java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
 < <tdclass=
 </tr>
 /java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
<td
<td class="tdleft">
 ,
</td>
<td class="tdleft">
 < =
 <tr
 td class=tdright"deletedPermutationMat: gap> deletedPermutationMat:= function
 <> localmat, ,ijava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
 <tr>
 <td class="tdright"> : mat 1. n- ]}{C>generatorsOfPerfectSpaceGroup>,
 <> j>n then
 </tr>

</td>
<td class="dleft"
 .
</td>
</tr>
</table>
</div>
]]></Alt>



These generators are returned by the function
<C>generatorsOfPerfectSpaceGroup</C>,
when the inputsare <>[g_1,g_2,\dots,g_k ]/>java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
<M>[ V(g_1), V(g_2), \ldots, Vs for the construction of the group via



<Example><![CDATA[
gap>for in[
> local d, result, i, m;
> d:= Length( Pgens[1] );
>=[java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
> for i in [ 1 .. Length( Pgens ) ] do representation
> m:= IdentityMat( d+1 );
> m{ [ 1 .. d ] }{ [ 1 .. d ] }:= with the effect that it requires much more spof ourcases. + {[1 .d :Vi;
> m[ d+1 ]{ [ 1 ..
> result[i]:= gap>verifyFactorGroup=function( ,id )
> od;
>:=IdentityMat(d1 )
>m d+ ] [1. ]}= ;
> Add( result, m );
 return result;
> end;> s:NormalClosure Subgroup g, [[Length(gens )]]))java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
]]></Example>

</Subsection>


<Subsection Label="subsect:constructfactors">
<Heading>Constructing the factor groups gens: GeneratorsOfGroup act )

The space group <M>S</M> acts on <M>&ZZ;^d</M>,
via <M>v \cdot M(g, t) = >
A (not necessarily faithful) representation ifnot IsRecord
can be java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0
<M>&ZZ;^d/(n &ZZ id) ) <ubsection=":constructfactors"
that is, by reducing the vectors modulo <M>n</M>.
For the &GAP; computations, we work instead The (id)java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
extending each vector in <M>&ZZ;^d ( ) M/MnT<M>
and onthese by> (stored) )
Multiplication followed by reduction modulo <M>n</M> is
the action function returned by <C>multiplicationModulo</C>
when this is called with argument <M>n</M>.

</

<Example><![CDATA[
gap> multiplicationModulo:= n -> function( v, g )->
> return List Pjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
]]></Example>

</>

In some of the examples,
the representation <><M the representation of <M>P</M> given
is> :List(gens, - ^ )
ofapermutation module modulo its trivial submodule.
For that,we providethe function <>deletedPermutationMatC,
cf. <Cite Key="HP89" Where="p. 269"/>.



<Example><![CDATA[
gap> deletedPermutationMat:= function( pi, n )
> local mat, j, i;
> mat:= <>< &<Key"P89/java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
> mat: of a permutatiof a permutation module modulo its trivial submodule.
> j:= n ^ pi;

> for i in [ 1 .. n-1 ] do
 [][j]=-1;
> od;
> fi;
> return mat;
> end;;
]]></Example>



After constructing gap> deletedPermutationMat:=<Subsection>
we verify that the groups fit to the character tables
&AP;Character Table Library and to the java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 36
 groupvia
<C>CTblLib.FactorGroupOfPerfectSpaceGroup</C>.

</>





<Example><![CDATA[
apverifyFactorGroup=(gens
> local sm, act, stored, hom;
> sm:= SmallerDegreePermutationRepresentation( Group( gens ) );
> gens:= List( gensonModulo 16);;
> act:= Images( sm );
> if return;
> CharacterTable( act ),
> CharacterTable( id ) ) ) then
> return "wrong character table";
> fi;
> ( )
>>
>hom=(stored,act,
> GeneratorsOfGroup( stored ), gens );
> if hom = fail or not IsBijective( hom ) then
> return "wrong<eading>Examples with point group <> sm:= SmallerDegreePermutationReGroup ))
> fi;
> There <>d 6/>
>end not ((
]]></Example>

/>


<Subsection Label="subsect:expl-A5> return" ;
<Heading>gap> a:= [ [ 0, 1,1 ,0 ]

</M>.
The generators of the point group are as follows
(ee&;Cite Key=HP89"Where=".272/)java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45

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

<>!java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
gap> a:= deletedPermutationMat( (1 0,0 ,0>fi;
gap> b:= return true
> [1 0 ,0,,0]]



In both cases

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

<Example><![CDATA[
gap> v:= [ [ 2, 2, 0, 0, 1 ], the <>L_1=&Z^<M%%%
]]></Example>

java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 4

nthefirst ,
the translation lattice is the sublattice <M>L = 2 L_1
<>.

</

<Example><![CDATA[
gap> tg>b=java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 25
]]>< : 10 ,0,0,0,1]

<>

java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 69
belongs/>
<M>M(4 L)Example><![CDATA[

/

<Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
funmultiplicationModulo(8)
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 1 <java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
gap> gap>v= [ [1 0, 1,0 0,0] 0 *a[1]];
gap>verifyFactorGroup(permgens,"P1/G2/L1/V2/ext4" );
true
]]></Example>

gap>g: Group(sgens );java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25

In the gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
thetranslation latticeisthesublattice M L_2/ of <M&Z^<Mjava.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
where <M>L_2</M> </

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



For the sake of simplicity, we gap> v:= [ [ 0, 1, 0, 0, 1, 0 ], 0 * a[1] ];;
to one on <M>L_2</M>, and we adjust also the vector system.

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

<Example><![CDATA[
ap>B: Basis( ^ength bas) bas);
gap> abas:= List(permgens"G1L1xample!DATA[
gap> bbas:= List( bas, x -> java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 13
gap> vbas:= List( v, x -> Coefficients( B,The generators thepointgroup the java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ [ 3/2, 1
]]></Example>

a: ,0 0,,0,-,1 ,1]>(4), the <.

 toworkwith matricesw necessarybecause
<C>multiplicationModulo</C><>!CDATAjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
we double both the vector system and the translation lattice.



<Example><![CDATA[
gap> vbas:= vbas * 2 0, 0, 0 1 0]java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
[ [>[1 ,1,,0 ,0,0 ;java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
gap> t:= 2 * t;
[ 4, 0, 0, 0, 0 ]
]>



The library character table with identifier <C>"P1/G2/L2/V2/
belongs to the factor
<M>M(8 L_2)</M>;
since we have doubled,
tion on an orbit modulo <>6/M>



<Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ abas, bbas ], vbas, t );;
gap> g:= Group( gap> g:= Group( sgens
gap> fun:= multiplicationModulo
gap>=Orbitg ,0bas= -- ,1java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
>:Listsx-Permutationxorb,);
gap> verifyFactorGroup( permgens, "P1/G2/L2/V2/ext4" );
true >
]]></Example>

</Subsection>


<Subsection Label="subsect:expl-L32">
<Heading>Examples with pointP

Thereare three examples with <> =6M
ndone example to one on <M>L_2</M>, and we adjust also the vector system.
The generators of the point group for the first three examples are>t=[1 0,0,0 0 0 0, 0 ];
(see <Cite

</

<Example><![CDATA[
gap> a:= [ [ 0, 1, 0, 1, 0, 0 ],
> [ 1, 0, 1, 1, 1, 1 ],
> [-1,-1,-1,-1, 0, 0 ],
> [ 0, 0,-1,-1,-1,-1 ],
> [1,1,1,1,0 1],
> [ 0, 0, 1, 0, 1, 0 ] ];;
gap> b:=[[-,0 0 0,,0,Subsection>
> [ 0, 0,-1, 0,-1, 0 ],
> [java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 67
> [ 0HeadingExample with point we double both the vector system lattice
 ,-,0 , 0 ]
> [ 1,0,0 0,0,0 ]];
]]></Example>

<java.lang.StringIndexOutOfBoundsException: Range [5, 3) out of bounds for length 15

The first java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
(that is b: 0-,0 0java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
and the translation 1 , ,, ,, ,
and the translation lattice is the java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 16



Thejava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
belongs to the factor group of <M>S</M> java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 0
<M>M(4 L_1)</M>, so we compute the( ] ;

</

<java.lang.StringIndexOutOfBoundsException: Range [17, 8) out of bounds for length 18
gap> v:= List( [ 1, 2 ], i -> 0 * a[1] );;
gap> t:= [ 1, 0, 0, 0, </
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 4 );;
0,,0 , ,1]java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
gap> orb:= OrbitThere are examples M <M>
gap> permgens:= List(> [0,0 0 0, 0 ]java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40

true
]>/xample

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

The second vector system is <M>V_2</<>!CDATA
and the translation a=[[0 1,,1,,0 ,



belongs to the factor group of 1 , , 0 >
<>8L_1)<M,so compute the action on an orbit modulo <M>8</M>.



<xample>!CDATAjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
gap> v:= [ [ 1, 0, 1, 0, 0, 0 ], 0 * a[1] ];;
gap
gap> sgens:= generatorsOfPerfectSpaceGroup
gap>g: Group(sgens );;
gap> fun:= multiplicationModulo( 8 );;
java.lang.StringIndexOutOfBoundsException: Range [7, 3) out of bounds for length 53
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P11/G1/L1/V2/ext4" );
true
]]></Example>



The third vector system is <M>V_3</M>,
and the translation lattice [0,0, 1,0,0,0 ]java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32



The library character table with identifier <C>"]>Example>
belongs the factor group of <M><M modulo the normal subgroup
<M>M(8 L_1)</M>, so we compute the action on an orbit modulo <M>8</M>.



<Example(hat , java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 74
gap> v:= [ [ 0, 1, 0lattice)
gap>t=[2,,
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
ap fun:=ultiplicationModulo(8 );java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
gap>orb:=Orbit(g, [1,0, 0,0 ,0 1 ] fun);java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
java.lang.StringIndexOutOfBoundsException: Range [45, 3) out of bounds for length 64
gap> verifyFactorGroup( permgens:java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 38
true
]java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13

<>

The The generators ,fun);
(see <gap> permgens:=10 ,,0 ,0,,0]];



<Example><![CDATA[
gap> a:= [ [ 1, 0, 0, 1, 0,
> [ 0,-1, 1, 0,-1, 0, 0, 0 ],
> [ 1, 0, 0, 1, 0,-1, 0, 0 ],
> [ 0,-1, 0,-1, 0, 1java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
> [ 1, 0,-1, 1, 1,-1, 0, 0 ],
> [ 1,-1,-1, 0, 0, 0, 1, 0 ],
> 0,-1,1,0,1,1,0,1 ]java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
> [ 1, 0,-1,
gap> b:= [ [ 1gExample>[java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
> [ 0,-1, 0, 0, 0, 0, 1,-1 ],
> ,0,1 ,1,1 0 0]java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
> -,1 ,-1 2,,1 ,
> [ 0, 0, 0,-1, 0, 0, 0, 0 ],
> [ 0,-1, 0,-1,-1, 1, 1,-1 ],
> [ 1-,0, 0,0 0,0, ,
> [ 1, 0, 0, 0, 0, 0, 0, 0 ] ];;
]]></Example>



The vector system is the trivial vector system <M>V_1</M>java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 13
and the java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 4



The library character table with identifier <C>"P11/G4/L1/V1/ext3"</C>
belongs to the factor java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 4
<M>M(3 L_1)</M>, so we compute The library character table with identifier <C>"P11/G1/L1/V3/ext4"</C>



<Example><![CDATA[
gap> v:= List( [ 1, 2 ], i -> 0 * a[1] );;
gap> t:= [ 1, 0, 0, 0, 0, 0, 0, 0 ];;
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 3 );;
gap> seed:= [ 1, 0, 0, 0, 0, 0, 0, 0, 1 ];;
gap> orb:= Orbit( g, seed, fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P11/G4/L1/V1/ext3" );
true
]]></Example>

<>/>

>[
< =:xpl
<Heading>Example with point group SL<Mgap t=[1 ,0,0 0,0 0 ];

There is one example with <M>d = 8</M>.
The generators of the point group are as follows
(see <Cite Key="HP89"Example<!CDATA[

<ga> =[1,0,0,0 ,0 0 0, 1 ];

<Example><![CDATA[
gap> a:= gap> t:= [ 1, 0, 0,0,0,0 ];
gap b: ,1 ,0 0,0, 0 ,
> [ 0, 0, 1, 0, 0, 0, 0gap>g: Group( sgens );
> [-1, 0, 0, 0, 0, 0, 0, 0 ],
> [ 0, 0, 0, 0, 0, 0,-1, 0 ],
 ,0-1 ,0,0,0]
> [ 0, > verifyFactorGroup
> [ 0, 0, 0, 0, 1, 0, 0, 0 ],
> [ 0,0,0 0 0,0 0,1]java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
]/>



e 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>B(2,8)</M> in <Cite Key="HP89" Where="p. 269"/>.



<Example><![CDATA[
gap : [ ,- , ,
> [ [0 ,,,0 000 ,
> [ 0, 0, 1, 1, 0, 0, 0, 0 ],
0java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
[ 1 ,,0,
> [ 0, 0, 0, 0, 0, 1, 1, 0 ],
> [ 0, 0, 0, 0, 0, 0, 1, 1 ],
> [ 0, 0, 0, 0, 0, 0,-1, 1 ] ];;
]/java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 13



For the sake of simplicity, we rewrite the action to one on <M>L_2</M>.



<Example><![CDATA[
gap> B:= Basis( Rationals^Length( bas ), bas );;
gap> abas:= List( bas, x -> Coefficients( B, x * a ) java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
gap> bbas:= List( bas, x -> Coefficients( B, x * b ) );;
]<Example>



The table with identifier <C>"P12/G1/L2/V1/ext2"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(2 L_2)</M>.
The action on an orbit modulo <M>2</M> is not faithful,
its kernel contains the centre of SL<M>(2,7)</M>.
We can compute a faithful representation by acting on pairs:
ne , 0 ,1 00, 0]]java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
of the point group.



<Example><![CDATA[
v: List([1 2 ] i- 0 *]);
gap> t:= rjava.lang.StringIndexOutOfBoundsException: Range [14, 12) out of bounds for length 45
gap=01- , ,,,]
gap> g:= Group( sgens );>ap BasisRationals(bas ,);
gapgap>abas: ( bas,x-Coefficients0],
gap>funpairs=function( pair java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
> return [ fun( pair[1], g ), pair[2] * g ];
> end;;
gap> seed:= [ [ 1, 0, 0, 0, 0, 0, 0, 0, 1 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
> [ 1, 0, 0, 0, 0, 0, 0, 0, 0 ] ];;
gap> orb:= Orbit( g, seed, funpairs );;
gap<>(2 L_2<M.
gap> verifyFactorGroup( permgens[1,0,0 0,,0,0 ,0 ]];
true
]]></Example>

</Subsection>


<ubsection="ubsect:-23L32"
<Heading>Example with point group <M>2^3

Thereis one examplewith <java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 39
The generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 297"/>).



gap> a:= PermutationMat( ]<Example
gap> gapjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
gap> c:= DiagonalMat( [ -1, -1, 1, 1, -1, -1, 1 ] );;
]]></Example>

<>

The systemis >java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
and the>: , -CoefficientsB2 b ));
java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 69
which M(7/>innbsp<Key" Where=p 269".



<Example
gap> bas:= [ [ 1, 1, 0, 0, 0, 0, 0 ],
> [ 0, 1, 1, 0, 0, 0, 0 ],
> [ <
> [ 0, 0, 0, 1, 1, 0, 0 ],
> [0 0,0 0,java.lang.StringIndexOutOfBoundsException: Range [61, 62) out of bounds for length 61
> [ 0, 0, 0, 0, 0, 1, 1 ],
> [ 0, 0, 0, 0, 0,java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 18
]></Example>



For the sake of simplicity, we rewrite the action to one on <M>L_2</M>.



<<[[
gap> B:= Basis( Rationals^Length( bas ), bas );;
> : List(bas java.lang.StringIndexOutOfBoundsException: Range [0, 24) out of bounds for length 0
gap> bbas:= List( bas, x -> Coefficients( B, x * b ) ); java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 58
gap> cbas:= List( bas, x -> Coefficients( B, x * c ) );;
]]></Example called M(27<M> <ite=HP89"Where=".269">



The library character java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 70
 thegroup > java.lang.StringIndexOutOfBoundsException: Range [66, 50) out of bounds for length 66
<2<M
P>


<Examplegap =generatorsOfPerfectSpaceGroup, ,,t);;
gap> vgap>g:= Group( sgens );;
gap> 0 ,0,0,0-,1 ;
gap> sgens:= generatorsOfPerfectSpaceGroup( [ abas,bbas,cbas ], v, t );;
gap> g:= Group( ]]<Example>
gap> fun:= multiplicationModulo( 2 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 0, 1 ], fun );;
gapFor thesake simplicity, we rewrite the action to one java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 0
gap> permgens<Examples<>
gap> verifyFactorGroup( permgens, "P13/G1/L2/V1/ext2" );
true
]]>/Example>

<Subsection>


<Subsection Label="subsect:expl
<Heading>Examples with point group <M>A_6</M>

twoexampleswith Md 10/.
In both cases, the generators> [0 0,1,,0>
java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 45



<Example
 0 , 0 0-1 0 java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
 ,, 0 ,,java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
[,,0 ,,0 ,0 ,
 1,,0 ,0, 0, 0, 0, 0 ],
> [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
> [ 0, 0,]>/
> [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],
> [ 0, 0, 0,P
> [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ] ];;
gap> c:= [ g> : (sgens - ( ,orb, fun ;java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
> [
> [ 0, 0, 0,]>
> [/java.lang.StringIndexOutOfBoundsException: Range [12, 7) out of bounds for length 38
> [ 0, 0, 0, 0, 0, 0, 0, 0, 0,-1 ],
> [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],
> [ 0, 0, 1, 0, 0, 0, 0, 0, 0, character tableidentifierCP41////C>
> [ 0, 0, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
> [ 0, 0, 1,-1, 0, 0, 0, 0, 0, 1 ],
 ,0- ,0 0 0 ];java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
]]></Example>



In both examples, the vector system is the trivial vector system <M>java.lang.StringIndexOutOfBoundsException: Range [0, 71) out of bounds for length 62
and the translation lattices are the lattices <M>L_2</M> and <M>L_5</M>,
respectively, which have the following bases.



<Example><![CDATA[
gap> bas2:= [ [ 0, 1,-1, 0, 0, 0, 0, 0, 0, 0 ],
> [ 0, 0, 1,-1, 0, 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 ],
> [ 0, 0, 0, 0, 0, 1, 0,-1, 0, 0 ],
> [ 0, 0, 0, 0 0, 0,0,1-1,java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47

> [ 0, 0, 0, 1, 0, 0, 0, 0, 0,-1
> [ 0, 1, 0, 0, 0, 0, 0, 1, 0, 0 ],
> [ 1, 0, 0, 0, 1, 0, 0, 0, 0, 0 ] ];;
java.lang.StringIndexOutOfBoundsException: Range [5, 3) out of bounds for length 47
> [ 1, 0,-1,-1,-1, 1, 1,-1,-1, 0 ],
>
> [ 1, 1, 0,-1, 0,-1, 1,-1, 1,-1 ],
> [-1, 0,-1, 1java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
> [ 0, 1,-1, 1, 1,-1, 1, 1, 0,-1 ],
> [g=Group );
> [ 1,-1, 0,-1, 1,-1, 1, 1, 0,-1 ],
> [,1-,1-,,1 ,, ,
> [ 1,-1,-1, 1, 1, 1, 0, 0,-1,-1 ] ];;
]]></Example>



orthe sakeof , werewrite the actiontoactionson <>L_2/M
and <M>L_5</M>, respectively.



<Example><![CDATA[
>B2:= Length ,java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
gap> bbas2:= ListThere M =<>
>= - (B2,x *c );java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
gap> B5:= Basis( Rationals^Length( bas5 ), bas5 );;
gap> bbas5:= List( bas5, x</
gap> cbas5:= List( bas5, x -> Coefficients( B5, x * c ) );;
]]></Example>



The library character table with identifier <C>"P21/G3/L2/V1/ext2"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(2 L_2)</M>, so we compute java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0



Example>!CDATA[
gap> v:= List( [ 1, 2 ], i -> 0 * bbas2[1] );;
gap> t:= [ 1, 0, 0,<Example>[java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
gap> sgens:= generatorsOfPerfectSpaceGroup( [ bbas2, cbas2 ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= gap> t:= [ 2, 0, 0, 0, 0, 0, 0, 0 0];
gap> seed:= [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ];;
gap> orb:= Orbit( g, seed, fun
gap permgens: List sgens, x- Permutation ,orbExample>!CDATA[
gap> verifyFactorGroup( permgens, "P21/G3/L2/V1/ext2" );
true
]]></Example>



The library character table with identifier <C>"P21/G3/L5/V1/ext2"</C>
dulo the normal subgroup
<M>M(2 L_5)</M>, so we compute the action on>[gap g: )



<Example><![CDATA[
>: v ;;
gap> g:= Group( sgens );;
gap> orb:= Orbit( g, seed, fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P21/G3/L5/V1/java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 0
true
]]></Example>

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

!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 37
<Heading>Examples with point group <M>L_2(8)< ,1 ,1 ,1 ,

here two examples
In both cases, the generators of the point group > [1,1,1,1,1,0-1,
(see <Cite Key="HP89" Where="p. 327"/>).



[
gap> a:= [ [ 0,-1, 0, 1, 0,-1, 1],
> [ 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,-1, 1, 0,-1, 0, 0],
> [ 1,-1, 0>[-1 ,1,1 ,,0 ,
gap> b:= [ [-1, 0, 1, 0,-1, 1, 0],
> [ 0,-1, 0, 1,-1, 0, 0],
>
> [ 0, 0,-1, 0, 0, 0, 0],
> ,,1 ,0 ,0,
> [-1, 1, 0,-1, 0, 0, 0],
> [-1,>0 *];
]<



both, ]<>
The translation lattice injava.lang.StringIndexOutOfBoundsException: Range [0, 7) out of bounds for length 0



<Example><![CDATA[
>v= [2 ,0 0,>: java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 25
>[
gap
]<



The library character table with identifier <C>"g>v=List , ,i 1);
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(3 L)</M>, so java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20

</

 inthisaction quitelong.
we choose a seed vector from the fixed space of an element of order <M>7</M>.



<Example><![CDATA[
java.lang.StringIndexOutOfBoundsException: Range [5, 3) out of bounds for length 62
gap> g:= Group( sgens );;
gap> aa:= sgens[1];;
gap> bb:= sgens[2];;
gap> elm:= aa*bb;;
gap> Order( elm );
7
gap ectionLabel=subsect:explU42>
[ [ 1, 1, 1, 1, 1, 1, <>Exampleswith <>2)</M></Heading><>2 )M,we an java.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 70
gap> fun:= multiplicationModulo( 9 );;
> =fun 2,0)
[ 5, 1, 1, 4, 4, 2, 0, 1 ]
gap> orb:<![DATA[
>permgens: List sgens, ->Permutation( x, java.lang.StringIndexOutOfBoundsException: Range [45, 3) out of bounds for length 34
gap> verifyFactorGroup( permgens, "P41/G1/L1/V3/gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
true
]]></Example>



The translation latticeThere are two exampleswith<Md=7<M.
<M>L = 6 &ZZ;^d</M>.

java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 4

<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
<>(6L)/> sowe computethe action onan orbitmodulo M></>



<Example><![CDATA[
gap> fun:= > [ 1,-1, 0, 1, 0,,0]];
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t </
gap> g:= Group( sgens );;
gap> [0-, ,0]
[ 14, 1, 1, 13, 13, 2, 0, 1 ]
)java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
p:(, - (, java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 64
java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 56
true
] java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 66

<>


<ubsection=ubsect:-M11>
<Heading>Example with point group <M>M_{11}</M></Heading>

There is one example with <Mjava.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
The generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 334"/>).



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 <M>L = 2 &ZZ;^d</M>.

<>

<Example><![CDATA[
gap> v:= [ 0 * a[1],
> [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 ] ];;
gap> t:= [ 2, 0, 0, 0, 0, 0java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 77
]]></Example>



The library character table with identifier

<M>M(2 L)</M>, so we<



<TA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ <>2^3<M,<M>23</M>,<>33/M>,and <M5^3</M,respectively.

gap> fun:= java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 18
gap> </ection>
gap> permgens:= List( sgens, x
>permgensP48G1L1V2/";
true
]]></Example>

</Subsection>


<Subsection Label="subsect:expl-U33">
<Heading>Example withchoices thesame

There is one example with <M>d = 7</M>.
The generators of the point in <Cite Key="JLPW95" Where="p. x"/>.
(see> : Orbit seed );



<<ap>verifyFactorGroup(permgens, "P41/G1/L1/V3/ext3" );
gap> a:= [ [ 0, 0,-1, 1t"ubsect:enerality_J3"/>givesan
> [ 1, 0,-1, 1, 1,-1, 0 ],
> [ 0, 1,-1, 0, 1, 0,-1 ],
> [0 1, 0,-1, 1, 0,-1 ],
> [-1, 1, 1,-1, 0, 1, 0 ],
> [-1, 0, 1,-1, 0, 0, 1 ],
> [ 0, 0, 0, 0, 0, 0, 1 ] ]( functions shownthissectionarebased java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
gap> b:= [ [ 0, 0, 0, 0, 0, 0, 1 ],
> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
>,, ,1-,
> [ t choice necessary
 0 1, <Example>![DATA[
> [-1, 1, 1,-1, 0, 1, 0 ],
> [-1, 0, 1,-1, 0, 0, 1 ] ];;
]]></Example>



Thesystem Mjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 4
etranslation lattice is<M> = 3&ZZ^<M>.



<Example><![CDATA[
gap> v:= [ [ 2, 1, 0, 0, 2, 1viapermutingthe preimages.
> 0 * b[1] ];;
gap>t= [ 3,0 0, 0, 0, 0, 0 ]];
]]>/xample

<>

The library character table with identifier <C>"P49/G1/L1/V2/ext3"</C>
belongs to the factor group of <M>S</M> gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ],automorphisms <tbl/>.
<M>M(3 L)</M>, so we compute the action on an orbit modulo <M>9</M>.



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



The orbits in this action are quite long.
we choose a seed vector from the fixed space of an element of order <M>12</M>.



<Example<!CDATA[
gap>aa: 1];
gap> bb:= sgens[2];;
>elm=aa*b;;
gap> Order( elm );
12
gap> fixed:= NullspaceMat( elm - aa^0 );
[ [ -1, -1,
gap> seed:= fun( fixed[2], aa^0 );
[ 0
gap> orb<>Remark:/E> Thanksto Lux discussionson this topic.
<>
gap> verifyFactorGroup( permgens, "P49/G1/L1/V2/ext3" notto java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 70
true
]]></Example>

</Subsection>

>[java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 18
<Subsection Label="subsect:expl-U42">
<Heading>Examples with point group <M>U_4(2)</M></Heading>

Therejava.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 71
Incases, the point groupare as
(see <Cite Key="HP89" Where="p. 336"/>).

</>

<Example><![CDATA[
gap> a:=[ [ 0, g under the automorphismsofthe ordinary table with the orbitof java.lang.StringIndexOutOfBoundsException: Range [71, 72) out of bounds for length 71
> [ 1, 0,-1, 0, 1, 0 ],
> [ 0, 0, 0,-1, 0, 1 ],
> [ 0, 0,-1, 0, 0, 1 ],
> ,00 0,1 0]
> 0
gap> b:= [ [ 0,-1, 0, 1, 0,-1 ],
> [0 1,0,-,1, 0],
> [ 0, 0, 1, 1, 0,-1 ],
> [ 0, 0, 0, 0,-1, 0 ],
> [ 0, 1 2 2 2 1
> [ 1, 0, 0, 0, 0, 0 ] ];;
]]></Example>



In both examples, the vector system is the trivial vector system <M>V_1</M>,
and the translation lattice is the java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 18



<Example><![CDATA[
gap> v:java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
gap> t:= [ 1, 0, 0, 0, 0, 0 ];;
]<



library character tablewithidentifier <C>"P50/G1/L1/V1/ext3"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(3 L_1)</M>, so we compute the action on an orbit modulo <M>3</M>.



<Example><![CDATA[
X.9 9 9 1 . .. . 1 - -1 - 3 1 . . ..
gap>> = Group(gens );
gap> fun:= multiplicationModulo( 3 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 1 ], fun );;
:(, -> ,
gap> verifyFactorGroup( permgens, "P50/G1/L1/V1/ext3" );
true
]]></Example>



The libraryjava.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 70
to > normal
<()/> computetheactiononan modulo <>4</>.



<Example><![CDATA[
: ( (=java.lang.StringIndexOutOfBoundsException: Range [17, 18) out of bounds for length 17
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 4 );;
gap> orb= Orbit( g 1 0 0,0 ,0 1] fun );java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
gap> permgens:= List( sgens, xrdi table <Mtbl<> givenby theorbitofafixedclass
gap/G1/L1/V1/")java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
true
>java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13

</Subsection>

<Example<[DATA[
<> remarkon the former case, we chooseoneof ,

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



<Example>![DATA[
gap> degrees:ote that the procedure sketched above does not tell
[ [ 1, 1 ], [ 2, 2 ], [ 3, 2or which classes by the choices
 [ 10, = sgens[2];;
gap> List( degrees, x -> x[1]
true
]]></Example>



java.lang.StringIndexOutOfBoundsException: Range [19, 5) out of bounds for length 66
gap>orb= Orbit>der, p modtbl taut,triv, admiss, fusion, maut,
<M>2^3</M>, <M>2^3</M>, <M>3^3</M>, and <M>5^3</M>, respectively # Find theprimes for which gap> verifyFactorGroup( permgens, "P49/2/xt3 )java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
This phenomenon has been described in <Cite Key="NR14"/>(eenbsp;Cite Key=H"> inconsider do

</Subsection>

</Section>


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

The term <Q>generality problem</Q> is used for problems java.lang.StringIndexOutOfBoundsException: Range [0, 66) out of bounds for length 58
onsistent choices of conjugacy classes of Brauer tables for the same
group, in different characteristics.
iven
in <Cite Key="JLPW95" Where=> 0,0 0 0,,0],>#under the actions

 orbits:= [];

Section <Ref Subsect="subsect:generalityproblems_list"/>
shows how to detect generality problems
and lists the known generality problems,
and Section <Ref Subsect="subsect:generality_J3"/> gives an example
that actually arose.


<Subsection Label="subsect:generalityproblems_list">
<Heading>Listing possible generality problems</Heading>

> #We have found a generality problem.
may involve generality problems.
(The functions shown in this section are based on &GAP; 3 code
that was originally written by Jürgen Müller.)

</

If the <M>p]]</Example>
of a group contributes to a generality problem
then some choice of conjugacy classes is necessary
in order to write down this table,
in sensesense that some symmetry of the corresponding ordinary tablethe <M>5</M>-modular character table of the group <M>2.A_5.2</M>.
<M>tbl</M>, say, is broken in <M>java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 0
This valuesM\ s{-3}/M>onjava.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 77
assume thega> :([a,b] v ;;
has been fixed.
All possible class fusions are obtained as the orbit of this
class fusion under the actions of table automorphisms of <there are table automorphisms which interchange each pairgap>permgens= List( sgens, x->Permutation( , orb fun ));
via mapping the images of the class fusion
(with the function <Ref Func="OnTuples" BookName="ref"/>),
and of the table automorphisms of <>
via permuting the preimages.
The case of broken symmetries occurs if and only if this orbit
splits into several orbits when only the action of the
table automorphisms of <M>mtbl</M> is considered.
Equivalently, symmetries are broken if and only if the orbit under
ofMmtbl/ notunderactionof
table automorphisms of <M>tbl</M>.





<><![DATA[
gap> BrokenSymmetries:= function( ordtbl, modtbl )
> local taut, maut, triv, fus, orb;
> taut:= AutomorphismsOfTable( ordtbl );
> maut:= AutomorphismsOfTable( modtbl );
> triv:= TrivialSubgroup( taut );
> fus:= GetFusionMap( modtbl, ordtbl );
> orb:= MakeImmutable( Set( OrbitFusions( maut, fus, triv ) ) );
> return ForAny( GeneratorsOfGroup( taut ),
> x -> ForAny( orb,
> fus -> not OnTuples( fus, x ) in orb ) );
> end;
]]></Example>



<E>Remark:</E> (Thanks to 

Notethat thislist may become longer as new Brauer tables become
available.
(For example, the prime <M>2</M> was added to the entries for
xtensions <)/>when <>2<M-odular table of<M>(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>
]]>
/java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
<Alt Not="HTML">
In March 2015,
</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> cand:= Filtered( Irr( m ), x -> x[1] = 110 );;
gap> Length( cand );
1
gap> slp:= AtlasProgram( "J3", "classes" );;
gap> 17a:= Position( slp.outputs, "17A" );
18
gap> info:= OneAtlasGeneratingSetInfo( "J3", Characteristic, 19,
> Dimension, 110 );;
gap> gens:= AtlasGenerators( info );;
gap> reps:= ResultOfStraightLineProgram( slp.program,
> gens.generators );;
gap> Quadratic( BrauerCharacterValue( reps[ 17a ] ) );
rec( ATLAS := "-1-b17", a := -1, b := -1, d := 2,
 display := "(-1-Sqrt(17))/2", root := 17 )
]]></Example>



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.



<Alt Only="LaTeX">Table~\ref{valuesJ3}</Alt>
<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|">
<Caption>Representations of <M>J_3</M> that may define <C>17A</C></Caption>
<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><M>1-b_{17}</M></Item>
<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></Item>
</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>+</M></Item>
</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>+</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
</java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 63

<java.lang.StringIndexOutOfBoundsException: Range [70, 2) out of bounds for length 70

Note> CharacterTable (2006)Heading
M5/> that thetwo classesC></C>and <C>17B</>occur
inpairsof Galois conjugate characters.



java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 68
 java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 65
but notcompatible with the three available characters in characteristic
<M> : [645121024,024,,64646464128,,,



<>[java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
gap> table:= [];;
gap> for pair in [ [ 2, [ 78, 80, 244
> [ 19, [ 110, 214, 706 ] ] ] do
> :=[];
> for d in pair[2] do
 : ( "" Characteristic,,
> Dimension, d );
> [[1,1]1,[2[,[11,1-]],[,][,][1java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
> );
> ;
> : (reps17a ] )
> java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 49
 ( .java.lang.StringIndexOutOfBoundsException: Range [63, 62) out of bounds for length 67
>>
 ;
gap>[(8 ^ E^1 L2()N ^6E 2^ I, L2(8) N 2^ E^ II,
[ [
 then we have
, -java.lang.StringIndexOutOfBoundsException: Range [40, 38) out of bounds for length 41
 [ 2, [java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 26
 ,-,
 [ 19, 214
 [ 19, 706, 1+b17> java.lang.StringIndexOutOfBoundsException: Range [22, 20) out of bounds for length 31
]/java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 13

<java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 51
in
n, straight
java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 20



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 gap> cen:= ClassPositionsOfCentre
2/>java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 23
then java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 64
and the classes of element orders <M>17</M> and ,7],
in the <M>2</M>-modular character table of the triple cover <M>3.J_3</M>
of <M> [ 2.HS 3 5,7 java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
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
<17/ the><M java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 69
<3.</> <M>32/,and <M>J3./M>.



We decide to change the <M>19</M>-modular character tables.
java.lang.StringIndexOutOfBoundsException: Range [25, 4) out of bounds for length 74
Brauer trees— have been described in <% java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 17
where explicit choices are mentioned thatThejava.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 80
Questions about the consistency with Brauer tables in other characteristic
had not been an issue in this book.
Ojava.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 71
<M>3.J_3</M> is mentioned.)
In fact, the book mentions java.lang.StringIndexOutOfBoundsException: Range [0, 31) out of bounds for length 2
utedjava.lang.StringIndexOutOfBoundsException: Range [39, 36) out of bounds for length 53
The inconsistency of Brauer character tables % > facttbls[1], facttbls[2] )
java.lang.StringIndexOutOfBoundsException: Range [54, 53) out of bounds for length 74
have been put together, and had not been detected until now.



<E>Remarks:</E>

<List>
<Item>
 Such a change of a Brauer table can Such a change of a Brauer table can injava.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
 from and to this table.
 Note that Brauer Note that Brauer tables table
 among possible fusions that are equivalent w. % then <M>G</M> is a central extension of this group bordertwojava.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
 table automorphisms of the ordinary table.
 In this particular case, in fact no class fusion had to be changed,
 see the sections <Ref Subsect="subsect:L2(16).4_in_J3.2"/> and
 Subsect=subsectgenerality/.
 !-- the relevant sections in <Cite Key="AmbigFus"/>. -->
</Item>
<Item>
 The change of the character tables affects the decomposition matrices.
 Thusthe files M</M>-odular decomposition
 matrices had to be updated, see
 <RL>http:/www.math.wth-.de/~homas./tbllib//tex//java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 61
</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.
< (3,,4,8,6,7,,9) ])]
</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
in the <M>11</M>- and <M>19</M>-modular character tables
are determined by unique Brauer characters that have different java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 61
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 question is whether the two Brauer tables are consistent
w...this property ornot



(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
<M>HN</M>.)

</

We start with G;CharacterTable since 2000.
There are exactly two ordinary java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 0
take different onthe classes<20A/>, <C>0B</>,
these are <M>\chi_{51}</M> and <M>\chi_{52}</M>.



<xample<!CDATA[
t:CharacterTable HN" );java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
gap> pos20:= Positions( OrdersClassRepresentatives( t ), 20 );
[ 39, </
gap> diff:= Filtered( Irr( t ), x -> x[39] <> x[40] );;
gap>List
[ 51, 52 ]
]]></Example>



These values are irrational and lie in the field that is generated by
the square root of 17E()27+(91)34+()45E(91^+91^+(91^+()75



<Example><![CDATA[
gap> List diff, x -> List( x{ [ 1, 39, 40 ] },
> CTblLib.StringOfAtlasIrrationality ) )<Subsection Label="subsect:error2F422">
[ [ "5103000", "2r5+1", "-2r5+1" ], [ "5103000", "-2r5+1", "2r5+1" ] ]
]><Example>



each of the characteristics < not ,
the <M>p</M>-modular reductions of <M>\chi_{51}</M> and <M>\chi_{52}</M>
decompose differently into irreducibles.
Notethat Galoisautomorphism of the ordinary character table
that maps<>\sqrt5<M>to <>\sqrt{5<M does not live in the
<M>11</M>- and <M>19</M>-modular Brauergap>t=CharacterTable2F4)2 );

</>

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>,
nd reduction of<>\chi_52<is
<M>\varphi_{39} + \varphi_{49}</M>,
with ,75 ,78, ,77,79 79,,86,85,85,88,
<M\varphi_{49}1)= 3\\,741,081</M>.



<Example><![CDATA[
gap> t11:= t mod 11;;
gap> rest11:= RestrictedClassFunctions( diff, gap> rest11:= RestrictedClassFunctions( diff, t11
gap> <>
gap> List( dec11, Set );
[[F32 ,,29 ]
gap> List( dec11, x -> Positions( x, 1 ) );
[ [ 40, 48 ], [ 39, 49 ] ]
gap> > od;
[ 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

<>\varphi_{42} + \varphi_{45}</M>,
with <M>\varphi_{42
<M>\varphi_{45}(1) = 2\,977\,075</M>,
and the reduction of <M>\chi_{52}</M> is
<M>\varphi_{33} + \varphi_{48}</M>,
<>varphi_{33}(1w&java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 29
<M>\varphi_{48}(1) = 3\,905\,670</M>.


gap> t19:= t mod 19<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 39
est19: (diff t19);
gap> dec19:= generatorsofpoint as
gap> List( dec19, Set );
] ]
gap> List( dec19, x -> Positions( x, 1 ) );
[ [ 42, 45 ], [ 33, 48 ] ]
gap> List( Irr( t19 ){ [ 42, 45 ] }, x -> x[1]gap>cbas:=List bas,x-> Coefficients(B,x *c) );
[ 2125925, 2977075 ]
gap> List( Irr( t19 ){ [ 33, 48 ] }, x -> x[1] );
[ 1197330,3905670]
]]></Example>



Frobenius-indicatorsalljava.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 76
are <M>+</M>.
This implies that <M>\chi_{51}</M> reduces orthogonally stably modulo <M>11</M>
but not orthogonallygap>: (sgens - Permutation ,orb, ));
whereas <M>\chi_{52}</M> reduces orthogonally stably modulo <M>19</M>
but not orthogonally stably modulo <M>11</M>.



<Example>![DATA[
gap> both casesthe java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 0
[ 1, 1, 1, 1 ]
gap> Indicator( t19, 2 ){ [ 33, 42, 45, 48 ] };
[ 1, 1, 1, 1 ]
]]></ 0 ,, , ,

In version up to 1.3.4 of the character table library,
this condition was not In this particular case, in no class fusion had to bechanged,
The reduction of <M>\chi_{51}</M> modulo both <M>11</M> and and

nd of M\{2/>moduloboth<M>1/M>and<M>9/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
<>20A<C> and <>20B</C>in characteristics<M></M> and <M>19
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 MJ_3M>>J3<> M>.<M> <>._.<M are
because there are no other generality problems for <M>HN</M> involving[ 0 ,0 ,1-,0 0 ,0],
<M>19</M>-modular table, and hence we are sure that this table will not
need to bechanged because of new solutions to generality problems.



2M java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 42
There are additionally two classes of element order <M>40</M> that have to be
swapped if <C>20A</C> and <C>20B</C> get swapped.
Thus wehave to change the <M>11</M>-modular table of <M>HN.2</M> accordingly.



Changes of this kind may affectalso derived character tables in the library.
In this case, the <M>11</M>-modular table with identifier <C>"(are unique 1, 1, 1, 0, 0,-1,-1 ] ];;
was changed as well.
Notethatjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 0
the Brauercharacters tellus how to <E>choose</E> the two Brauer characters tell us how to <E>choose</E> them
<Ref Func="ConstructIndexTwoSubdirectProduct" BookName="ctbllib"/in the other characteristic.

java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 13

</Section>


<Section Label="sect:derivebrauercharacters">
<Heading>Brauer Tables

In a few situations, one can derive
the <M>p</M>-modular Brauer character table of a group
character java.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 43



For quite some time, a methodtake differentvalues the <>20A/C,<C>20B</C>,
the Brauer characters of <M>p</M>-solvable groups
(to factor <></>modulo
<Ref Subsect="IsPSolvableCharacterTable" BookName="ref"/>).



The followingExample>!CDATA[
can be computedgap>:CharacterTableH P21//V1 ;


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

If a given ordinary character table <M>t</M>, say,
has been constructed from other ordinary java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 4
then g= ,1 ,,0- ]
of <M[0 ,,,1,1 ]java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
<Q>ingredients</
This happens currently in gap b: - ( ,x-> List( x{ ,3940],



<List>
<Item>
 <M>t</M> has been constructedP>
 <Ref Oper="CharacterTableDirectProduct" BookName="ref"/Pjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 and &GAP; can compute the <M>p</M>-modular Brauer tables
 thefactorsjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
</Item>
<Item>
 <M>t</M> has been constructed with
=java.lang.StringIndexOutOfBoundsException: Range [0, 36) out of bounds for length 0
 Mp/- Brauer
 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" BookName="ctbllib"/> set,
 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 51 2 1
 and &GAPgap> rest11:=RestrictedClassFunctions( diff, t11 );;
 of the relevant ordinary character table(s),
 which are library tables whose names occur in <M>l</M>.
</Item>

</Subsection>


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

[,java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
<
It can be read off from the set Irr<M>(B)</M>
charactersof <B</M>
whether all irreducible Brauer characters in <M>B</M> are restrictions
of ordinary characters to the <of ordinary characters to the <M
as follows.



If <M>B</M> has only one irreducible Brauer character then all ordinary
characters in <M>M\}1=3905670/>
So let us assume that <The generators of the point group are as follows
at least two irreducible java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

to the <M>p</M>-regular classes of <M>G</M>.



The block <M>B</M> java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
characters,
and the decomposition of and thetranslationlattice<M> =2 &;^d/>
characters is described by
an <M>|S|</M> by <M>|S| - 1</M> matrix <M>java.lang.StringIndexOutOfBoundsException: Range [0, 43) out of bounds for length 18
whose > [ ,, ,]<java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 13
occur in each columnjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
S ="Where" 2.5/,
which refers to <Cite Key="Dad66"/>.)

<

If all irreducible Brauer characters> java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 25
then the matrix <M>M</M> contains <M>|S| - 1</M> rows that contain
exactly one nonzero entry,
hence the remaining row consists only of
This means This means that
the sum of all other elements in <M>S</M>.
Conversely,
of allotherelements <S/>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>.



java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
one could state that all irreducible Brauer characters
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
<M>\circ</M
see <Cite Key="HL89" Where="Lemma
and the distribution to types is determined by Irr<M>(B)</M>.



The default method for <
that is contained in the &java.lang.StringIndexOutOfBoundsException: Range [0, 3) out of bounds for length 0
 pM
<M>G</M> are <M>G</M> are cyclic
whether all <M>p</M>-blocks of <M>G</M> have the whether all <M>p</M>-blocks of <M>G</M> have the property
(This>java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20



Examples>: fixed[2] aa^ )
are the <M>p</M>-modular> orb: Orbit ,seed );
gap ( ,"49G1//")



<Example><![CDATA[
gap> t:= CharacterTable( PSL( 2, 11 )java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 37
gap> modt:= t Thereare two examples with <M>d = 6</M>.
gap> modt <> fail;
true
gap
"computed using that all Brauer characters lift to char. zero"
]]></Example>

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

Another such example is the <M>5</M>-modular table
of the gap> b:= [ [ 0-, 0,,,1]



<Example e>
p:=java.lang.StringIndexOutOfBoundsException: Range [27, 25) out of bounds for length 36
gap> fromgroup:= CharacterTable( MathieuGroup( 11 ) );;
gap> DecompositionMatrix
[ [ 1, 0, 0,</>
 [ 0, 0
 [ 0, Heading>Brauer Tablesvia Construction Information</Heading>
 [ 0, 0,0,0 0,0 1,0, 0] [1 ,0 ,1,1,1,0 ]
 [ 0, 0, If java.lang.StringIndexOutOfBoundsException: Range [29, 10) out of bounds for length 50
gap> fromgroup mod P>
true
]]></Example>



<RefharacterTableDirectProductBookName"/
to characteristic zero but <and compute M<-java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 58
java.lang.StringIndexOutOfBoundsException: Range [22, 4) out of bounds for length 31
An example is the <M>2</M>-modular table of the Mathieu group <M>M_{11}</Itemjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6



<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 ], ef"java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 70
 ="ect: Tablesof .3/,
 [ 1, 1, 0, 0, 1 ] ]
gap> fromgroup mod 2;
fail
]]Pjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

</Subsection>

</Section>


<Section Label="sect:monstersubgroups">
<Heading>Information java.lang.StringIndexOutOfBoundsException: Range [0, 26) out of bounds for length 4


<Subsection Label="sect:no2U42inMonster">
<Heading>The Monster group does not contain subgroups of the type

Inof java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 55
group<>M</>java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
Benjamin Sambale was interested in possible embeddings of certain groups
<M>G</M> into <M>&M;</M> such that the decomposition matrices<
are known.
For a given <M>G</M>, the first steps were to compute the possible class
whether the corresponding
be.



Apparently, calling <Ref Func="PossibleClassFusions" BookName="ref"/>
with <>p/M-regular of <><>
when <M>G</M> is a small group such as <M>2.U_4(2)</M>.
 the
java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 25
This java.lang.StringIndexOutOfBoundsException: Range [22, 19) out of bounds for length 68
decomposability of restricted characters of <M>&M;</M> as an integral
 orbisjava.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 58
As a rule of thumb, if the number of classes of the subgroup is small
compared to the number of classes of>
and if the result consists of many candidates then it might be faster
to omit the decomposabilitywhich refers to < >local,,,;



<Example><![CDATA[
:CharacterTable(".(2) ;java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
gap> m:= CharacterTable( "M" );;
gap> sfusm:= java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0
gap> Length( sfusm );
2332
]]></Example>



<>

Looking at the (many) candidates,
wesee allmap 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_4are ordinary characters java.lang.StringIndexOutOfBoundsException: Range [4, 2) out of bounds for length 72
M&/>
We compute <Example><![CD



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





 java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 73
<M>2.U_4(2)</M>.

</Subsection>


<Subsection Label=
<Heading>X.2 1 1 11 1 1 - - 1- - - -

low</LinkText
<https/mathoverflownet//450255/>/>
which perfect central extensions of the simple group <. 5 5 1 - 1 2 -. 3- 1..
subgroups of the sporadic simple Monster group <M>&M;</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( java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 15
gap>[0 =()7+)java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
Size ) (simp ;
gap> SortBy( ext, Size );
gap> names:= Listgap>fromgroupmod5< fail;;
[ "L3(4)", "2.L3(4)", "3.L3(4)", "2^2.L3(4 M><M its
 "4_2.L3(4)", "6.L3(4)", "(2x4).L3(4)", java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 "12_1.L3(4)", "12_2.L3(4)", "4^2.L3(4)", "(2x12)
 "(4^2x3).L3(4)" ]
]]></Example>



fact that<>G</>isE></E>isomorphic to a subgroup of <M>&M;</M>
is shown in <Cite Key="HW08"/> (at the end of this paper).





And embeddingscentral MG<> <>&;/M>
can be established using known since we are not free to choose an arbit class fusion the

<List>
<Item>
 <M>2.G < forthegivenordinarycharacter.
</Item>
<Item>
 <M>2^2.G < He < 3.Fi_{24}',that to
</Item>
<Item>
 <M>6.G < 2.G_2(4) < 6.Suz < 3^{1+12}_+.2Suz < &M;</M>.
</Item>
</List>

Note that <M>G</M> is a subgroup of <M>U_4(3)</M> but not of <M>2.U_4(3)</M>,
<M>3.G</M> is a subgroup of < : [;
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> : mod ;
and the negative ones from the following computations.



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



java.lang.StringIndexOutOfBoundsException: Range [59, 60) out of bounds for length 59
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</admiss: Stabilizer java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
Clearly the case <M>3C</M> cannot occur,
and <C>3B</C> is excluded by the fact that no class fusion between <M>3.G</M>
and the <M>3B</M> normalizer <M>3^{1+12}_+.2Suz.2</M> is possible.



<Example><![CDATA[
gap> t:= CharacterTable( "MN3B" );
CharacterTable( "3^(1+12).2.Suz.2" )
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>&M;</M> then it is contained in
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>).
Again we use <Ref Func="PossibleClassFusions" BookName="ref"/>
to list all candidates for the class fusion,
but here we prescribe the central involution of the <C>2A</C> or <C>2B</C>
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_1.L3(4)", "4_2.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)" ]
gap> invcent:= List( [ "MN2A", "MN2B" ], CharacterTable );
[ 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
> := CharacterTable( name );
> ords:= OrdersClassRepresentatives( s );
> invpos:= Filtered( ClassPositionsOfCentre( s ), i -> ords[i] = 2 );
> for i in invpos do
> for t in invcentdo
> init:= InitFusion( s, t );
> if init = fail then
> continue;
> fi;
> init[i]:= 2;
> fus:= PossibleClassFusions( s, t, rec( fusionmap:= initgap java.lang.StringIndexOutOfBoundsException: Range [51, 49) out of bounds for length 61
>> java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 1
> if fus <> [] then
> Add( cand, [ s, t, i, fus ] );
> ;
> odand the <M</M> normalizer <>3^{1+12}+2Suz2<M ispossible.
> od;
> od;
gap> List( cand, x -> x{ [ 1 .. 3 ] } );
(4. ,java.lang.StringIndexOutOfBoundsException: Range [51, 49) out of bounds for length 70
 , [ CharacterTable( "(2x4).L3(4)" ), CharacterTable( "2.B" ), 2 ],
 [ java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 55
]]></Example>





(Note that we have called <Ref Func="PossibleClassFusions" BookName="ref"/>
with the option <C>decompose:= false</C>, in 11P 1a 2a 3a 5 5 6a7 10a 10 1 15a 15java.lang.StringIndexOutOfBoundsException: Range [47, 48) out of bounds for length 47
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 to a <C>2A</C> element of <M>&M;</M>.
Since we get candidates only for two out of the three central java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
we see that <M>(2 \times 4).G</M> does notjava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 20



Thus it turns out
this way.
Namely, these character-theoretical criteria leave the possibility
that <M>4_1.G</M> may occur as a subgroup of



 seeif the <><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>,
centralizerof M<> in<>2^124}+C<M
<>2{}\ Co_3<>
We see that <M>4_1.G</M>,
if it exists as a subgroup of <M>2^{1+24}_+.Co_1</M>,
must lie inside thesubgroup <M>[2^{24}].Co_3</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 ]
> ClassPositionsOfPCore , ;
true
gap> siz:= SizesCentralizers( t )[5] / 2^24;
495766656000
gap> mx:= Filtered( List( Maxes( CharacterTable( " [2.A10, 5,7 ]],
> CharacterTable ),
> x -> Size( x ) mod siz = 0 );
[ CharacterTable( "Co3" ) ]
gap> Size( mx[1] ) ,5 ]
true
]]></Example>

2Alt(4, 2 3,5,7]]java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61



I do not see a character-theoretic argument that could disprove
the existence of such an <M>4_1.G</M> type.Fi22.2 ,13 ]],

</Subsection>


<Subsection Label="sect:2BM12">
<Heading>The2.uz., ,7 13]java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61

Consider a maximal subgroup <M>H</M> of type <M>(3^2:2 \times O_8^+(3)).S_4</M>
in the sporadic simple Monster group.
The character table of <M>H</M> has been contributed by Tim Burness.
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 L()21 5 ]

/

<Example><[2 [2 ,5 ] ]
gap> tblH:= CharacterTable( "(3^2:2xO8+(3)).S4" );
java.lang.StringIndexOutOfBoundsException: Range [16, 14) out of bounds for length 37
gap> N:= ClassPositionsOfSolvableResiduum( tblH );;
gap> tblF:= tblH / N;;
gap> Size( tblF );
432
gap> known:= NamesOfEquivalentLibraryCharacterTables( tblF );
]
gap> Filtered( GroupInfoForCharacterTable( known[1] ),
> x -> x[1] = "SmallGroup" );
[ [ "SmallGroup", [ 432, 734 ] ] ]
]]></Example>



(Note that the precomputed
<Ref Attr="GroupInfoForCharacterTable" BookName="CTblLib"/>
information about &GAP; library character tables means that exactly
one isomorphism type of groups fits to the character table of <M>F</M>.)



We compute that <M>O_3(F) \cong 3^2</We compute that <M>O_3(F) \cong 3^2</M,7]]
thus <>H<M has a subgroupM></>ofthetype <M>O_8^+3.S_4</M>,
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( ComplementClassesRepresentatives( G, P ) );
1
]]></Example>



We can derive the character table of < 35 ],
and compute that the group structure of <M>V</M> is
<M>(2 \times O_8^+(3)).S_4</M>.
For that, we consider the element orders of the unique normal subgroup
of order <M>2 |O_8^+(3)|</M> in <M>V</M> [ Fi23 3,17, 23 ]]
If this normal subgroup would not be isomorphic with <M>2 \times O_8^+(3)</M>
itwould one thestructures<>+3.1/>java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
>,
24<.



<Example><![CDATA[
gap> tblV:= tblH / ClassPositionsOfPCore( tblH, 3 );;
gap> ord:= 2 * Size( tblH ) / Size( tblF );
9904359628800
gap> classes:= SizesConjugacyClasses( tblV );;
gap> 2N:= Filtered( ClassPositionsOfNormalSubgroups( tblV ),
> l -> Sum( classes{ l } ) = ord );;
gap> Length( 2N );
1
gap> Set( OrdersClassRepresentatives( tblV ){ 2N[1 [ 1, 7, 11, 19]]
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 18, 20[[ ,,19 java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
gap> Set( OrdersClassRepresentatives( CharacterTable( "O8+(3)" ) [ (7).3 3 7,19]]
[ [ 2., [ 2 7 ]
gap> Set( OrdersClassRepresentatives( CharacterTable( "O8+(3).2_1" ) ) );
[ 1, 2, 3, 4, 5, 6, 7, 8, 9 [3 ,11 , ,
 ,36 40]
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 Monster group shows that <M>V</M>
centralizes a <C>2A</C> element in the Monster,
hence <M>V</M> is a subgroup of a maximal subgroup of the type <M>2.B</M>.



<Example><![CDATA[
gap> tblM:= CharacterTable( "
gap> VfusM:= PossibleClassFusions( tblV, tblM );;
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 theH">
or <M>V</M> is equal to the ,

<Example><![CDATA[
gap> tblB:= CharacterTable( "B" );;
gap> mxB:= List( Maxes( tblB ), CharacterTable );;
gap> cand:= Filtered( mxB, s -> Size( s ) mod ( Size( tblV ) / 2 ) = 0 );
[ CharacterTable( "Fi23" ), CharacterTable( "O8+(3).S4" ) ]
]]></Example>

</

The former possibility is excluded from the fact that the factor of <M>V</M>
center notadmita fusion <>{23</>



<Example><![CDATA[
gap> Length<Example><![CDATA[
0
]]></Example>



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



Thus
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" );;
>java.lang.StringIndexOutOfBoundsException: Range [50, 44) out of bounds for length 66
java.lang.StringIndexOutOfBoundsException: Range [4, 5) out of bounds for length 4
java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 42
true
]]></Example>

In order to add the table to the library,
we have
 subgroupM(O_8+3.4/> >/>java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
the maximal subgroup <M>(3^2:2 \times O_8^+(3)).S_4</M> of <M>M</M>,
such thatjava.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 23
via <M>O_8^+(3).S_4</M> and <M>2.B</M> are compatible, <M>\ldots</M>



<Row
gap>tblU=CharacterTable
gap>The following computations showthat the given line program
CompositionMapsGetFusionMap(, tblB,
> GetFusionMap( lib, tblU ) ) =
> CompositionMaps( GetFusionMap( tbl2B, tblB ),
> GetFusionMap( lib, tbl2B ) );
true
]<>[java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 18



<M>\ldots</M> and that the compositions of fusions from <M>V</M> to <M>M</M>
via<>(22 \ O_8+3)S_4<M and<>2./>are compatible.



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

</Subsection>
</Section>
<

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