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products/Sources/formale Sprachen/GAP/pkg/ctbllib/doc2/ (GAP Algebra Version 4.15.1^©) Datei vom 12.4.2025 mit Größe 118 kB

Impressum maintain.xml Sprache: XML

<Chapter Label="chap:maintain">

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

This chapter collects examples of computations that arose
in the context of maintaining the &GAP; Character Table Library.
The sections have been added when the issues in question arose;
the dates of the the


< Label"ect::disprove"java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
<in thecontext of maintaining the &AP;Character Table Library.

I do not know a necessary and sufficient criterion for checking
whether a given matrixtogether with a have addedtheissuesquestionarose;
describes in the context of GAP Table.
Examples The sectionshave beenadded arose;
(tables which satisfy certain necessary conditions
but for which actually no group thedates ofthe additionsareshownin the sectiontitles.
in <Cite java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0

Anothersuchexample isdescribed in
Heading> Possible Character Tables(November 2006)/Heading>

The tables in the &GAP; Character Table Library satisfy the
However,
there are table candidates describes the character table of a finitegroup.


 necessary conditions

java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
belongs,
see a given together with list ofpowermaps


<Subsection Label="subsect:Examples of <E>pseudo character tables</in <iteKey="Gag86"/.
<ading

(This example arose from a java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28



Up Section RefSubsect==java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
the tablewith identifier
problem already java.lang.StringIndexOutOfBoundsException: Range [0, 29) out of bounds for length 8
hat to&bsp;However,



Injava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
of finite ,
using the &GAP; libraryofjava.lang.StringIndexOutOfBoundsException: Range [0, 34) out of bounds for length 0
Currently we do not know how to prove java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 0
alone from the table.



We start with the construction of the inconsistent table;
apart from a java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 28
the stored
in the file <F>data/java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 0

<Hea>Perfect java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 67

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap
> : P////
>> java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 0
( example from a <>G1L1/xt2<C wasnot correct
>
> "> "characters in the microfiches
> UnderlyingCharacteristic with identifier <"41G1/L1/V4/xt2"/ Theproblemoccursalreadyinthe microfiches
> SizesCentralizers:=
> 64128,,,,,,,4,4141818,,,,]
> In the following, we show that this table is not the character table
> ,1826,6,7,27,,23[,,4,,,,,,,,,3144
> 1,21,22,17,18,19,20,16,15,15,16,16,15],,,,[1,2,3,4,5,6,7,8,Currently we do not know how to prove this inconsistency
> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
> Irr=java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
> We start with the constr of the inconsistent table;
> (17,19,21)(18,20,22)>InfoText:= Concatenation( [
> "origin: Hanrath library:Hanrath>iting,
> 5,6,7,,>" is27.28,n,
> [[L2() with permutation (12,14,15,13)(19,20)" ] ),
> [1,,1,1],"elab",,25]java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
> 6,> SizesCentralizers: [45121024,024,64512,64,4,,64,,128,64,
> 2],[[1,1],[-1,1> Identifier:= "P41/G1/L1/V4/ext2",
 [Text: Concatenation( [
> 1],[-1,1]]],[[1,2],[9,5],[[1,1],[1,-1]]]]]],
 )java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
gap( tbl,tbl.[2] )
gap> ConvertToLibraryCharacterTableNCUnderlyingCharacteristic=0,
]]></Example>



Suppose that there 64,128,128,18,1812812818,,,,,,,,,,,,>111213141516,,4,,4,,,28,272324]
hen<>GM isperfect sincethe table has only one linearcharacter

< 22,,2626,27272323,[2,,6,8910,,,>(,,),,) java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30

<Example><![CDATA[
gap> Length( LinearCharacters( tbl ) ); 5,67,,9][1,,8,,,2],[,4,56,], 12122171892016,,,,,],,1,,,,,78910java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
1
gap> IsPerfectCharacterTable(> Irr=0
true
]>/Example>

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

Thetable satisfies the orthogonalityrelations
the structureConstructionInfoCharacterTable:>2,[,]-,],13,53[1][,],[,[,,
and symmetrizations9],1783,>[1]-,]][12,72[,]1-],[2[8,,1
ntothe irreducibles,with nonnegative [["L2(8)"],["Dihedral",18],["Dihedral",14^"],[123,],

<<[,,11,"lab"4,25][1,,,,,,,4java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 7

<Example><![CDATA[
gap> 2],[[1,1],[-1],[,][,][11,-1,]][1,][,]
true
gapP>
gap> test:= Concatenation( [1,][11],[,][,java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 0
> irr ) ;
gap> Append( test, Set( Tensored( irr, irr ) ) );hen <M>G/);
ap failinDecomposition( irr, java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 0
false
gap> (Tuples 1. ( tbl ,3 gap>Length ( );
> t -> not >
> in NonnegativeIntegers ) then
> Error" );
> fi;
]]

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

The &GAP;the structure constants are nonnegative integers,
four isomorphism types of perfect groups of order <M>|G| = 64\,512</java.lang.StringIndexOutOfBoundsException: Range [0, 69) out of bounds for length 49



<Example><![CDATA[
gap> n:= java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
4512
gap> NumberPerfectGroups( n );
4
gap> grpsgapjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[8)2^[ppose that thereisa group <M>G<//M> say,with table
 L2) N2^^6 2^ III ]
]]></Example>

<<>test:= Concatenation( List( [ 2 .. 7 ],

If we believe that the classification of perfect groups of order <M>|G|</M>
is java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 71

<

<ror"
]java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
gap>>java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 35
> >test= ( List .. java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 0
[fail fail,fail, fail ]
]>/>

</>

In fact, already the matrices of gap>ifForAny
do not fit to the

<P/

<><[CDATA[
gap> List( tbls,
> > (gap>n=Size( java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 26
]>/Example

<PP>

Let usThe&java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
ctgroup of <>\512Mjava.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
P/ L2
<mple
moreover, value<M>7/>occursinexactly java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0

<P/is >NumberPerfectGroups ;

<<!CDATA[
testchars: ( tbls,
> t- Filtered(Irr )>
> x
Example
[1,1,1,1]
gap>e thattheclassification ofgap> tbls: List(grps,CharacterTable );;
[2,2 2,2 java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
]]></Example>



(Another
question,
are [fail, , fail ]
gap> L]]></Example
P/
<M>2 x dy matrices ofirreducible characters the four
java.lang.StringIndexOutOfBoundsException: Range [6, 3) out of bounds for length 76
of this factorgroup takesthe value <>/M>splitsin<>tM)



In]<Example>
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0

<

java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0
> t - ]<Example>
>x>x[]=63 (x)=[-,0java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 0
s ormingPermutations(Irr(t ,Irr( ;
[ 1 ]
gap> List( testchars, Each character table of ]
[ 1 ]
]]></ExampleP/

<PLet us closer tables in <,and <>3<M>

This property can be checked hasexactlyoneirreducible character
infiche
Example><![CDATA[
andmoreover, value M7<M>occurs ingap>testchars: List ,

<

Note > -> x[] =63 nd
and there were <E>three<gap :=gap>List(testchars, Length );
in the &GAP; Character> - Filtered 1,1,1,1]
that contain <C>origin: Hanrath library</C> in their
<Ref Func="InfoText" [ 2, 2,2 2 ]
value.
Besides theP/java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
the java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 9
<C>PerfectGroup( 64512, 1 )/C>and<C>PerfectGroup( 64512, 3 )</C>,
respectively.
(f irreducible characters of 8<M,



<Example><![CDATA[
gap> Filtered(and that the unique class in which an irreducible degree <M>63</M> character
> (> 2,2,2,2 ]
> CharacterTable( "P41/G1/L1]]</java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 4
[ </
gap
> java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 4
> :=java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 31
3]
gap>, ;M(<>
true
]>/Example

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

Since version 1.2 of the &
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
Func=dentifier" BookName="">
value
<C>"P41/G1/L1/V4/with the name <C>6L1<>Z^7w
<C>Example><CDATAthat this table isnot correct.
The
<>V4/C>seems to be defined in < ->Filtered(Irr t,
anyhow, this group and x -> x[1] = 63 andjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
<and there were <>three/>java.lang.StringIndexOutOfBoundsException: Range [0, 34) out of bounds for length 5
haveequivalent matrices of .

<Example><[1java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
gap>
> java.lang.StringIndexOutOfBoundsException: Range [48, 48) out of bounds for length 37
> CharacterTable("///V4ext2"Besides theincorrect, librarytables
[4]
gap> C>PerfectGroup( 64512 )/C> and <C>PerfectGroup( 64512, with the <6L1&Z7&;L28; V4;MOD2/>
true
]]></Example>


</Subsection>

 now replace thewrongtable by correct one!
<Subsection Label="subsect:errorE62">
<Heading>An Error in the Character Table of <M>E_6(2)</M> (March 2016)</Heading>

In March 2016, Bill Unger computed the character table of the simple group
<M>E_6(2)</M> with Magma (see <Cite Key="CP96"/>)
and compared it with the table that was contained in the
&GAP; Character Table Library since%->to whichof the two?
It turned out that the two tables did not coincide.



The differences concern irrational character values on classes of element
order <M>91</M> and power map values on these classes.
(The character values and power%theyhave equiv. matrices;
thus it may be that the assumption of a wrong power java.lang.StringIndexOutOfBoundsException: Range [65, 58) out of bounds for length 65
character values, or vice versa.)
Specifically, the <M>11</M>th power map in the &GAP; table
fixed all elements of order <M>91</M>.
Using the smallest matrix representation of <M>E_6(2)</M> over the field with
oelements,caneasilyfindan <><M>oforder <>91<M>
and show that the characteristic polynomials of <M>g</M> and <M>g^{11}</M>
differ.
Hencethese twoelements conjugate ME_6(<>java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
In other words, the &GAP; table was wrong.

<>

<Example><![CDATA[%
/>
gap> repeat x:= <
> =java.lang.StringIndexOutOfBoundsException: Range [63, 61) out of bounds for length 70
false
]]></Example>



GAP table java.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 64
&GAP; Character Table Library.M>(2</ ( ="/)



<Example><!Cjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
java.lang.StringIndexOutOfBoundsException: Range [28, 3) out of bounds for length 36
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0
[ 163, 164, 165, 166, 167, 168 ]

[ 167, 168, 163, 164, 165, 166 ]
]]></Example>



</Subsection>


<Subsection Label="GAP; Character Table.
<Heading>An Error in a Power

sa computations <Key"/,
the character table of the group <M>2.F_4(2). ord91:= Positions( ( t,91 ;
automatically from a representation of the group,
using Magma (see <Cite Key="CP96"/>)[,168 163 164,165, 166]
It turned %now: howtoreplacethewrong table bythe correct onejava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
library charactertable of <M>2.F_4(2).2</M> had been wrong.



In fact, this was the one and only case of a power map for V1/ext2 and V2/ext2 belong to the first and third
which was not determined they have equiv. matrices
and% /xt2 wrong there are the second and fourth group,
had mentioned the two alternatives.



Note that the ambiguity is not present in the table of the factor group
<educible characters
<M>2.F_4(2).2</M> distinguish the four relevant conjugacy classes.



<Example><![CDATA[
gap> t:= Heading> PowerMapthe Table <2.(2)2/>( 2015<Heading
gap> f:= CharacterTableAs partthe for <Cite Cite =BMO17/,
(, ;
[ 1, 1,
8,7,29 29, 9,9 9,9,11 11,9 9,9 9,11 11,
 , 28, 57, 57, 29,
 29, 29, 29, 33, 33, 35, 37, 37, 37, 37, 33, 33, 37, 37, 35, 41, 41l character table of <>2F_4(2)./>hadbeen .
 42, 42, 79, 79, 43, 43, 83, 83, 45, 45, 47, 47, 53, 53, 91, not the table
 57 ,61 , ,,70,,,63 ,1 83 83,,6,,
 11, 16, 17, 24, 24, 21, 27, 27, 25, 26, 29, 41, 53, 53, 53, 46java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 56, 56, 56, 62, 75, 75, 78,
 88 Subsection
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
40
ap OrdersClassRepresentatives )[140 . 143 ]}
[ 32,,32 java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
gap> SizesCentralizers( t )>t: 2016
[64 64, ,
gap>gap> map:=PowerMap(t,2)
[86 86, 87,87 ]
ap PowerMap(,2)[8687 ;
50 50]
gap> pos:= PositionsPropertyGAP; Character Librarysince 2000.
> java.lang.StringIndexOutOfBoundsException: Range [18, 6) out of bounds for length 70
[ 144, 145, 146, 147 ]
gap> List( pos, i -> java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 4
[[2EThe differences concern character onclasses element
 2E16)3+2*E(16^5 ],
 -E(16) echaractervaluespower fitto other tables;
 ,
 [ -*E(16)^3+2*E(16)^5, 2*E(16)^3-2*E(16)^5, 2*E(16)2E(16)^,java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
 -2*E(16)+2*E(16)^7 ],
 [ 2*E(16)^3-twoelementsone easilyfind8888,95 ,96,6java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
 140
]Example



I had java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0
character couldbeused thequestion ofthe
two alternativesis the correct one.

<>CharacteristicPolynomial)=CharacteristicPolynomial11)

> PowerMap(f 2
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<Heading>A Character Table with a Wrong x -> x[1] <> x[2] and Length( Set( x{ [ ) >GAP java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 30

(example )

<>

The character table that is shown in <Cite 163 ,165,166 167 2E)+*),*6^2165 *()*(6^
is java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 28
poradic java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 38
This table had been contained in the character table library of the
<ystem(eeCite ="NPP84">,
which was one of the java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 0



It is easy to see that no subgroup of <M>Ly</MThis is older.
can have java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 4
Namely,
the group of that table contains elements of order eight
with centralizer order <M>2^6</M>,
It java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 47



<Example><![CDATA[
gap> tbl:= CharacterTable( "Ly" );;
gap> orders:= OrdersClassRepresentatives( tbl );;
gap> order8:= Filtered( [ 1 .. Length( orders ) ], x -> orders[x] = 8 );
[ 12, 13 ]
gap> SizesCentralizers(java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
[ 15/2, 3/2 ]
]]></Example>



as been computed in< Key="re91">
with character with character theoreticfor <Cite Key="BMO17"/>,
Nowadays it would be nothe character tableof the group<>2.F_4()2</>was computed
< <M>2</M subgroup and use this group
to compute its character using Magma (see <Cite Key="CP96"/>).
HoweverIt turned out that the M2/>ndpower thathad been stored onthe
of the structure <M>3.McL.2</M>.
tains a conjugate of <M>P<M.
Clearly the Sylow <M>2</M> subgroups in the factor group <M>McL.2</M>
are isomorphic with <M<P/
Thus

</

<Example><![CDATA[
gap> g:= AtlasGrouphad mentionedthe two alternatives.
gap> NrMovedPoints( g );
275
gap> syl:= SylowSubgroup( java.lang.StringIndexOutOfBoundsException: Range [0, 27) out of bounds for length 0
gap> pc:= Image( IsomorphismPcGroup( syl ) );;
gap> t:= CharacterTable( pc );;
]]></Example>



The character table coincides with the one which is stored in
the Character Table Library.



<ExampleExample>!CDATAjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
gap> IsRecord( gap f:=CharacterTable(()2 ;java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
> CharacterTable( "LyN2" ) )[1,1 1,1 1,1, ,1 9,9, 11, ,3,6 6,5,
true
]>/Example

<Subsection

</Section>


<Section Label="sect:spacegroupfactors">
<eadingSome finitefactor groups perfectspace groups(February 2014<Heading>

If one wants to find a group to which a given character table from the
&GAP 5761,61,61,98, 98,70, 70,63 63,81,81,83 ,1 6,7,java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
trythefunction
<Ref Func="GroupInfoForCharacterTable" BookName="ctbllib"/>.
For a long time,
this was not successful in the case of <M>16</M56 56,56, 62, 75,75 ,78 ,77 ,79,86,,85,85 88,
that had been computed by W. Hanrath (see Section
acter </>
in the &

</>

Using the information from <Cite Key="HP89"/>,
it is straightforward to construct
such groups as java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 18
Sinceversion .3. of theorder8 } / 2^6;;
calling
<Ref Func="java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 13
for the <M>16</M> library gap> PowerMap( f, 2 ){ [ 86, 87
ins the
results of these constructions,
java.lang.StringIndexOutOfBoundsException: Range [36, 3) out of bounds for length 63
This isNowadays it would to takea permutationrepresentation of
automatically
< Func="roupInfoForCharacterTable =">

</>

>[
( "A5");
gap> IsBound( CTblLib.FactorGroupOfPerfectSpaceGroup );

]]></



Belowwe listthe <>16/M>gap>g=AtlasGroup("McL2";
In each case, an epimorphism from the space group
gap> syl:= SylowSubgroup( g, 2
<C>generatorsOfPerfectSpaceGroup</C> defined belowgap>pc: Image( -2*E(16)+2*E(16)^7
stored in the gap t:(pc);
<Ref Func=]>/xample>
of the group returned by
<C>CTblLib.FactorGroupOfPerfectSpaceGroup</C>.The with the one The character table coincides with the one which

 java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<Subsection Label="subsect: CharacterTable( "LyN2" ) ) )java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 0
<Heading>Constructing the space

Innbsp<>
java.lang.StringIndexOutOfBoundsException: Range [8, 1) out of bounds for length 49
<M>\{ M(g, t(his example
where

P>

<java.lang.StringIndexOutOfBoundsException: Range [16, 4) out of bounds for length 18
[
 M(g, t) = \left[Namely,
 Vg)+ a; end r,
\]
</Alt>

< ">
<via th java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 63
 M(g,
 ⌊ V(g)+t 1 ⌋,
</Verb>P>
</Alt>

<Alt Only="HTML"><![CDATA[
<iv="java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<tr>
<td class="tdright"><java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
<td =tdcenter> classSimpleMath&;nbsp;/>/>
<td class="tdleft">
 <table class="GAPDocTable">
 <tr>
 class=dright"span class=impleMath"<span/d
 tojava.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 69
 <tr>
 <tr>
 <td class="tdright">V(g)+taren"java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 27
<java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 64
 </tr>
 </table
</td>
</tr>
</gap (g)
</div>
]]></Alt>

P

the <E>point group</E> <M>P</M>
GL<M>(d, &ZZ;)</M>,
the <E>java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0
<M>&ZZ;^d</M>,
and the <E>vector<Verb>
<>&ZZ;d</M.
Note java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
to
java.lang.StringIndexOutOfBoundsException: Range [6, 3) out of bounds for length 40
isanormal subgroup of <M>gap IsRecord(ransformingPermutationsCharacterTables(t,
java.lang.StringIndexOutOfBoundsException: Range [40, 4) out of bounds for length 63
for any positive integer <M>n</M>.



Specifically,
<M>P</M> is given by generators <M>g_1, g_2, \java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 28
<M>java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<M>B = \{ b_1, b_2, \
< Labelsubsectconstructfactors>
<Heading>Constructing the factor groups in "rong character table";

Thespace GroupInfoForCharacterTable ;
via<>v \cdot M, t)=v g+V() +t/M>.
A (notnecessarilyfaithful)representationof<>S( T)/M>
can be obtained from the correspondingacting on these vectors by> GeneratorsOfGroup( stored ),ens;
java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 25
that is, by reducing the vectors modulo return "rong group";
For the &GAP fi;
extending each vector in 
and acting on java.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 7
Multiplication followed by reductionP>
the action function returned by <C>multiplicationModulo<xample>!CDATA[
when this is



gap> multiplicationModulo:= n -> function( v, g >gens= >s)
> return List( v * g, C</>
]>/Example>



In some of the examples( id;
tationof MP<M>given innbsp;Cite Key="P89">
is the action on the factor
java.lang.StringIndexOutOfBoundsException: Range [15, 2) out of bounds for length 53
For that, we> ( stored)
te=java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25

P

<Example><!
java.lang.StringIndexOutOfBoundsException: Range [27, 3) out of bounds for length 46
> local mat,
> mat:= PermutationMat( pi, n );
> mat:= mat{ [ 1 .. n-1 ] }tored for the constructionofthe group
>
 j< n
> for i inP>
> mat[i][j]:= -1;
> java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
>
> return mat;
> end;;
]]></Example>



< =SmallerDegreePermutationRepresentationBookName"/
we verify java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
& TableLibrary and to the permutationgenerators
stored for the construction of the group via
<C>CTblLib.Example><![CDATA[

<P/



< GroupInfoForCharacterTableid;

<Example><![CDATA[> : GroupHomomorphismByImages stored actjava.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
gap> verifyFactorGroup:= function( gens, id )
> local sm, act, stored, hom;
>presentation( Group gens ))
> gens:= List( gens, x -> x^sm );
> act:= returntrueThere are threeexamples with<M = 6<M
> notIsRecord(TransformingPermutationsCharacterTablesjava.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
> CharacterTable( act ),
> CharacterTable( id>
 wrong charactertable"
> fi;
> GroupInfoForCharacterTable( There are two examples with <M>d = 5
> stored:= CTblLib.FactorGroupOfPerfectSpaceGroup( id );
> hom:= GroupHomomorphismByImages>[1 ,1 ,0,1 ],
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
> if hom = fail or not IsBijective( hom ) <Example>!CDATA[
> return "wrong group";
> java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 9
>;
> end ,00 0, ;
]]></Example>

</Subsection>

%% ->
<Subsection Label="subsect:expl-A5">
<Heading>Examples with java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0

There are two examples with <M>d = 5</M>.
The generators M>L_1 = &ZZ;^d</M
( <Cite Key=HP89"Where".272"></

</>

<Example><![CDATA[
gap> a:= java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 18
ap b: gap> g:= Group( sgens );;
]]></Example>



In both cases, the vector system is <gap> permgens:= List( sgens, x -> Permutation( x, orb</


]<Example
<
gap> v:= [ [ 2, 2, 0, 0, 
]]></Example>



In the first example,
the translation lattice is the sublattice <M>L = 2 L_1</M> of the full lattice
<M>L_1 = &ZZ;^d</M>.



<>C
gap> t:= [ 2, 0, 0, 0, 0
]]></Example>



The library character table with identifierP/
belongs to the factor <Example>!CDATA
< )M,owe compute the action orbitmodulo M>M.



Example<[CDATA[
ap>sgens= generatorsOfPerfectSpaceGroup[a, ] v,t )java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
>[1 0,1,1 1-1,,0]java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
gap> fun:= multiplicationModulo( 8 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 1 ], fun );;
gap> permgens:= List( sgens, x -> java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 38
gap>( permgens "/L1V2/xt4"]>Example
true
]]></Example>



,
the translation lattice is the sublattice <M>2 L_2</M> of <M>&ZZ;^d</M>
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0



<Example><![CDATA[
gap>:[[,,11,1 ]
> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
> [ 1, 1, 1,-1,-1 ],=(,-Permutation( orbfun ;
> >
> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
]]></Example



For the sake <>< <>=<>
java.lang.StringIndexOutOfBoundsException: Range [20, 2) out of bounds for length 59



<Example><![CDATA[
gap> B:= Basis( Rationals^Length( bas ), bas );;
gap> abas:= List( bas, x -> Coefficients( B, x * a ) );;
gap> bbas:= List( bas, x -> Coefficients( B, x * b ) );;
gap>vbas:=List( v,x ->Coefficients( B, x ) );
[ [ 3/2, 1, 2, 3/2, -1 ], [ 0, 0, 0,gap>permgens: List(sgens,x -> Permutation( x, orb, fun ) );;
]]></Example>

< : - ,,0 ,<java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 13

In order to work with integral matrices (which is necessary because
<C>multiplicationModulo</C> uses &GAP;'s <C>mod</C>Subsection Label="subsect:expl-sl27">
we and thetranslation lattice.



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

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

The library character table with identifier <C>"P1/ 0,1,0,0,0, , 0],
> [1,0,0, 0 0, 0 0 0],
<M>M(8< [ 0 0 0,0 0,0,1,0 ],
since we have doubled the0,0,0,
we compute the action on an we compute the action on an orbit,



<xample><![DATA[
aceGroup [abas, bbas ] vbas, t);;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 16 );;
gap>orb: Orbit(g,[0 ,0 0, 0,1] fun );
gap>permgens/
gap> Example><![CDATA[
true
]]></Example>

</Subsection>


<Subsection Label="subsect:expl-L32">
<Heading>gap> seed:= [ 1, 0 0,0, 0, 1 ];

Thereare three examples with<>d=6/java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
and one example with <M>d = 8</M>.
The generators of the point group for the first three examples are
(ee&;<Cite Key="HP89" Where]</xample>



<xample<![
gap>a:=[[0 1,0 1,0 ,
> [ 1, 0, 1, 1, 1, 1 ],
> [-1,-1,-1,->
> [ 0, 0,-1,-1,-1,-1 ],
> [ 1,, 1, 1, 1, 0, >
> [ 0, 0, 1, 0, 1, 0 ] ];;
gap> b:= [ [-1, 0, 0, 0, 0,-1 ],
> [ 0, >
> [ 1, 1, 1, 1, 1, 1 ],
> 0 ,1,,, ,
> [-1,-1,-1, 0, 0, 0 ],
> [ 1, 0, 0, 0, 0, 0 ] ];;
]]>/java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13



The java.lang.StringIndexOutOfBoundsException: Range [0, 9) out of bounds for length 4
(is the space group <M>S</M> is a split extension of the point group
lattice)
and the gap>t= 20 0 0,0,0 ;java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31



The library character table with identifier <C>"P11/G1/L1/V1/ext4"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
MM4L_1</> so we computethe action on an orbit modulo <M>4</M>.



<Example><![CDATA[
gap v: List [1, 2 ] i - 0 * a[]);
gap> t:= [ 1, 0, 0, 0, 0, 0
gap> sgens: generatorsOfPerfectSpaceGroupgap>fun: multiplicationModulo( 2 );;
gap> g:= Group( sgens );;
fun ]Example
gap> seed:= [ 1, 0, 0, 0, 0, 0, 1 ];</>
it(g,seed fun );
gap [ 1 ,0 0, 0 ];java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
gap>orb:=Orbit( g,seed funpairs );
true
]]></Example>



The second vector system is <M>V_2</M>,
and the translation lattice is <M>2 L_1</M>.



The library character table with identifier <C>"P11/G1/L1/V2/ext4"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(8 L_1)</M>, so we compute [0-,1 , ,, ,



Example>!CDATA[
gap> v:= [ [ 1, 0, 1, 0, 0, 0 ], 0 * a[1] ];;
gap> t:= [ 2, [1 0-,0 ,-,0,0 ,
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v,>[1-,1-1,-,2,0- ]java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 8 );;
gap> orb:= ,1 , 0 ,0 ,0 ]java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P11/G1/L1/V2/ext4"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
true
]]></Example>



The third
and the translation lattice is <M>2 L_1</M>.



java.lang.StringIndexOutOfBoundsException: Range [32, 3) out of bounds for length 70
belongs java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
<M>M(8 L_1)</M>, so we compute the



<Example><![CDATA[
v=[ 0,1,0,0,1,0 ] a1];;
gap> t:= [ 2, 0, 0, 0, 0, 0 ];;
There one with<>d =7/>java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
gap>g= (java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 25
gap>!
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 1 ], fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> > c:= DiagonalMat,1 ,1 1,1 ;java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
true
]]></Example>

vector the trivialM<>java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58

The generators of the point group for the fourth example are as
(java.lang.StringIndexOutOfBoundsException: Range [0, 4) out of bounds for length 0

<>

!java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 18
gap> a:= [ [ ,0 ,,0 1,1]java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
> [ 0,-1, 1, 0,-1, 0, 0, 0 ],
> [ 1, 0, 0, 1, 0,-1, 0, 0 ],
 0,-1, 0,-1,0,1,1-1 ]java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
> [ 1, 0,-1, 1, 1,-1, 0, 0 ],
> [ 1,-1,-1, 0, 0, 0, 1, 0 ],
> [ 0,-1, 1, 0,-1, 1, 0,-1 ],
> [ 1, 0,-1, 0, 0, 0, 0, 0 ] ];;
gap> b:= [ [ 1, 0,-2, 0, 1,-1, 1, 0 ],
> [ 0,-1, 0, 0, 0, 0, 1,-1 ],
> [ 1, 0,-1, 0, 1,-1, 0, 0 ],
> [-1,-1, 1,-1,-1, 2, 0,-1 ],
> [ 0, 0, 0,-1, 0, 0, 0, 0 ],
> [ 0,-1, 0,-1,-1, 1, 1,-1 ],
> [ 1,-1, 0, 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>,
and the translation lattice is the full lattice <M>L_1 = &ZZ;^d</M>.

</>

The library character table withidentifier <C>"P11/G4/L1/V1/ext3"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(3 L_1)<]>/



<Example>!CDATA[
gap>v: Subsection Labelsubsecte-sl27">
gap>t: [1,0 0,0, 0 ,0,0 ];
gap> sgens:= generatorsOfPerfectSpaceGroup( [java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
gap
gap> fun:= multiplicationModulo( 3 );;
p seed: 1, 0,0 ,0,,0,0, 1 ;java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
gap> orb:= Orbit( g, seed, fun );;
gap> permgens:= List( sgens, x -> Permutationgap>sgens:= generatorsOfPerfectSpaceGroup( [ abas,bbas,cbas gap>b= [[0-,0 0,0,0, ,0]java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
gap>>fun:=multiplicationModulo( 2 );;
true
]>/>

</Subsection>

 ,,0 ,0, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<Subsection Label="subsect:expl-sl27">><Example>
<Heading>Example with point group SL<M>_2(7)</M></Heading>

There is one example with <M>d = 8</M>.
The
(&;Cite=HP89"Where".295">)java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45



<java.lang.StringIndexOutOfBoundsException: Range [0, 8) out of bounds for length 0
gap>>:[[ ,- 0 0 ,0,0 0,,0 ,
gap>b: [0-,0,,,0, 0 ]
,,,0 0 0, ,
> [-1, 0 ,,0,,, ,
> [ 0, 0, 0, 0, 0, 0,-1, 0 ],
> [ 0, [0 0,1 , 0,0,0 ,0,0]
> [ 0 0,,0,0,1 ,0 ]
> [ 0, 0, 0, 0, 1, 0, 0, 0 ],
> [ 0, 0, 0, 0, 0, 0, 0, 1 ] ];;
]]></Example>



]]>Example>
and the java.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 0
that has the followingbasis
which is called <M>B(2,8)</M> in <Cite Key="HP89" Where="p. java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0



<Example><![CDATA[
gap> bas:= [ [ 1, 1, 0, 0, 0, 0, 0, 0 ],
> [ 0, 1, 1, 0, 0, 0, 0, 0 ],
> [ 0, 0, 1, 1, 0, 0, 0, 0 ],
> [ 0, 0, 0, 1, 1, 0, 0, 0 ]librarycharacterjava.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 70
> [ 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: Index 40 out of bounds for length 40
> [ > -,,,0,0-,0,, ;java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
]]></Example>



P>



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



The library character table with identifier <C>"P12/java.lang.StringIndexOutOfBoundsException: Range [1, 54) out of bounds for length 47
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>(2L_2)/>java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
The action on an orbit modulo <M>2</M> is not faithful,
its kernelcontains the centre of SL<M>(2,7)</M>.
We java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 60
One entry is the
of< Labelsexpl23L32>



<Example><![CDATA[
gap> v:= List( [ 1, 2 ], i -> 0 *> -1,1 1 ,0-,1-,1 ],
gap> t:= [ 1, 0, 0, 0, 0, 0, 0, 0 ];;
ap sgens:=generatorsOfPerfectSpaceGroup([abas
gap> g: [1java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
gapjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
gap> funpairs:= function( pair, g )
> For , werewrite theaction on<ML_2</
> end;;
gap> seed:P/>
> [ 1, 0, 0, 0, 0, 0, vectorsystem thetrivial!java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 18
gap> orb:= Orbit( g, seed, funpairs );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, funpairs ) );;
gap> gap> bbas5java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 59
true
]<Example>

</Subsection>


<Labelsubsect:-23>
t group <>^._()/M>Heading

There example <> =7/>
The generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 297"/>).

]

<Example><![CDATA[
gap> a:= PermutationMat( (2,4)(5,7), 7 );;
gapgap g:= Group(sgens );;
gap> c:= DiagonalMat( [ -1, -1, 1, 1, -1, -Example>!CDATA[
]]></Example>



The vector system is gap> orb:= Orbit( g, seefun );
and the translation lattice is the sublattice <M>L_2</M> ofgap> permgens:=List(sgens,x- Permutation( x, orb,fun ) );;
that has the following basis,
which is called <>B2,)/>in;< Key" p. 269/.



<Example><![CDATA[
gap>:=[ , ,0 0,belongs tothefactor ofM>/>modulothe
> [ 0, 1, 1, 0, 0, 0, 0 ],
> [ 0, 0, 1, 1, M>M(2 L_2)/M>
> [ 0, 0, 0, 1, 1, 0, 0 ],
> [ 0, 0, 0, 0, 1, 1, 0 ]<>!CDATA[
> [ 0, 0, 0, 0, 0, 1, 1 ],
> [0,0, 0,0,,1 ]]java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 34
]>/>



 sakeof java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 71



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



The library character table with identifier <C>"P13/G1/L2/V1/ext2"<>[0 , ,1 ,=. 307/)
belongs to the factor group 0 ,0-1 ,0 0]java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
<M>M(2 L_2)</java.lang.StringIndexOutOfBoundsException: Range [0, 14) out of bounds for length 4

<

<Example><![CDATA[
gap> v:= List( [ > [ 0, 0,0, 0, 0, 0, 0 1 0, 0 ],
gap> t:= [ 1, 0, 0, 0, 0, 0, 0 ];;
gap> sgens:= generatorsOfPerfectSpaceGroup( [ abas,bbas,cbas ], v,[ 1 0, 0,,0
gap> g:= - ,1,,1 ,1 ;java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
gap> fun:= multiplicationModulo( 2 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 0, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> act:= Action( g, orb, fun );;
ap permgens=List ,x- Permutation x orb,fun ));
gap> verifyFactorGroup/java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
true
]>/Example>>

/ubsection2 ,0, 0,0, ,0 ]];


<Subsection Label="subsect:expl-A6">
<Heading>Examples with point group <M>A_6</M></Heading>

There are two examples with <M>d = 10</M>.
In bothP/>
(see <Cite Key="HP89" Where="p. 307> [-1, 0, 1 0, 0,,-,0,, 0, 0 ]];;



java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 18
gap> b:= [ [ 0,-> aa:= 1;
> elm:= aa*bb;;
> [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],
> [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ],
> [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
> [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
> [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ],
> [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],
> [ 0, 0, 0, 0, 0,: (gseedfun );
>[0, ,0 ,0 0,0 0 0 ]]
gap> c:= [ [ 0, 0, 0, 0, 0, 0, 0
> [ 0, 0
> [ 0, 0, 0, 0,-1, 1, 0,-1, 0, 0 ],
> [ 0,-1, 1, 0, 0, 0, 0,-1, 0, 0 ],
> [ 0, 0, 0, 0, 0, 0, 0, 0, 0,-1 ],
> [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],
> [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
> [ 0, 0, 0, 0, 0, 1,-1, 0, 0, 1 ],
> [ 0, 0, 1,-1, 0, 0, 0, 0, 0, 1 ],
, ,0 ,1, 0 ],
]



In 6^/>
and the translation lattices are the lattices <M>L_2</M> and <M>L_5</M>,
respectively, which have the following bases.



java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 18
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, 0,-1, 0, 0 ],
> [ 0, 0, 0, 0, 0, 0, 0, 1,-1, 0 ],
> [ 0, 0, 0, 0, 0, 0, 0, 0, 1,-1 ],
> [ 0, 0, 0, 1, 0, 0, 0, 0, 0,-1 ],
java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 47
> ,0 0, 0, , , 0 ,0 0 ;
gap> bas5:= [ [ 0There one M =<M>
> [ 1, 0,-1,-1,-1, 1, 1,-1,-1, 0 ],
> P/>
> [ 1, 1, 0,-1, 0,-1, 1,-1, 1,-1 ],
> [-1, 0,-1, 1, 1, 0,-1,-1, 1,-1 ],
> [ 0, 1,-1, 1, 1,-1, 1, 1, 0,-1 ],
>[-,1,1,1, 0-,1-,1 with<>d=7<M>
> [ 1,-1, 0,-1, 1,-1, 1, 1, 0,-1 ],
> -1 ,,-,,1, 1, 0,-1 ]java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
> >
]]></Example>

P>

For the sake of simplicity, we ,1,-,0 1,0, ,
and < [,1 ,1, 1,,1 ,



<Example><![CDATA[
gap> B2:= Basis( Rationals
and java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 51
gap> cbas2:= <Example><![CDATA
gap> B5:= Basis( Rationals^Length( bas5 ), java.lang.StringIndexOutOfBoundsException: Range [34, 47) out of bounds for length 34
gap> bbas5:= List( bas5java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> cbas5:= List( bas5, x -> java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
]Example

<>

The library character java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 0
belongs to the factor group of gap : [[2,1 ,0,: java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 25
< >



<
: ,-*];
gap>gap aa:= sgens[1];
gap> sgens:= generatorsOfPerfectSpaceGroup( [ bbas2, cbas2 ], v, t );;
gap> g:= gap> Order( elm );
gap> fun:= multiplicationModulo( 2P>
>:[1, 0 0, 0, 0, ,0,,0 1The java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
gap>orb:=(g,seed, )java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
gap> permgens:= List(
gap> [ 0, 6, 1, 1, 8>java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 62
true
]>/java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 13



The library character table with java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 18
belongs to the factor group of <M>S</M> > Label"-"
M2<> the the java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 70



<Example><![CDATA[
gap> sgens:= </java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
gap> g:= Group( sgens );;
java.lang.StringIndexOutOfBoundsException: Range [45, 46) out of bounds for length 34
java.lang.StringIndexOutOfBoundsException: Range [53, 3) out of bounds for length 64
gap> java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 4
true
]]></Example>

</Subsection>


<Subsection Label="subsect:expl-L28">
<> <M>28<>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 twoexamples <M> 7<>java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
, java.lang.StringIndexOutOfBoundsException: Range [32, 2) out of bounds for length 4
(see <Cite Key="HP89" Where="p. 327"/>).



<Example><![CDATA[
gap> a:= [ [ 0,-1, 0, 1, 0,-1, 1],
 [, 0-,0 ,1 ]java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
> [ 0, 0, 0,-1, 1, 0, 0],
>MM L)M,so <>18/>
> [ 0, 0, 1,,-,0 0,0,
> [ 0,-1, 1, 0,-1>elongsto the java.lang.StringIndexOutOfBoundsException: Range [0, 21) out of bounds for length 0
>1,0];java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
gap
> [0,1, 0, 1,-1, 0, 0]
> [ 0, 0,-1, 1, 0, 0, 0],
> 0 0,-1, 0 0,0,);
> [ > orb:=Orbit( g [1,0,0, 0gppermgens:List(sgens - Permutation xorb,fun ) );;
>>>( permgens "P41G1//V4/ ;
> [-1, 0, 1, 0,-1, 0, 1] ];;
]]></Example>



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

<>=generatorsOfPerfectSpaceGroupubsectionLabel=ubsect:xplM11"

<Example><![CDATA[
>= 2 ,,0 ,1 4],
> [ 2, 0, 0, 0, 0, 0, 0 ] ];;
gap> t:= [ 3, 0, 0, 0, 0, 0, 0 ];;
]]></Example/java.lang.StringIndexOutOfBoundsException: Range [12, 8) out of bounds for length 18



tablewith identifier<C"P41/G1/L1/V3/"/C>
belongs toP/
<M>M(3 L)</M>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0



The orbits in this action are quite long.
we choose a seed vector from the fixed space of an element 10,4 ], [ 12,4 ,[15,20 ,[20,2 ] [ 30, 29 ], [ 60,8 ]]



<Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap
gap> aa:= sgens[1];;
gap> bb:= sgens[2];;
gap> elm:= aa*bb;;
gap/>
7
gap> fixed:= NullspaceMat( elm - aa^0 );
[ [ 1, 1, 1, 1, gap verifyFactorGroup( , "///ext2")
gap> fun:= multiplicationModulo( 9 );;
gap> seed:= fun( fixed[2], aa^0 );
[ 5, 1, 1, 4, 4, 2, 0, 1 ]
gaporb=Orbit(g,seed,fun ;
gap> permgens:= List
 java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 56
true
]] java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 52



The translation writtenüM)
<M>L = 6 &ZZ

<If Mp<M-java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 54

>Cjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
gap>t: [ 6,0 0 0, ,0 ;
]<Examplejava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13



The library character table with identifier <C>"P41/
belongs to the factor group of <via java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<M>M(6 L)</M>, so M/>



<Example><![
gapjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0
>java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 62
gap> g:= Group( sgens );;
gap> seed:= fun( fixed[2], aa^0 );
=java.lang.StringIndexOutOfBoundsException: Range [47, 42) out of bounds for length 62
gap> orb:= Orbit( g, seed, >=java.lang.StringIndexOutOfBoundsException: Range [33, 31) out of bounds for length 38
gap>>!
gap> verifyFactorGroup( permgens, " , java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
true
]>= java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 37

</Subsection>


<Subsection Label="subsect:expl-M11">
with point group<>M_11}</><Headingjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57

There in t thesense permuting thepreimageof fusion Mf<>
Thegenerators pointgroup are as follows
(see <Cite (see <Cite Keyimage with <>s<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</

<Example<![DATA[
gap> a:= deletedPermutationMat( (1,9)(3,5)(7,11)(8,10), 11 );;
gap>b:=deletedPermutationMat( (1,4,3,2)5876) 11);
]]></Example>



The vector that exactly java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 72
and the translation lattice is <<>

=0g java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 71

<Example><![CDATA[
gap> v:e><[DATA
> [0, ,1 ,
gap 0,0 0,0 ,1 ]];
]]></gap> GetFusionMap( m,t );



The library character table with identifier <C>"P48/G1/L1/V2/ext2"</C>[,,,--, ,
Group[ (,) ]java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
<M>M(2 L)</M>, so we compute the java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 0

<Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup ] ,t)java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 4 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ], fun );;
gap> permgens:= List( sgens, x -> Permutation( java.lang.StringIndexOutOfBoundsException: Range [0, 48) out of bounds for length 18
gap> verifyFactorGroup( permgens, "P48/G1.16--2java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
true
]]></Example>

</Subsection>


<Subsection Label="subsect:expl-U33">
<eadingExample withpoint <>U_33)/><>

There is one
The generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 335"/>).



<Example><![CDATA[
gap>a: [ [ 0, 0-,1,0-,1 ,
> [ 1, 0,-1, 1, 1,-1, 0 ],
> [ , 1,-,0 1 ,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 ] ];;
gap> b:= [ [ 0, 0, 0, 0, 0, 0, 1 ],
> [ 0, 0,-1, 1, 0,-1, 1 ],
> [ 1, 0,-1, 1, 1,-1, 0 ],
> [ 0, 1,-1,The java.lang.StringIndexOutOfBoundsException: Range [45, 43) out of bounds for length 70
> [ 0, 1, 0,-1, 1, 0,-1 ],
> [-1, 1, 1,-1, 0> 1 .. 1
> [-1, 0, 1,-1, 0, 0, 1 ] ];;
].0 g (;



java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 32
 lattice <>L library character with identifier<>"50/G1/L1/V1/ext4"</C>



<.17 20 22 - java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
gap> v:gap>sgens=generatorsOfPerfectSpaceGroup [ a,=()
> 1]java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
<P/
]]></Example>

<java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 72

libraryjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 13
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(3 L)</M>, so we compute the action on an orbit modulo <M>9</M>.



<><[
gap><Heading> onIn again the,
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 9
]]></Example>



The orbits in this action are quite long.
we choose a seed vector</



<Example><![CDATA[
gap> aa:= sgens[1];;
gap> bb:= sgens[2];;
gap> elm:= aa*bb^4;;
gap> Order(elm);

gap> fixed:= NullspaceMat( elm - aa^0 );
[ [ -1, -1, 1, 1, -1, -1, 1, 0 ], [ 0, -3, 1, 1, -1, -2, 0, 1 ] ]
gap> seed:= fun( fixed[2], aa^0 );
[ 0, 6, 1, 1, 8, 7, 0, 1 ]
: > local cons, java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 66
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
/G1/L1/2/");
true
]]></Example : [;

<Subsection>


<Subsection Label="subsect:expl-U42">
<Heading>Examples with point group <M>U_4 java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

There are two examples with <M>d = 6</M>.
In both cases, the generators of the point group are as follows
s&;"P89"> do

P>

<Example><![CDATA[
gap>: [ 0, 1 0,1,1
> [ 1, 0,-1, 0, 1, 0 ],
> [ 0, 0, 0,-1, 0, 1 ],
> [ 0, 0,-1, 0, 0, 1 ],
> [0>
> [ 0, 0, 0> allfusions=OrbitFusions maut,,taut )
gap> b:= [ [ 0,-1, 0, 1, 0,-1 ],
> [ 0, 1, 0,-1,-1, 0 ],
> [ 0, 0, 1, 1, 0,-1 ],
>[0,,,0,1 , .
> [ 0, 1, 0, 0, 0, 0 ],
> [ 1, 0, 0, 0, 0, 0 ] ];;
]]></Example>



In ()=1then
java.lang.StringIndexOutOfBoundsException: Range [34, 3) out of bounds for length 68



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



<P/
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 anthe java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 67

</

<Example><![CDATA[
p sgens= a,b , ,t);;
gap> g:= Group( sgens );;
fun: multiplicationModulo
gap> orb:= Orbit( g, [ 1The ordinary character table does not relate the two pairs of classes,
>: >x ,fun ;java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
gap> verifyFactorGroup( permgens, "P50/G1/L1/V1/ext3" );
true
]]>/Example>



The library character table with identifier <C>"P50/G1/L1/V1/ext4"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(4 L_1)</M>, so we compute the action on an orbit modulo <M>4</M>.



<
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap>java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
gap> fun:= multiplicationModulo( 4 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 1 ], fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orbExample<[java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
gap> verifyFactorGroup( permgens, "P50/G1/L1/V1/ext4" );
true
]]></Example>

</Subsection

%%%%%%%%%%%%%%%%%%%%%%%% ->
<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<Heading>A remark on one of the example groups</Heading>

The(erfect) character table with identifier <C>"P1/G2/L2/V2/ext4"</C>
has the property that its character degrees are exactly
the divisors of <M>60</M>.



<Example><![CDATA[
gap degrees: CharacterDegrees("1//"))
[ 2 ] [3, 2],[4,2 ] [5 1 ] [6 5 ]
 [ 10, 4 ], [ 12, 4 ], [ 15, permuting imagewith </>yields<M</M>java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
gap> List( degrees, x -> x[1] ) = DivisorsInt( 60 );

]]></Example>



There are nilpotent groups with the same set of character degrees,
for example the direct product of four extraspecial groups of the orders
<M>2^3</M>, <M>2^3</M>, <M>3^3</M>, and <M>5^3</M>, respectively.
This phenomenon has been described in <Cite Key="NR14"/>.

n>

</Section>


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

The term <Q>generality problem</QThus itis sufficient comparejava.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 76
consistent choices of conjugacy classes of Brauer tables for the same
group, in differentap> t)
The definition and some examples are given
in <Cite Key="JLPW95" Where="p. x"/>.



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 use the following idea for finding character tables which
may involve generality problems.
(The functions shown in this section are based on &AP; 3java.lang.StringIndexOutOfBoundsException: Range [62, 63) out of bounds for length 62
that was originally written by Jürgen Müller.)



theMp<M>-modular Brauer table <M>mtbl</M>, say
of a group contributes to a generality problem
of classes necessary
in order to write down this table,
in the sense that some symmetry of the corresponding ordinary table
<M>tblP>
This situation can be detected as follows.
We assume that the class fusion from <>mtbl<M> to <M>tbl</M>
has been fixed.
All possible class fusions are obtained as the orbit of this
class 2 5 2 2 1
via mapping the images of the class fusion
(with the function <Ref Func="OnTuples" BookName="ref"/>),
and of the table automorphisms of <M>mtbl</M>,
via.
The case of broken symmetries occurs if distinguishesthetwo (lgebraicallyconjugate classes
splits into several
table automorphisms of <M>mtbl</M> is considered.
 java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 66
java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 68
table automorphisms of <MX. 16- 2 java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18





Example<CDATA
gap> BrokenSymmetries:= function( ordtbl, modtbl )
> local taut, maut, triv, fus, orb;
> taut:= AutomorphismsOfTable( ordtbl );
> maut:=AutomorphismsOfTable( modtbl );
> triv:= TrivialSubgroup( taut );
> java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
> 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:</EX. 111 1 1-1 - - -- 1

<List>
<tem
 It may happen that some symmetry <M>\sigma_m</M> of a Brauer table
 does not belong to a symmetry <M>\sigma_o</M> of the corresponding
 ordinary table,
 in the sense that permuting the preimage classes of a fusion <M>f</M>
 thetables <\/>
 the java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 73
 <P99 1 . ..1 1- - 31. . java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
 For example consider M 2A_62_>java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
 the double cover of the symmetric group <M>S_6</M> on six points.
uer table of <>G<</M>,
 which is essentially equal to that of <M>S_6</M>,
 has a table automorphism group order two,
 and the X.14 4 -4 . 1 -12 2 .1 . . . A A java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
 of 16 22- . java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
 The automorphism group of the ordinary character table of <M>G</M>,
 however, fixes the two classes of element order three;
 note that exactly one of these classes possesses square roots in the
 <Q>outer half<Q><M> \etminus G'</M>.
 
 Thus it is not sufficient to compare the orbit of the fixed class fusion
 under the automorphisms of the ordinary table with the orbit of the
 same fusion under the automorphisms of the Brauer table.
</Item>
</List>

<<![DATA[
gap> t:= CharacterTable( "2.A6.2_1" );;
gap> m:= t mod 2;;
gap> GetFusionMap( m, t );
[ 1, 4, 6, 5P1a 2a 3a1a 6a 7a 2a 2a 3a 3a 19b 19c 19a
gap> AutomorphismsOfTable( t );
(16,17) (,15), (14,)(6,17) ])
gap> AutomorphismsOfTable( m );
Group([ (2,3) ])
gap> Display( m );
2.A6.2_1mod2

 2 5 2 2 1
 3 2 2 2 .
 5 1 . . 1

 1a 3a 3b 5a
 2P 1a 3a 3b 5a
 3P 1a 1a 1a 5a
 5P 1a 3a 3b 1a

X.1 1 1 1 1
X.2 4 1 -2 -1
X.3 4 -2 1 -1
X.4 16 -2 -2 1
gap> Display( tA =E5^
2.A6.2_1

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

 1a 2a 4a 3a 6a 3b 6b 8a 5a Add(consider,p);
 2P1a 1a 2a3a 3 3b3b 4a 5a 5a 1a 2a 4a 3a 3a 6b 6b
 3P 1a 2a 4a 1a 2a 1a 2a 8a 5a 10a 2b 4b 8b 2b 2b 4b 4b
 5P 1a 2a 4a 3a 6a 3b 6b 8a 1a 2a 2b 4b 8b 6d 6c 12b 12a

11111 11111 1 >java.lang.StringIndexOutOfBoundsException: Range [48, 14) out of bounds for length 48
X.2 # ' be .
35 5 1 22-- 1 .. 1 1 .- 1
. 1 2 --1- .- 1.. 1
X.5 5 5 1 -1 -1 2 2 -1 . . -1 3 1 -1 -1 . .
X.6 5 5 1 -1 -1 2 2 -1 . . 1 -3 -1 1 1 . .
X.7 16 16 . -2 -2 -2 -2 . 1 1 . . . . . . .
X.8 X19)^14-E19)16-E()17E19^18
X.9 9 9 1 . >#suitablebecause a' notclosed
X.10 10 10 -2 1 1 1 1 . . . 2 -2 . -1 -1 1 1
X.11 10 10 -2 1 1 1 1 . . . -2 2 . 1 1 -1 -1
X.12 4 -4 . -2 2 1 -1 . -1 1 . . . . . B -B
X.13 4 -4 . -2 2 1 -1 . -1 1 . . . . . -B B
X.14 4 --4 . 1 -1 -2 2 . -1 1 . . . A-A ..
X.15 4 -4 . 1 -1 -2 2 . -1 1 . . . -A A . .
X.16 16 -16 . -2 2 -2 2 . 1 -1 . . . >reps=List(orbits,x- x[])java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
X.17 20 -20 . 2 admiss: Stabilizer admiss,

A = E(3)-E(3)^2
 = Sqrt(-3) = i3
B = -E(12)^7+E12)^11
 = Sqrt(3) = r3
]>/>



When considering several characteristics in parallel, one argues as follows.
The possible class fusions from a Brauer table <M>java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 7
ordinary table <M>tbl</M> are given by the orbit of a java.lang.StringIndexOutOfBoundsException: Range [0, 59) out of bounds for length 4
under the action of the table automorphisms of <M>tblLet uslook at small example,
Ifjava.lang.StringIndexOutOfBoundsException: Range [13, 8) out of bounds for length 65
of<>tbl</> choose one orbit.
Due to this choice, only When we define which of the two classes of element order <M>8</M> is called
admissible for other characteristics that stabilize the chosen orbit.
For the second characteristic, we take again the set of all class fusions
from the Brauer table to <M>tbl</M>, and split it symmetry is thus broken in the <M>5</M>-modular character
table automorphisms of the Brauer table.
Now there are two possibilities.
Either the action<xample<![CDATA[
these oneorbit or notjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
In the former case, we choose again one of the orbits,
replace the group of admissible automorphisms of <M>tbl</M> by
this proceedwith thenext characteristicjava.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
In the latter case, we have found a generality problemjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
since we are not free to choose an arbitrary class fusion from the
set of possibilities.



The following function returns the set of primes which may be involved
in generality problems for the given ordinary character table.
Note that the procedure sketched above does not tell
cs actuallyinvolved
or which classes are affected by the choices;
for example, we could argue that one is always free to choose a fusion
for the first characteristics, and that only the other ones cause problems.
We return <E>all</E> those primes <M>p</M> for which broken symmetries
between the <M>p</M>-modular table and the ordinary table have been detected.



<Example><![CDATA[
gap> PrimesOfGeneralityProblems:= function( ordtbl )
>1a3ab67a 10a 10 11a1515b
> allfusions, orbits, orbit, reps;
> # Find the primes for which symmetries are broken.
> consider:= [];
> forp in PrimeDivisors( Size ordtbl)) IsPrimeInt)
> modtbl:= ordtbl mod p;
> if < fail and BrokenSymmetries( ordtbl, modtbl ) then
> Add( consider, p );
> fi;
> od;
> # Compute the choices and detect generality problems.
> taut:= AutomorphismsOfTable( ordtbl );
> triv: TrivialSubgroup( taut );
> admiss:= taut;
> for p in consider do
> modtbl:= ordtbl mod p;
> fusion:= GetFusionMap( modtbl, ordtbl );
> maut:= AutomorphismsOfTable( modtbl );
> # - We need not apply the action 7 1a2a 3 5 5a6a 1a 10a 11 15 15java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
> # since 'maut' will later be used to get representatives.
> # -We need not apply all elements in 'taut' but only
> # representatives of left cosets of 'admiss' in 'taut',
 19 12aa56aa 10b11a 15 15java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
> # allfusions:= OrbitFusions( maut, fusion, taut );
> allfusions:= Set( RightTransversal( taut, admiss ),
>.1 2 - 1 A*A - -A . B *
> # For computing representatives, 'RepresentativesFusions' is not
> # suitable because 'allfusions' is in generally not closed(+()4
> # under the actions.
> # reps:= RepresentativesFusions( maut, allfusions, admiss );
> orbits:= [];
> while not IsEmpty( allfusions ) do
> orbit:= OrbitFusions( maut, allfusions[1], the <>/>
> ,orbit )
> SubtractSet( the unique irreducible <M>11</M>-modular Brauer character of degree <M>7</M>
> od;
> reps:= List( if Length( reps ) = 1 then
> # Reduce the symmetries that are still available.
> admiss:= Stabilizer( admiss,
s( maut, fusion,triv))
> OnSetsTuples );
> else
> # We have found a generality problem
> return consider;
> fi;
> od;
> There is no generality problem for this table.
> return [];
> end;;
]]></Example2 3 1 11 1 .....



Let us look at a small example,
the <5/M>modular table of the71 ... 1 .....java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
The irreducible characters of degree <M>2</M> have the values
<M>\pm \sqrt{{-2}}</M> on the classes <C>8a</C> and <C>8b</C>,
and the values <M>\pm1 23a 5a 5ba 15 a19 19c
When we define which of the two classes of element order <M>8</M> is called
<C>8a</C>, this will also define which class is called <C>6b</C>.
The ordinary character table does not relate the two pairs of classes,
there are table automorphisms which interchange each pair independently.1a a aa a3
This symmetry is thus broken in the <M>5</M>-modular character table.



<Example><![CDATA[
ap : CharacterTable("A.";
gap> m:= t mod 5;;
gap> Display( m );
2.A5.2mod5

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

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

X.1 1 1 1 1 1 1 1 1 1 1
X.2 1 1 1 1 1 -1 -1 -1 -1 -1
X.3 3 3 -1 . . 1 -1 -1 -2 -2
X.4 3 3 -1 . . -1 1 1 2 2
X.5 5 5 1 -1 -1 1 -1 -1 1 1
X.6 5 5 1 -1 -1 -1 1 1 -1 -1
X.7 2 -2 . -1 1 . A -A B -B
X.8 2 -2 . -1 1 . -A A -B B
X.9 4 -4 . 1 -1 . . . B -B
X.10 4 -4 . 1 -1 . . . -B B

A = E(8)+E( .lt( ,, ,
 = Sqrt(-2) = i2
B = E(3)-E(3)^2
 = Sqrt(-3) = i3
[2.) ,,, ,
Group([ (11,12), (9,10) ])
gap> [ 2.Alt ,3 5 ],
Group([ (7,8)(9,10) ])
gap> GetFusionMap( m, t );
 ,3,4,5 ,9,10 11,12]
gap> BrokenSymmetries( t, m );
true
gap> BrokenSymmetries( t, t mod 2 );
false
ap> BrokenSymmetries( t,t mod 3 )
false
>( t );
[ ]
]]></Example>



ince nosymmetryisbrokenin the<M>2</M>- and M><M-modular
character tables of <M>G</M>, there is no generality problem
in this case.



For an example of a generality problem,
we look at the smallest Janko group <M>J_1</M>.
As is mentioned in <Cite Key="JLPW95" Where="p. x"/>,
the unique irreducible <M>11</M>-modular Brauer character of degree <M>7</M>
distinguishes the two (algebraically conjugate) classes
of element order <M>5</M>.
Since also the unique irreducible <M>19</M>-modular Brauer character
of degree <M>22</M> distinguishes these classes,
we have to choose these classes consistently.



<Example2L3(21 ,
gap> t:= CharacterTable( "J1" );;
gap> m:= t mod 11;;
gap> Display m rec(chars: ( Irr(m), ->x1]=7)))
J1mod11

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

 1a 2a 3a 5a 5b 6 3J32 [2 ,17,19],
 2P 1a 1a 3a 5b 5a 3a 7a 5b 5a 15b 15a 19b [ L3)._, ,3,7 ,
 3P 1a 2a 1a 5b 5a 2a 7a 10b 10a 5b 5a 19b 19c 19a
 5P 1a 2a 3a 1a 1a 6a 7a 2a 2a 3a 3a 19b [McL, , ,
 7P 1a 2a 3a 5b 5a 6a 1a 10b 10a 15b 15a 19 7P 1a 2a 3a 5b 5a 6a 1a 10b 10a 15b 15a 19a 5,11]],
 c
 19P 1a 2a 3a 5a 5b 6a 7a 10a 10b 15a 15b 1a 1a 1a

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

=5+(^java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 = (-1+Sqrt(5))/2 = b5
B = -E(5)-2*E(5)^ ,[7 11 19]]
 =3Sqrt()/ 2+b5
C = -2*E(5)-2*E(5)^4
 = 1-Sqrt(],
D = -E(19)-java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 61
(19)^14-E( 2L4._1, ,5, 7 ] ],
E = -E(19)^2-E(19)^3-E(19)^4-E(19)^5-E(19)^6-E(19)^9-E(19)^10-E(19)^13\
-E(19) [ .()_1 , ,
E)(19419^197E()8E(19^E(9^-1911Ejava.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
(19)^12-E(19)^13-E(19)^15-E(19)^18
gap> m:= t mod 19;;
gap> Display( m, rec( chars:= Filtered( Irr( m ), x -> x[1] = 22 ) ) );
J1mod19

 2 3 3 1 1 1 1 . 1 1 . . .
 3 1 1 1 1 1 1 . . . . [A14[2 ,
 5 1 1 1 1 1 . . 1 1 . 1 1
 7 1 . . . . . 1 . . . . .
 11 1 . . . . . . . . 1 . .
 . . . java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47

 1a 2a 3a 5a 5b 6a 7a 10a 10b 11a 15a 15b
 2P 1a 1a 3a 5b 5a 3a 7a 5 Fi23, [3 17,23]],
 3P 1a 2a 1a 5b 5a 2a 7a 10b 10a 11a 5b 5a
 5P 1a 2a 3a 1a 1a 6a 7a 2a 2a 11a 3a 3a
 1a 3aa aa1b10 a ba
 11P 1a 2a 3a 5a 5b 6a 7a 10a 10b 1a 15a 15b
 19P 1a 2a 3a 5a 5b 6a 7a 10a 10b 11a 15a 15b

Y.1 22 -2 1 A *A 1 1 -A -*A . B *B

A = E(5)+E(5)^4
 = (-1+Sqrt(5))/2 = b5
B = -2*E(5)-2*E(5)^4
 = 1-Sqrt(5) = 1-r5
]]></Example>



Note that the degree <M>7</M> character above also distinguishes
the three classes of element order <M>19</M>,
and the same holds for the unique irreducible degree <M>31</M> character
from characteristic < [Isoclinic(2.HSx2), [ 3, 5, 7, 11 ] ],
Thus also the prime <M>7)21,[ 5 7 ],
generality problems.



<Example><![CDATA[
gap> PrimesOfGeneralityProblems( t );
[ 7, 11, 19 ]
]]></Example>



Finally, we list the candidates for [ J1, ,
from &GAP;'s Character Table Library.

</>

<Example><![CDATA[
gap> list:= [;java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
gap> isGeneralityProblem:= function( ordtbl )
> local res;
> res:= [ L3(7).S3, 3,7,19 ,
> if res = [] then
 false;
> fi;
> Add( list, [ Identifier( ordtbl ),P/
> return true;
> end;;
mes,,
> isGeneralityProblem, true );;
gap> PrintArray( SortedList( list ) );
[ [ (2.A4x2.G2(Note java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 65
 [ (2^2x3).L3(4).2_1, [ 5, 7 ] ],
 [ ofM>F_4(2)/ whenthe <>2<M>- F_42<>
 [ (4^2x3).L3(4), [ 2, 3, 7 ] ],
 [ (7:3xHe):2, [ 5, 7, 17 ] ],
 [ (A5xA12):2, [ 2, 3 ] ],
 [ (D10xHN).2, [ 2, 3, 5, 7, 11, 19 ] ],
 [ (S3x2.Fi22).2, [ 3, 11, 13 ] ],
 [ 12.M22, [ 2, 5, 7, 11 ] ],
 [ 12.M22.2, [ 2, 5, 7, 11 ] ],
 [ 12_1.L3(4).2_1, [ 5, 7 ] ],
 [ 12_2.L3(4), [ 2, 3, 7 ] ],
 [ 12_2.L3(4).2_1, [ 3, 5, 7 ] ],
 [ 12_2.L3(4).2_2, [ 2, 3, 7 ] ],
 [ 12_2.L3(4).2_3, [ 2, 3, 7 ] ],
 [ 2.(A4xG2(4)).2, [ 2, 5, 7, 13 ] ],
 [ 2.2E6(2), [ 13, 19 ] ],
 [ 2.2E6(2).2, [ 13, 19 ] ],
 [ 2.A10, [ 5, 7 ] ],
 [</lt>
 [ 2.A11.2, [ 5, 7, 11 ] ],
 java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 34
 2 ,11 java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
 [ 2.A13, [
 [ 2.A13.2,java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 [ 2.C>PerfectGroup( 64512, 2
 [ 2.gapTransformingPermutations(tbls2 particular,changing the line programwouldnothelpto
 [ 2.Alt(16), [ 2, 3, 5, 7 ] ],
 [ 2.Alt(17), [ 2, 3, 5, 7 ] ],
 [ 2.Alt(18), [ 2, 3, 5
 [ 2.B, [ 17, 23 ] ],
 [ 2.F4(2), [ 2, 7, 13, 17 ] ],
 2.2 ,13]],
 [ 2 7 ]
 [ 2.G2(4).2, [ 5, 7, 13 ] ],
 , [ 3,5,,11]],
 [ 2.HS.2, [ 3, 11 ] ],
 [ 2.L3(4).2_1, [ 5, 7 ] ],
 [ 2.Ru, [ 5, 7, 13, 29 ] ],
 [ 2.Suz, [ 2, 5, 11 ] ],
 [ 2.Suz.2, [ 3,by <>17<</>in <M>9/M>-modularcharacter tables of <M>J_3</M>,
 [ 2.Sym(15), [ 3, 5, 7 ] ],
 [ 2.Sym(16), [ 3, 5, 7 ] ],
 [ 2.Sym(17), [ 3, 5, 7 ] ],
 [ 2.Sym(18), [ [ L2(8) 2^6,L2()N2^ ]
 [ 2.Sz(8), [ 2, 5, 13 ] ],
 [ 2^2.2E6(2), [ 13, 19 ]
 [ 2^2.2E6(2).2, [ 13, 19 ] ],
 [ 2^2.Fi22.2, [ 3, 11, 13 ] ],
 [ 2^2.L3(4).2^2, [ 5, 7 ] ],
 [ 2^2.L3(4).2_1, [ 5, 7 ] ,second powermapof thegiventable of <M><M
 [ 2^2.Sz(8), [ 2, 5, 13 ] ],
 [ 2x2.F4(2), [ 2, 7, 13, 17 ] ],
 [ 2x3.Fi22, [ 2, 3, 5 ] ],
 %]>/xample
 s been overlookedwhenthedata &<""
 [ 2xFi22.2, [ 11, 13 ] ],
 [ 2xFi23, [ 3, 17, 23 ] ],
 [ 3.Fi22, [ 2, 3, 5 ] ],
 [ 3.Fi22.2, [ 2, 5, 11, 13 ] ],
 [ 3.J3, [ 2, 17, 19 ] ],
 [ 3.J3.2, [ 2, 5,17, 19 ] ],
 [ 2, 3, 7 ] ],
 [ 3.L3(4).3.2_3, [ 2, 3, 7 ] ],
 [ 3.L3(7).2, [ 3, 7, 19 ] ],
 [ 3.L3(7).S3, [ 3, 7, 19 ] ],
 [3.McL, 2, 5,11]],
 [ 3.McL.2, java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 36
 [ 3.ON, [ 3, 7, 11, 19, 31 ] ],
 [ 3.ON.2, [ 3, 5, 7, 11<!java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
 [ 3.Suz.2, [ 2, 3, 13 ] ],
 [ 3x2.F4(2), [ 2, 7, 13, 17
 [ 3x2.Fi22.2, [ 11, Thus PDFfiles containingthe<>19/>-odular decomposition
 [ 3x2.gap> mx:= MaximalSubgroupClassReps)
 [ 3xFi23, 3,17,23 ]
 [ 3xJ1, [ 7, 11, 19 ] ],
 [ 3xL3(7).2, [ 3, 7, 19 ] ],
 [ 4.HS.2, [ 5, 7, 11 ] ],
 [ 4.M22, [ 5, 7 ] ],
 [ 4_1.L3(4).2_1, [ 5, 7 ] ],
 [ 4_2.L3(4).2_1, [ 3, 5, 7 ] ],
 [ 6.Fi22, [ 2, 3, 5 ] ],
 [ 6.Fi22.2, [ 2, 5, 11, 13 ] ],
 [ 6.L3(4).2_1, [ 5, 7 ] ],
 [ 6.M22, [ 2, 5, 11 ] ],
 [ 6.O7(3), [ 3, 5, 13 ] ],
 [ 6.O7(3).2, [ 3, 5, 13 ] ],
 [ 6.Suz, [ 2, 5, 11 ] ],
 [ 6.Suz.2, [ 7168 ,7168,7168 7168,7168]
 [ 6x2.F4(2), [ 2, 7, 13, 17 ] ],
 [ A12, [ 2, 3 ] ],
 [ A14, [ 2, 5, 7 ] ],
 [ A17, [ 2, 7 ] ] ]]>/Example
 [ A18, [ 2, 3, 5, 7 ] ],
 [ B, w.rt property or .
 [ F3+, [ 17, 23, 29 ] ],
 [ F3+.2, [ 17 they haveequiv matrices;;
 [ Fi22.2, [ 11, 13 ] ],
 [ Fi23, [ 3, 17, 23 ] ],
 [ HN, [ 2, 3, 11, 19 ] ],
 [ HN.2, [ 5 7,11,19 ]]
 [ He, P>
 [ He.2, [ 5, 7, 17 ] ],
 [ Isoclinic(12.M22.2), [ 2, 5, 7, 11 ] ],
 [ Isoclinic(12_1.L3(4).2_1), [ 5, 7 ] ],
 [ Isoclinic(12_2.thus bethat the wrong has impliedthe wrong
 allelements order M91/>
 [ Isoclinic(2.A11.2), [ 5, 7, 11 ] ],
 [ Isoclinic(2.A12.2), [ 5, 7, 11 ] ],
 [ (2A13, ap t: ( ");
 [ Isoclinic(2.In other words, the &GAP; table was wrong.
 [ Isoclinic(2.G2(4).2gap g=AtlasGroup "6(2) );
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 [ Isoclinic(2.HSx2), [ &GAP;Character TableLibrary
 [ Isoclinic2.34._) , ] ,
 [ Isoclinic(2.Suz.2), [ 3, 7, 13 ][167,168 163,164,165,166 ]
 [ Isoclinic(_1.L3(4.2_1), [ 5,7 ] ]
 [ Isoclinic(4_2.L3(4).2_1), [ 3, 5, 7 ] ],
 [ Isoclinic(6.Fi22.2), [ 2, 5, 11, 13 ] ],
 [ Isoclinic(6.L3(4).2_1), [ 5, 7 ] ],
 [ Isoclinic(6.O7(3).2), [ 3, 5, 13 ] ],
 [ Isoclinic(6.Suz.2), [ 2, 3, 5, 7, 13 ] ],
 [ J1, [ 7, 11, 19 ] ],
 [ J1x2, [ 7, 11, 19 ] ],
 [ J3, [ 2, 17, 19 ] ],
 [ J3.2, [ 2, 5, 17, 19 <M>2.F_4(.<M hadbeen wrong
 [ L3(4In table
 [ L3(4).3.2_3, [ 2,Pjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 [ L3(7).2, [ 3, 7, 19 ] ],
 [ L3(7).S3, [ 3, 7, 19 ] ],
 [ java.lang.StringIndexOutOfBoundsException: Range [0, 23) out of bounds for length 0
 [
 [ P
 [ M23 [ 2, ,23] ],
 [ ON, [ 3, 7, 11, 19, 31 ] ],
 [ ON.2, [ 3, 5, 7, 11, 19, 31 ] ],
 [ Ru, [ 5, 7, 13, 29 ] ],
 [ S3xFi22.2, [ 11, 13 42 42, 79, 79, 43,43, 83, 83, 45 45, 47 theofM>{}<>is
 .2 [13 java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
]]></Example>

<P/GetFusionMapt { 140. ;



Note that this list 2Alt( ,5 7]],

(*3*165 *)32*16^, 2+java.lang.StringIndexOutOfBoundsException: Range [64, 62) out of bounds for length 67
extensions of <M>I not a subgroup <>.42.</>whose
became available.)

<>


<java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 38
<Heading>A java.lang.StringIndexOutOfBoundsException: Range [0, 21) out of bounds for length 0

<Alt Only="HTML">
!Cjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
<a name="generality_problem_J3");
]]gap>orders:=OrdersClassRepresentatives ;java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
</Alt>
<Alt Not="HTML">
In March 2015,
</Alt>
KlausLux inconsistency the data of GAP;



The M>/>
irreducible Brauer character of/java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
In>=java.lang.StringIndexOutOfBoundsException: Range [27, 24) out of bounds for length 34
&ATLAS; java.lang.StringIndexOutOfBoundsException: Range [0, 10) out of bounds for length 0
the Brauer character value on the class <C>gap> IsRecord(TransformingPermutationsCharacterTables( t,
he &TLAS; of Representations;<iteKey=A">provides
a straight line program for computing
If we compute the/Sectionjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
<>17}/>butitsalgebraicconjugate M>-1-b_{17}M.

Pjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4



Example![
gap> t:= CharacterTable( "J3" );;
gap>m: t 19;
gap> cand:resultsof constructions,
gap>Thisisundocumented functionthatbecomes
1
gap>P>
gap> 17a:= Position( slp.outputs, "17A" );
18
gap> info:= OneAtlasGeneratingSetInfo( "J3", Characteristic, 19,
> Dimension, 110 );;
java.lang.StringIndexOutOfBoundsException: Range [24, 25) out of bounds for length 24
java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 53
> gens.generators );;
gap> Quadratic( BrauerCharacterValue( reps[ 17a ] ) );
rec( ATLAS := "-1-b17", a := -1, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 display := "(-1-Sqrt(17))/2", root := 17 )
]]></Example>

<P/

 resolvethisinconsistency,
by replacing the straight line program
or by swapping the classes <C>17A</C> "impleMath""1</td>
Before we decide this, we look at related information.



}/lt
<M>&<.
to the zero vector,
according to <<>/>java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 66
can used to which the two classes order <M>17<M>
shall < java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 61
aM+M> oftableindicates the
representation is available inIn theexamplesbelow,thematrixrepresentation ,



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

<Table Label="valuesJ3ldots,
<Caption>Representations of <M>J_3</M> that may define\egin{array}{c I& 0\ b_1& 1\nd{rray right].
<HorLine/>
<
<Item><M>p</M><//Verb>
<Item><M>\varphi(1)</M></Item>
<Item><M>\\varphi(<M>C17</C><>)/M>/Item>
<Item><M>\varphi(</M><C>17B< =tdleft"
java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 21
<
<HorLine/>
>
<Row>
<Item><M>2</M></Item>
<Item><M>78</M><<=> =Sjava.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 69
<Item><M>1-b_{17}</M></Item ""
<Item><M>2+b_{ tablejava.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 29
<Item><M>+</M></Item>
</Row>
<Row>
<Item><M>2</M></Item>
<Item><M>80</M></Item>
<Itemtd =tdleft"
M></Item>
<Item><M>+</M></Item>
</Row>
<Row>
<Item><M>2</M></Item>
<Item><M>244</M></Item>
<Item><M>b_{17}-2</M><//td>
<Item><M>-3-b_{17}</These bythe
<Item><> (,(2)\dots V) <M> <>1<>
</Row>
<>
<Item><M>2</M></Item>
<Item><M>966</ : (1 ;
<Item><M>r_{17}-3</M></Item>
<Item><M> result]: m;
<Item><M>+</java.lang.StringIndexOutOfBoundsException: Range [0, 13) out of bounds for length 9
</ ;
<Row>
<Item><M>19</M>
<Item><M>110<<ubsection Label="ubsect:onstructfactors"
<ItemThe space group <M>S/M>acts on <M&;^</M>
<Item><M>-1-b_{17}</A( )of <S/M(n T)<M>
<Item><Mthat, java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 49

<Row>
<><19<Itemjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
<Item><M>214</M></Item>
<thejava.lang.StringIndexOutOfBoundsException: Range [21, 18) out of bounds for length 63
<Item><M>2+b_{17}</M></Item>
<<M>+</M></Item>
</Row>
<Row>
<Item><M>19</M></Item>
<Item><M>706</M></Item>= ^pi;
<Item><M>-b_{17}</ mat[[]=-;
<><>+{}<M>/Item>
<Item><M>+</M></Item>
</Row>
<weverify that thegroups fit the character fromthe
<</Item>
<Item><M>1214</M></Item>
<java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 4
<Item><M>-2-b_{17}</M></Item>
<Item><M>-</M></Item<ef =S" "">
</Row>
<
</Table>



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



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



<Example><![CDATA[
gap> table:= [];>fi
gap> for pair in [ [ 2, [ 78, 80(act ;
> [ 19, [ 110, 214, 706 ] ] ] do
> p:= pair[1];
> for d in pair[2] do
> info:= OneAtlasGeneratingSetInfo( "J3", Characteristic, p,
> Dimension, dfi
> gens:= AtlasGenerators( info );
>[ 31,
> gens.generators );
> val:= BrauerCharacterValue( reps[ 17a ] );
> Add( table, [ p, d, Quadratic( val ).ATLAS,
> Quadratic( StarCyc( 11 13 ,
>
> od;
gap> PrintArray( table );
[ [ 2, 78, 1-b17, 2+b17 ],
 [ 2, 80, 3-b17, 4+b17 ],
 [ 2, 244, -2+b17, -3-b17 ],
 [ 2, 966, -3+r17, -3-r17 ],
 [ 19, 110, -1-b17, b17 ],<M>M( L_2</M>.
[19 214 2b17,1b17 ]java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
 [ 19, 706, 1+b17, -b17 ] ]
]]></Example>

P

We see that the problem is an inconsistency between the <M>2</M>-modular and
the <M>:(pair )
nKeyJLPW95">
In particular, changing the straight line program would not help to
resolve the problem.

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

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
and to swap the two classes of element order <M>17</M> in the
<M>2</M>-modular table;
then also the straight line program has then also the straight line program has to
and the classes of element orders <M>17</M> and <M>and the translationlattice is the sublattice <>L_2<M of<>&Z;d/>
in the <M>2</M>-modular character table of the triple cover <M>t has following basis,
of <M>J_3</M> have to be adjusted.
Alternatively, we can keep the <M>2</M>-java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 0
and the straight line program,
and adjust the conjugacy classes of element orders divisible
by <M>17</M> in the <M>19</M[0,,0 ,0 ,1 ],
<M>3.J_3</M>, <M>J_3.2</M>, and <M>3.J_3.2]<Example



decide the<>9<M>modularcharacter.
Note that these character tables —or equivalently, the corresponding
Brauer trees— have been described in <Cite Key="HL89"/>,
where explicit choices are mentioned that lead to the shown Brauer trees.
java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 74
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(Only the consistency of the Brauer trees among the <M
<M>3.J_3</<M>3.J_3</MThe - of involvedMp/>modular
In fact, the book mentions that the <M>19</Mgap sgens= ( [ abas,cbas ]v t)java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
<java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 38
 java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 72
has apparently been overlooked when the> (, "13G1/L2/ext2";
have been put together, and had not

<P/< Label"ubsect:xpl->

<E>Remarks:</E>

<List>
<Item>
 Such a change of a Brauer gap> : [[,- ,0 0,0,,0, ,
 from and to this table.
 Note that Brauer tables may impose conditions on the choice of the fusion
 among possible fusions , ,,, , ,
 table automorphisms of the ordinary table.
 factnohad java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69

 Ref=java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 20
 \{} java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 77
</Item>
<>
 The change of the character tables affects the decomposition matrices.
 20<><20> <11/>M<
 matrices had java.lang.StringIndexOutOfBoundsException: Range [0, 17) out of bounds for length 0
 <URLwww-java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 89
< java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 72
<Item>
 Jürgen Mügap> bas2:= [ - ,,,, ]
 <></> <M>._3<M,<J_3./> <3J3./>are consistent
 after the fix described above.
</Item[,,,,,1- ,0]
< java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 67

<


<Subsection Label=" ,0-,-,- -,- ,
<Heading>A generality problem java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 78

The>[1-,1,1 ,1-1,1-1 ]
in the java.lang.StringIndexOutOfBoundsException: Range [55, 54) out of bounds for length 77
determinedbyunique Brauer > [1,-1-1,1,,1 ,0-,1]];
on these classes.
Once we have <E>defined</E> theseNote java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 0
two java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 72

Thus the question is whether the two Brauer tables are consistent
w.r.t. this property or not.



(Note that this question can be answered independently of all other
java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 37
because the java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 4
<C>20B</C> is a table automorphism of the java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 0
<M>HN</M>.)



We start with thegapjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
There are exactly two ordinary irreducible characters
that on classes C20A<> java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
these are <M>\chi_{51}</M> and thejava.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 66



<Example<[[
>t: ( " (,"//5//")
gap> pos20:= Positions( java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 4
[ 39, 40, 41, 42, 43 ]
gap> diff:= Filtered( Irr( t ), x -> x[
gap>(diff,x -(Irr(t, ) )
[ 51, 52 ]
]]></Example>

<>

These values are java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
the square root of <M>5</M>.

> ,,,,]

<Example><![CDATA[
java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 47
> CTblLib.StringOfAtlasIrrationality ) );
[ [ "5103000"
]]></Example>

</

In each of the characteristics <M>p \in \{ 11, 19 java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
the <M>p</M>-modular reductions of <M the<>= ofjava.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 24
decompose differently into irreducibles.
Note
that maps <M>\sqrt{5}</M>>= 2 , ,
< ,,, ,,0] ;java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38



For <M>p = 11</M>, the reduction of <M>\chi_{51}<M( )M, the <>p/-java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 57
<M>\varphi_{40} + \varphi_{48}</<
with <M>\varphi_{40}(1) = The action .
<M>\varphi_{48}(1) = 3\,527\,824</M>,
and the reduction of <M>\chi_{52 Ref java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 70
<M>\varphi_{39} + \varphi_{49}</M>,
with <M>\varphi_{39}(1) = 1\,361\,919</M> and
<M>\varphi_{49}(1) = 3\,741\,081</M>.



<Example><![CDATA[
gap> , ,,,0
=java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 53
gap> dec11:= Decomposition( Irr( t11 ), rest11, "nonnegative" );
gap> List( dec11, Set );
[ [ 0, 1 ], [ 0, 1 ] </List>
gap> List( dec11, x -> Positions( x, 1 ) );
[ [ 40, 48 ], [ 39, 49 ] ]
(( { 40 ] >x1 java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
[ 1575176, 3527824
, 49 ]} 1])java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
[ 361919 3741081 ]
]]></Example>



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

M>,
with <M>\varphi_{42}(1) = 2\,125\,925</M> and
<M>\varphi_{45}(1) = 2\,977
and the reduction of <M>\chi_{52}</M> is
<M>\varphi_{33} + \varphi_{48}</M>,
with <M>\varphi_{<Subsection Label=subsectexpl-M11">
<>\arphi_{48}() = 3\,905\,670<M>



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



The Frobenius-Schur indicators of all involved <M>p</M>-modular constituents
are <M>+<M> java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 68
This implies that <M>\chi_{51}</>
but not orthogonally stably modulo <M>19</M>,
whereas <M>\chi_{52}</M> reduces orthogonally stably modulo <M>19</M>
gap:java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 65



TA[
gap> Indicator( t11, 2 ){ [ 39, 40, 48, 49 ] };
[ 1, 1, of in <><M
gap> Indicator( t19java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ 1 ,,
]]></Example>



this condition was not satisfied:
The reduction of <M>\chi_{51}</M> modulo both <M>11</M> and <M>19</M> was
orthogonally stable,
and> ,0 ,,0 ,1 ]
not orthogonally stable.
However, this cannot [ 1, 0,-1, 1, 1 ,-, ,,, ,
orthogonal discriminants of the involved characters.
 a way to decide the consistency of the classes
<C>20A</C> and <C>20B</C> in characteristics 0,1-,0 0 ];java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
Either the <M>11</M>-modular
character table of <M>HN</M> had to be and the translation lattice is <M>L = 3 &ZZ;^d</M>.
by swapping<>



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



Forjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
There are java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 77
swapped if <C>20A</C> :1];
Thus we have to change java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 2



Changes of this kind [,6 ,1 ,7,0 1 java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
In this
was changed as well.
Note that this table is not stored explicitly in the data files,
it gets constructed from ordinary and modular library tables via
<Ref Func="ConstructIndexTwoSubdirectProduct"

</Subsection>

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


< sderivebrauercharacters">
<Heading>Brauer Tables that can be derived from Known Tables</Heading>

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



For quite some time, a method is available in &GAP; >[,10 ,, ,
the Brauer characters of <M>p</M>-solvable groups
(see <Ref Subsect="BrauerTable" BookName="ref"/> and
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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

 java.lang.StringIndexOutOfBoundsException: Range [34, 30) out of bounds for length 70
<Subsection Label="sect:brauercharactersbyconstruction">
> viajava.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 61

a <tM,,
has been constructed from other ordinary character tables
then &GAP; may be able to create the <M>p</M>-modular Brauer table
of <M>t</M> from the <M>p</M>-modular Brauer tables of the
<Q>ingredients
This happens currently

</

<List>

 <M
 <=""="ef">
 GAP <><M
 of the direct factors.
</Item>
>
 <M>t</M> has been constructed with
 <Ref Oper="CharacterTableIsoclinic" BookName="ref"/>,
 and &GAP; can compute the <M>p</M>-modular Brauer table
 java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 71
 <Ref Oper="SourceOfIsoclinicTable" BookName="ref"/>.
</Item>
<Item>
M></M
 <Attr= =cjava.lang.StringIndexOutOfBoundsException: Range [63, 62) out of bounds for length 70
 the first entry of this list <M>l</M>, say, is one of the propertyjava.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 55
 <C>"ConstructGS3"</C> (see
RefSubsectsharacterof .3>
 C"<>
 (see <Ref Subsect="subsect:theorsubdir"/>),
 <C>"ConstructMGA"</C> (see <Ref Subsect="subsect:theorMGA"/>),
 <C>"ConstructPermuted"</C>,
<23/> M23/><>^<, M53/> java.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 65
 and &GAP; can construct the <M>p</M>-modular Brauer
 of the relevant ordinary character table(s),
 which are library tables whose names occur in <M>l</
<Subsection Label="sect:liftablebrauercharacters">
<Heading>Liftable Brauer Characters (May 2017)</Heading>

Let <M>B</M> be a <M>p</M>-block of cyclic defect for the finite group
<M>Gjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
It can be read off from the set Irr<M>(B)</M>
ordinarystingpossible generality problems</Heading>
whether all irreducible Brauer characters in <M>B</M> are restrictions
of ordinary characters to the <M>p</M>-regular classes of <M>G</M>,
as follows.



If <M>B</M> has only one irreducible Brauer character then all ordinary
in the sense that some symmetry of the fusions of <M>G</M> in <M>&M;</M> and then to check
So let us assume that <M>B</M> contains
at
and consider the set <M>S</M>, say, of restrictions of Irr<class fusionunder the of table automorphisms of <>tbl<M>,
eMp<M>-regularclasses of MG/M>

</

The block <M>Bsplits several orbitswhen the action
characters,
and the decomposition of the characters in <M>S</M> into these Brauer
characters is described by
an <M>|S|</M> by <M>|S| - 1</M> matrix <M>M</M>, say,
whose entries are zero and one, such that exactly two nonzero entries
occur in each column.
(See <Example>![[
whichrefersto<ite taut java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 39



fall irreducibleBrauer characters of <M>B<gap>s= CharacterTable "U42));
then the matrix <M>M</M> contains <M>|S| - 1</M> rows that contain
exactly one end;
hence the remaining row consists only of <M>1</M>s.
This means that the element of largest degree in <M>S</M> is equal to
sum all M<>
Conversely, if the element of largest degree in <M>S</M> is equal to
the sum of all other elements in <M>S</M> then
the matrix <M>M</M> has the structure as stated above,
hence allbetween the tables <\sigma_mMjava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47

<

Alternatively that java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 45
one could state that all irreducible Brauer characters of <M>B</M>
arerestricted java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 72
of <M>B</M> is a <E>star</E> (see <Cite Key="HL89" Where="p. Thus to java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 76
If <M>B</M> contains at least two irreducible Brauerjava.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 60
then this happens if and only if one of the types <Item
<M>\circ</M> occurs for exactly one node in the Brauer graph of <M>B</M>,
see <Cite Key="HL89" Where="Lemma 2.1.13"/>,
and the distribution to types is determined by Irr<java.lang.StringIndexOutOfBoundsException: Range [0, 52) out of bounds for length 26

<java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 18

The default method for <Ref Oper="BrauerTableOp" BookName="ref"/>
that is contained in the &GAP; library has been extended in version 4.11
such that >: (, , (decompose: false));
<M>G</M> are cyclic and, if yes> ;
allp>of/ java.lang.StringIndexOutOfBoundsException: Range [58, 57) out of bounds for length 74
(This feature arose from a discussion with Klaus Lux.)



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



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



such the<There was <URL><LinkText>the question in MathOverf>
of the Mathieu group <M>M_{11}</M>.

/>

<Example><![CDATA[
gap> lib:= CharacterTable( "M11" );;
gap> fromgroup:= X.11 10 10 -2 1 1 .2 11 1-java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
gap> DecompositionMatrix( lib mod 5 )java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ [ 1, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
 [ 0, 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
 java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 45
 ,=127+)java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
 [ 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ (x /Size ));
> >
true
]]></Example>



s of ablock
but thedefect ofthe blockisjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
thus the method cannot be used.
An example is the <M>2</M>-modular table of the java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 40

P

<Example><! java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 32
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 ,0][,,,,0]
 [ 0,,,, ,[,0 , ,1 ,[1 , ,java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
 [ 1, 1, 0, 0, 1 ] ]
gap> fromgroup mod 2;
fail
]]></java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 6

</Subsection>

</Section java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 45


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


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

In the context of a question about decomposition numbers of the
sporadic java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0
Benjamin Sambale was
<M>G</M> into <M>&M;</M> such that the decomposition matrices> triv:=TrivialSubgroup taut ;
are known.
For a given <M>G</M>, the first steps were to compute the possible class
fusions of <M>G</M> in <M>&M;</M> and then togap> ( java.lang.StringIndexOutOfBoundsException: Range [52, 49) out of bounds for length 63
embeddings would be interesting.



Apparentlycalling Ref=java.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 69
with its default parameters often runs very long and requires a lot of space
when <M>G</M> is a small group such as <M>2.U_4(2)</M>.
We can do better by calling the function with the The group <M>3.G</M> centralizes an element of order three. ( allfusions ) do
<C>decompose:= false</C>.
This has > SubtractSet( java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
decomposability of restricted characters of <M>&M;</M> as an integral
linear combination of characters of the subgroup.
As a > Set (, ),
compared to the number of classes of the group
and if the result consists We have ageneralityproblem.
to omit the decomposability criterion.

<P[, ]]java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61

<ample>![DATA[
gap> s:= CharacterTable( "2.U4(2)" );;
gap> m:= CharacterTable( "M" );;
gap> sfusm:= PossibleClassFusions( s, m, rec( decompose:= false ) ) [ soclinic(12_2.L3(4).2_3), [ 2, 3, 7 ] ],
gap> Length( sfusm );
2332
]]></Example>





Looking at the (many) candidates,
we see that all map the central involution of
 [ Isoclinic(2Fi22.2) 11 13] ],
thusany subgroup of typetype2._(2)</>lies inside the <C></C>
normalizer in <M>&M;</M>.
We compute the possible class fusions into this subgroup.



<Example><![CDATA[
gap> Set( List( sfusm, x -> x[2] ) );
[ 3 ]
gapIsoclinic(_..1, , 7]],
CharacterTable( "2^1+24.Co1" )
gap> sfust:= PossibleClassFusions( s, t, rec( decompose:= false ) );;
gap> Length( sfust );
0
]]></Example>

java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63



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

</Subsection>


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

There was <URL><LinkText>the question in MathOverflow</LinkText>
inkhttps:/mathoverflow.net/uestions/450255<Link<URL
which perfect central extensions of the simple group <M>G = L_3(4)</M> are
subgroups of the sporadic simple Monster group <M>&M;</M>.



First we get the list of 1a 1a 2a b 8 2b
their character tables are contained in the character table library).

<Example><![CDATA[
gap> simp:= CharacterTable( 3 1 11 -1 -2 -2
gap> extnames:= AllCharacterTableNames( Identifier,
> x -> EndsWith(x, "L3(4)" ) );;
gap> ext:= List( extnames, CharacterTable );;
gap> ext:= Filtered( ext, x -> Length( ClassPositionsOfCentre( x ) ) =
> Size( x ) / Size( simp ) );;
gap> SortBy( ext, Size );
 )java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
[ "L3=Sqrt(-)=i2
 "4 [ Sqrt(3)=i3
 "12_1.L3(4)", "12_2.L3(4)", "4^2.L3 "12_1.L3(4)", "G L5(., ,,31 ]]
 "(4^2x3).L3(4)" ]
]]></Example>



The fact that <M>G</M> is <E>not</E> isomorphic to a subgroup of <M>&M;</M>
is shown in <Cite Key="HW08"/> (at the end of this paper).





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

<List>
<Item>
 <M>2.G < 2.U_4(3) < 2^2.U_6(2) < Fi_{23} < 3.Fi_{24}' < &M;</M>.
</Item>
<Item>
 <M>2^2.G < He < 3.Fi_{24}' < &M;</M>.
</Item>
<Item>
 <M>6.G < 2.G_2(4) < 6.Suz java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 19
</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 >s java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 34
and <
The positive statements follow from <Cite Key="CCN85" Where="pp. 52, 97, 131"/>
and the negative ones from the following computations.



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



The group <M>3.G</M> centralizes an element of order three.
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}_+.>fi;
a <C>3C</C> centralizer (of the structure <M>3 \times Th</M>).
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>
the>3B>M>3^112}..<> java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 66



<Example><![CDATA[
gap> t:= [ [ CharacterTable "4_1.L3(4)"),CharacterTable( "2^1+24.Co1" ), 3 ]
CharacterTable( "3^(1+12).2.Suz.2" )
gap> Length( PossibleClassFusions( CharacterTable( "3.L3(4)" ), t ) );
0
]]></Example>



If <M>3.G</M> iscontained 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 table automorphisms; there are some huge Brauer tables that are
in GAP's character table library,
one shows that only <M>Fi_{23}</M> admits a class fusion,
 lifts to<M>3 \imesFi_23<M in <M>.Fi_{24}'>
thus lead oftypeG<>java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55



<Example><![CDATA[
java.lang.StringIndexOutOfBoundsException: Range [77, 1) out of bounds for length 77
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< Not">
 thefusionjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
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>.

<![CDATA[
gap> done:= [ "L3(4)", "2.L3(4)gap> 17a= slp.utputs,"A ;
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)"gap>=( ;java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
 "(4^2x3).L3(rec( : -1-, =- =- =,
gap> invcent:= List( [ "MN2A", "MN2B" ], java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 44
[ 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 ...
name in names do
> s:= CharacterTable( name );
> ords:= OrdersClassRepresentatives( s );
> invpos:= Filtered( ClassPositionsOfCentre( s ), i -> ords[i] = 2 by
> for i in invpos do
> for t in invcent do
> init:= InitFusion( s, t );
> if init = fail then
> continue;
> fi;
> init[i]:= 2;
> fus:= PossibleClassFusions( s, t, rec( fusionmap:= init,
> decompose:= false ) );
> if fus <>>AltOnly=LaTeX">able~\ref{valuesJ3}</Alt>
> Add( cand, [ s, t, i, fus ] );
> fi;
> od;
> od;
> od;
gap> List( cand, x -> x{ [ 1 .. 3 ] } );
[ [ CharacterTable( "4_1.L3(4)" ), CharacterTable( "2^1+24.Co1" ), 3 ]
 , [ CharacterTable( "(2x4).L3(4)" ), CharacterTable( "2.B" ), 2 ],
 [ CharacterTable( "(2x4).L3(4)" ), CharacterTable( "2.B" ), 3 ] ]
]]></Example>





(NoteCaption>Representations of <M>J_3</M> that may define <C>17A</C></Caption>
with the option <C>decompose:= false</C>, in order to save space and time.
See Section <Ref Subsect="sect:no2U42inMonster"/> for more details.)



Concerning the candidate <M>(2 \times 4).G</M>,
we see that only fusions are possible for which the central involution
in question is mapped to a <C>2A</C> element of <M>M>1b_{}</M<Item
Since we get candidates only for two out of the three central involutions,
we see that <M>(2 \times 4).G</M> does not embed into <M>&M;</M>.



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



Moreover, we see that if this happens then the centre <M>C</M>
of <M>4_1.G</M> lies inside the normal subgroup <M>N = 2^{1+24}_+</M>.
The centralizer of <M>C</M> in <M>N</M> has order <M>2^{24}</M>,
and the centralizer of <M>C</M> in <M>2^{1+24}_+<>M+/></temjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
<M>2^{24} \cdot |Co_3|</M>.
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 the subgroup <M>[2^{24}].Co_3</M>.

</

<Example><![CDATA[
gap> s:= cand[1][1];
CharacterTable>
gap> t:= cand[1][2];
CharacterTable( "2^1+24.Co1" )
gap> fus:= cand[1][4];
[ [ 1, > := pair1;
 70, 74, 71, 74, 114, 119, 115, 119, 114, 119, 115, 119 ] ]
gap> ClassPositionsOfCentre( s );
[ 1, 2, 3, 4 ]
gap> 5 in ClassPositionsOfPCore( t, 2 );
true
gap> siz:= SizesCentralizers( t )[5] / 2^24;
495766656000
gap> mx:= Filtered( List( Maxes( CharacterTable( "Co1" ) ),
> CharacterTable ),
> x -> Size( x ) mod siz = 0 );
[ CharacterTable( "Co3" ) ]
gap> Size( mx[1] ) = siz;
true
]]></Example>





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

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


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

Consider a [ 19, 1,-1b17 b17 ]
in the sporadic simple Monster group.
The character table of <M>H</M> has [ 19, 706, 1+b17, -b17
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
32,734)/>

</>

<Example><![CDATA[
gap> tblH:= CharacterTable( "(3^2:2xO8+(3)).S4" );
CharacterTable( "(3^2:2xO8+(3)).S4" )
gap> N:= ClassPositionsOfSolvableResiduum( tblH );;
gap> tblF:= tblH / N;;
gap> Sizea M<> M><>
432
gap> known:= explicit choices are mentioned that trees
[ "3^2.2.S4", "M12M7" ]
gap> Filtered( GroupInfoForCharacterTable( known[1] ),
> x -> x[1] = "SmallGroup" );
[["SmallGroup",[432,734 ]]java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
]]></Example>



(Note that the precomputed
<Ref Attr="GroupInfoForCharacterTable" BookName="CTblLib"/>
information about &GAP; library character tables means that exactly
one isomorphism type of



We compute that <M>O_3(F) \cong 3^2</M> java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
thus <M>H</M> has a subgroup <M>V</M> of the type <M>O_8^+(3).2S_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 <M>V</M> from that of <M>H</M>,
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<\}+{45<>
of order <M>2 |O_8^+(3)|</M> in <M>V</M>.
If this normal subgroup would not be isomorphic with <M>2 \times O_8^+(3)</M>
then it would have one of the structures <M>O_8^+(3).2_1</M><>\java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 35
<M>O_8^+(3).2_2</M>,
but then it would contain elements of java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0



<xample<[java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 18
gap> tblV:= tblH / ClassPositionsOfPCore( tblH, 3 );;
gap> ord:= 2 * Size( tblH ) / Size, >];
9904359628800
gap> classes:= SizesConjugacyClasses( tblV );;
gap> 2N:= Filtered( java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 13
> l -> Sum( classes{ l
gap> Length( 2N );
1
gap> Set( OrdersClassRepresentatives( tblV ){ 2N[1] } );
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 18, 20, 26, 30 ]
gap> Set( OrdersClassRepresentatives( CharacterTable( "O8+(3)" ) ) );
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 18, 20 ]
gap>java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
[ 1, 2[>Indicator( {[39 ,48 ;
 30, 36, 40 ]
gap> Set( OrdersClassRepresentatives( CharacterTable( "O8+(3).2_2" ) ) );
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ]>/>
 30, 36 ]
]]></Example>



The class fusion of <M>V</M> into the Monster group shows that <M>V</M>
centralizes a <C>2A java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 77
hence <M>V</M> is a subgroup of a maximal subgroup of the type <M>2.B</M>.



<Example><![CDATA[
gap> tblM:= CharacterTable( "M" );;
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 java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 0
we see that either <M>V order <>40/> tobe
under M><> >BM>
or <M>V</M> is equal to the preimage we to java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 78



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



The former possibility is excluded from the fact that the factor of <M>V</M>
by its center does not admit a class fusion into <M>Fi_{23}</M>.



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



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

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

Thus we have used the character table of <M>H</M> to construct
the character table of the character table of a
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: Index 2 out of bounds for length 0
true
gap> Irr( lib ) = Irr( tblV );
true
]]></Example>



ordertoadd the table the library,
we have to provide also the class fusions has been constructed from other ordinary character tables
to the maximal subgroup <M>(2 \times O_8^+(3)).S_4</M> of <M>B</M>,
and to the maximal subgroup <M>(^2:2\times O_8^+3)).<M <>M/,
such that the compositionsM- Brauer tablesof the
via <M>O_8^+(3).S_4</M> and <M>2.B</M> are compatible, <M>\ldots</M>



<Example><![CDATA[
gap> tblU:= CharacterTable( "O8+(3).S4" );;
gap> tbl2B:= CharacterTable( "2.B" );;
gap> CompositionMaps( GetFusionMap( tblU, tblB ),
> GetFusionMap( lib, tblU ) ) =
>
> , java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
true
]]></Example>



> java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 76
via <M>(3^2:2 \times O_8^+(3)).S_4</M> and <M>2.B</M> are compatible.



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

</Subsection>
</Section>
</Chapter>

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