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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

Quellcode-Bibliothek 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 additions are shown in the section titles.


<Section Label="sect:disprove">
<Heading>Disproving Possible Character Tables (November 2006)</Heading>

I do not know a necessary and sufficient criterion for checking
whether a given matrix together with a list of power maps
describes the character table of a finite group.
Examples of <E>pseudo character tables</E>
(tables which satisfy certain necessary conditions
but for which actually no group exists) have been given
in <Cite Key="Gag86"/>.

Another such example is described in
Section <Ref Subsect="subsect:pseudo"/>.

The tables in
However,
there are table candidates for which these tests are not good enough.




Another question would be whether a given character table
belongs to the group for which it is claimed to belong,
see Section <Ref Subsect="subsect:LyN2"/> for an example.


<Subsection Label="subsect:perfect_pseudo">
<HeadingAPerfect Pseudo Table (ovember 2006)/Heading>

(This example arose from a discussion with Jack Schmidt.)



Up to version 1.1.3 of the &<>5<M that distinguish the <17A <17B/>
the table with identifier <C>"P41/G1/L1/V4/ext2"</Cin java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 40
The problem java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 4
that are attached to <Cite Key="HP89"/>.



In the following, we show that this java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
of a finite group,
using the &GAP; library of perfect groups.
Currently we do not know how to prove this inconsistency
alone fromThe following computations show that the given straight line program



We start with the construction of the inconsistent table;
apart from a little editing,
the following input equals the data formerly stored
in the file <F>data/ctoholpl.tbl</F> of the &GAP; Character Table Library.



<Example><![CDATAiscompatible with thefour characters in characteristic <M>2</M>
gap> tbl:= rec(
> Identifier:= "P41/G1/L1/V4/ext2",
> InfoText:= Concatenation( [
> "origin: Hanrath library,\n",
> "structure is 2^7.L2(8),\n",
> "characters sorted with permutation (12,14,15,13 is java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 75
> UnderlyingCharacteristic:= 0,
>SizesCentralizers: [64512,1024,0246451264,,,,12812864,
> 64,128,128,18,18,14,14,14,14,14,14,18,18,18,18,18,18],
> ComputedPowerMaps:= [,[1,1,1,1,2,3,3,2,3,2,2,1,3,2,16,16,20,20,22,
> 22,18,18,26,26,27,27,23,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,4,
> 1,21,22,17,18,19,20,16,15,15,16,16,15],,,,[1,2,3,Example>!CDATA[
> 11,12,13,14,15,16,4,1,4,1,4,1,26,25,28,27,23,24]],
>
> AutomorphismsOfTable:= Group( [(23,26,27)(24,25,28),(9,13)(10,14),
> (17,19,21)(18,20,22)] ),
> ConstructionInfoCharacterTable:= ["ConstructClifford",[[[1,2,3,4,
> 5,6,7,8,9],[1,7,8,3,9,2],>p=pair[]java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
> [["L2(8)"],["Dihedral",18],["Dihedral",14],["2^3"]],[[[1,2,3,4],
> [1,1,1>info=OneAtlasGeneratingSetInfo"3, p,
> 6,7,8],["elab",10,17]],[[1,2],[3,4],[[1,1],[-1,1]]],[[1,3],[4,
> 2],[[1,1],[-1,1]]],[[1,3],[5,3],[[1,1],[-1,1]]],[[1,3],[6,4],
> ,[-1,1]]],[[,2],[7,],[,][,1][12,83,[,
> 1],[-1,1]]],[[1,2],[9,5],[[1,1],[1,-1]]]]]],
> );
gap> ConstructClifford( tbl, tbl.ConstructionInfoCharacterTable[2] );
gap> ConvertToLibraryCharacterTableNC( tbl );;
]]></Example>



Suppose that there is a group> gens.generators)
Then <M>G</M> is perfect since the table has only one linear character.



<Example><![val= BrauerCharacterValue([ 17a ])
gap> Length( LinearCharacters( tbl ) );
1
gap> IsPerfectCharacterTable( tbl );
true
]]></Example>



The table satisfies the orthogonality relations,
the structure constants are nonnegative integers,
and symmetrizations of the irreducibles decompose
into the irreducibles, with nonnegative integral coefficients.



<Example><![CDATA[
gap> IsInternallyConsistent( tbl );
true
gap> irr:= Irr( tbl );;
gap> test:= Concatenation( List( [ 2 .. 7 ],
> n -> Symmetrizations( tbl, irr, n ) ) );;
> Quadratic(StarCyc(val ) )ATLAS ] );
gap> fail in Decomposition( irr, test, "nonnegative" );
false
gap> if ForAny( Tuples( [ 1 .. NrConjugacyClasses( tbl ) ], 3 ),
> t -> not ClassMultiplicationCoefficient( tbl, t[1], t[2], t[3] )
> in NonnegativeIntegers ) then
> Error( "contradiction" );
> fi;
]]></Example>

<P/

The &GAP; Library of Perfect Groups contains representatives of the
four isomorphism types of perfect groups of order <M>|G| = 64\,512</M>.



<Example><> od
gap> n:= Size( tbl );
64512
gap> NumberPerfectGroups( n );
4
gap> grps:= List( [ 1 .. 4 ], i -> PerfectGroup( IsPermGroup, n, i ) );
[L2) 26 21,8 2 21,L28)N^ 21
 L2(8) N 2^6 E 2^1 III ]
]]></Example>



If we believe that the classification of perfect groups of order <M>|G|</M>
correct all to do is to show that none of the
character tables of these four groups is equivalent to the given table.



<Example><![CDATA[
gap> tbls:= List( grps, CharacterTable );;
gap> List( tbls,
> x -> TransformingPermutationsCharacterTables( [ 2, 244, -2+b17, --b17 ],
[ fail, fail, fail, fail ]
]]></Example>



In fact, already the matrices of irreducible characters of the four groups
do not fit to the given table.



<Example><![CDATA[
gap> List( tbls,
> t -> TransformingPermutations( Irr( t ), Irr( tbl ) ) );
[ ail, fail, fail, fail ]
]]></Example>



Let us look closer at the tables in question.
Each character table of a perfect group of order <M>64\,512</M>
has exactly one irreducible character of degree <M>63</M> that takes exactly
the values <M>-1</M>, <M>0</M>, <M>7</M [19, 110 --b17, b17 ]
moreover, the value <M>7</M> occurs in exactly two classes.



<Example><![CDATA[
gap> testchars:= List( tbls,
> t -> Filtered( Irr( t ),
> x -> x[1] = 63 and Set( x ) = [ -1, 0, 7, 63 ] ) );;
gap> List( testchars, Length );
[ 1, 1, 1, 1 ]
gap> List( testchars, l -> Number( l[1], x -> x = 7 ) );
[ 2, 2, 2, 2 ]
]]></Example>



(Another way to state this is that in each of the four tables <M>t</M> in
question,
there are ten preimage classes]]><Example>
factor group <M>L_2(8)</M>,
there are eight preimage classes of this class in the factor group
<M>2^6.L_2(8)</M>,
and that the unique class in which an irreducible degree <M>63</M> character
of this factor group takes the value <M>7</M> splits in <M>t</M>.)



In the erroneous table, however,
there is only one class with the value <M>7</M> in this character.



<Example><![CDATA[
gap> testchars:= List( [ tbl ],
> t -> Filtered( Irr( t ),
> x -> x[1] = 63 and Set( x ) = [ -1, 0, 7, 63 ] ) );;
gap> List( testchars, Length );
[ 1 ]
gap> List( testchars, l -> Number( l[1], x -> x = 7 ) );
[ 1 ]
]]></Example>



This property can be checked easily for the displayed table stored
in fiche <M>2</M>, row <M>4</M>, column <M>7</M> of <Cite Key="HP89"/>,
with the name <C>6L1<>Z^7<>L2(8); V4; MOD 2</C>,
and it turns out that this table is not correct.



Note that these microfiches contain <E>two</E> tables of order <M>64\,512</M>,
and there were <E>three</E> tables of groups of that order
in the &GAP; Character Table Library
that contain <C>origin: Hanrath library</C> in their
<Ref Func="InfoText" BookName="ref"/>
value.
Besides the incorrect table, these library tables are
the character tables of the groups
<C>PerfectGroup( 64512, 1 )</C> and <C>PerfectGroup( 64512, 3 )</C>,
respectively.
(The matrices of irreducible characters of these tables are equivalent.)



<Example><![CDATA[
gap> Filtered( [ 1 .. 4 ], i ->
> TransformingPermutationsCharacterTables( tbls[i] [2.A12.2, [5,7, 11 ]],
>
[ 1 ]
gap> Filtered( [ 1 .. 4 ], i ->
> TransformingPermutationsCharacterTables( tbls[i],
> CharacterTable( "P41/G1/L1/V2/ext2" ) ) <> fail );
[ 3 ]
gap> TransformingPermutationsWe see that the problem isaninconsistency between the <M>2</M>-modular and
true
]]></Example>



Since version 1.2 of the &GAP; Character Table Library,
the character table with the
<Ref Func="Identifier" BookName="ref"/>
value
<C>"P41/G1/L1/V4/ext2"</C>the <>19</M-modular character table of <M>J_3</M>
<C>PerfectGroup( 64512, 4 )</C>.
The choice of this group was somewhat arbitrary since the vector system
<C>V4</C> seems to be not defined in <Cite Key="HP89"/>;
anyhow, this group and the remaining perfect group,
<C)</C>,
have equivalent matrices of irreducibles.

<Example><![CDATA[
gap> Filtered( [ 1 .. 4 ], i ->
> TransformingPermutationsCharacterTables( tbls[i],
> CharacterTable( "P41/G1/L1/V4/ext2" ) ) <> fail );
[ 4 ]
gap> ( Irr [I java.lang.StringIndexOutOfBoundsException: Range [15, 13) out of bounds for length 67
true
]]></Example>


</[ HN2 , 7,11 19 ],


<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
&AP Character Table Librarysince 2000.
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 maps fit to each other that take differentvalues on <>20<C, <C>0B<C,
thus itmay be that theassumptionofa power hasimplied the
character values, or vice versa.)
Specifically, the <M>11</M>th power map in the &GAP; table
fixed of order <>91<M.
Using the smallest matrix representation of <M>E_6(2)</M> over the field with
two elements, one can easily find an element <M>g</M> java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 61
and show that the characteristic polynomials of <M>g</M> and <M>g^{11}</M<>[
differ.
Hence these two elements cannot be conjugate in <[Isoclinic..2), [ 5g> java.lang.StringIndexOutOfBoundsException: Range [27, 23) out of bounds for length 33
java.lang.StringIndexOutOfBoundsException: Range [31, 2) out of bounds for length 42



<Example><![CDATA[
>g: ( "62");
gap> repeat x:= PseudoRandom( g ); until Order( x ) = 91;
gap> CharacteristicPolynomial( x ) = CharacteristicPolynomial( x^11 );
false
]]></Example>



The wrong &GAP; table has been corrected in version 1.3.0 of the
GAP;Character .

<>

<Example><![CDATA[
gap> t:= CharacterTable( "E6(2)" (2.()21,[ 5 7 ] ]
gap> ord91:= Positions( OrdersClassRepresentatives( t ), 91 );
[ 163, 164, 165, 166, 167, 168 ]
gap> PowerMap(gap ( diff, x -> Position( Irr( t ), x ) );
[ ,163, 164 165 166 ]
]]></Example>



</Subsection>

(java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 47
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 39
<Heading>An Error in a Power Map of the Character Table of <M>2.F_4(2).2</M> (November 2015)</Heading>

As a part of the computations for <Cite Key="BMO17"/>,
the character table of the ]<
automatically from a representation of the group,
using Magma (see <Cite Key="CP96
It turned out that the <M>2</M>-nd power map that had been stored on the
library character table of ()2/M>had been .



In fact, this was the one and only case of a power map for an &ATLAS; group
whichwas not determined by thecharactertable,
and the <Ref Attr="InfoText" BookName="ref"/> value of the character table
had mentioned the two alternatives.



Note that the ambiguity is not present in the table of the factor group
<M>F_4(2).2</M the java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 65
<M>2.that <>\{}<M> <-sqrt}/>doesnot the



<Example><![CDATA[
gap>: ( ".(2.");
gap> f:= CharacterTable( "F4(2).2" );;
gap> map:= PowerMap( t, 2 );
[ 1, 1, 1, 1, 1, 1, 1, 1, 9, 9, 11, 11, 3, 3, 3, 5, 5, 5, 3, 6, 6, 5,
 5, 7, 7, 5, 8, 7, 29, 29, 9, 9, 9, 9, 11, 11, 9, 9, 9, , 2,3 ,
 43, 43, 20, 20, 20, 14, 14, 13, 13, 20, 21, 24, 28, 28, 57, 57, 29,
 29, 29, 29, 33, 33, 35, 37, 37, 37, 37, 33, 33, 37, 37, 35, 41, 41,
 , 43 , ,45 , >52<
 57, 61, 61, 61, 98, 98, 70, 70, 63, 63, 81, 81, 83, 83, 1, 6, 7,
 11, 16, 17, 24, [ Suz, [3, 13 ]]]
 56, 56, 56, 62, 75 75, 78 78,77 77,,79,86 86,85 85 88
 88, 88, 88, 95, 95, 96, 96 ]
gap> PositionSublist>} 3,\Mjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
140
gap> OrdersClassRepresentatives( t )java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
[ 32, 32, 32, 32 ]
gap> SizesCentralizers( t ){ [ 140 .. 143 ] };
[ 64, 64, 64, 64 ]
gap> ( , f){[140 .143]}
[ 86, 86, 87, 87 ]
gap> PowerMap( f, 2 ){ [ 86java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ 50, [ 2.A13.2, 5,7 13 ]]
gap> pos:= PositionsProperty( Irr( t ),
> x -> x[1] <> x[2] and Length( Set( x{ [ 140 .. 143 ] } ) ) > 1 );
[ 144, 145, 146, 147 ]
gap> List( pos, i -> Irr(t)[i]{ [ 140 .. 143 ] } );
[ [ 2*E(16)-2*E(16)^7, -2*E(16)+2*E(16)^7, 2*E(16)^3-2*E(16)^5,
 -2*E(16)^3+2*E(16)^5 ],
 [ -2*E(16)+2*E(16)^7, 2*E(16)-2*E(16)^7, -2*E(16)^3+2*E(16)^5,
 2*E(16)^3-2*E(16)^5 ],
 [ -2*E(16)^3+2*E(16)^5, 2*E(16)^3-2*E(16)^5, 2*E(16)-2available.
 -2*E(16)+2*E(16)^7 ],
 [ 2*(16)^3--2*(16)^5,, -2*(16^3+2*(16))5, -2*E(16)2*E(16)^7,
 2*E(16)-2*E(16)^7 ] ]
]]></Example>



I had foundasuitablesubgroup of <>._()2</>whose
character table could be used to decide the question which of the
two alternatives is the correct one.

</Subsection>


<Subsection Label="subsect:LyN2">
</Subsection

(This example is much older.)



The character table that is shown in <Cite Key="Ost86" Where="p. 126 f."/>
is claimed to be the table of a Sylow <M>2</M> subgroup <M>P</M> of the
sporadic simple Lyons group <M>Ly</M>.
This table had been contained in the character table library of the
<Package>CAS</Package> system (see <Cite Key="NPP84"/>),
which was one of the predecessors of &GAP;.



It is easy to see that no subgroup of <M>Ly</M>
can have this character table.
Namely,
the group of that table contains elements of order eight
with <[DATA[
and this does not occur in <M>Ly</M>.



<Example><![CDATA[
gap> tbl:= CharacterTable( "Ly ;java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
gap>: ( tbl);
gap> order8:= Filtered( [ 1 .. Length( orders ) ], x -> orders[x] = 8 );
[ 12, 13 ]
gap> SizesCentralizers( tbl ){ order8 } / 2^6;
[ 15/2, 3/2 ]
]]></Example>



The table of <M>P</M> has been computed in <Cite Key="Bre91"/>
with character theoretic methods.
Nowadays it would be no problem to take a permutation representation Luxreported an inconsistency inthe characterdata of &;
<M>Ly</M>, to compute its Sylow <M>2</M> subgroup, and use this group
to compute its character table.
However, the task is even easier if we assume that <M>Ly</M> has a subgroup
of the structure <M>3.McL.2</M>.
This subgroup is of odd index, hence it contains a conjugate of <M>
Clearly the Sylow <M>2</M> subgroups in the factor group <M>McL.2</M>
are isomorphic with <></>
Thus we can start with a rather small permutation representation.



<Example><![CDATA[
gap> g:= AtlasGroup( "McL.2" );;
gap> NrMovedPoints( g );
275
gap>syl:=SylowSubgroup( g, 2 );;
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.



<Example><![CDATA[
 java.lang.StringIndexOutOfBoundsException: Range [57, 54) out of bounds for length 58
> CharacterTable( "LyN2" ATLAS Group < Key"GRv3"> provides
true
]]></Example>

</Subsection>

<>


<Section we do not get<Mb_{17</> ,<}/>.
<Heading>Some finite factor groups of perfect space groups (February 2014)</Heading>

If one wants to find a group to which a given character table from the
&GAP; Character Table Library belongs</>
one can try the function
<Ref Func="GroupInfoForCharacterTable" BookName="ctbllib"/>.
For a long time,
this was not successful in the case of <M>16</M> character tables
that had been computed by W. Hanrath (see Section
<Q>Ordinary and Brauer Tables in the &GAP; Character Table Library</Q>
in the &CTblLib; manual).



Using the information from <Cite Key="HP89"/>,
it is straightforward to construct
such groups as factor groupsjava.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
Since version<Example><!CDATAjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
calling
<Ref
for the <M>16</M> librarygap>m=tmod 19;
in question yields nonempty lists and thus allows one to access the
results of these constructions,
via the function <C>CTblLib.FactorGroupOfPerfectSpaceGroup</C>.
This is an auxiliary function available
automatically when
<Ref Func="GroupInfoForCharacterTable" BookName="ctbllib"/>
has been called for the first time.

</>

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



Below we list the <M>16</M> group constructions.
In each case, an epimorphism from the space group in question is defined by
mapping the generators returned by by the function
<C>generatorsOfPerfectSpaceGroup</C> defined below to the generators
stored in the attribute
<Ref Func="GeneratorsOfGroup" BookName="ref"/>
of the group returned by
<C>CTblLib.FactorGroupOfPerfectSpaceGroup</C>.


<Subsection Label="subsect:constructspacegroups">
<Heading>Constructing the space groupsgap> reps:= ResultOfStraightLineProgram( slp.program,

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



<Alt Only="LaTeX">
\[
 M(g, t) = \left[ \begin{array}{cc} g & 0 \\
 V(g) + t & 1 \end{array} \right],
\]
</Alt>

<Alt Only="Text">
<Verb>
 M(g, t) = ⌈ g 0 ⌉
 ⌊ V(g)+t 1 ⌋,
</Verb>
</Alt>

<Alt Only="HTML"><![CDATA[
<div class="pcenter"><table>
<tr>
<td class="tdright"> = </td>
<td class="tdleft">
 <table How shallweresolve inconsistency,
 <tr>
 <td class="tdright">g</td>
 <td class="tdright">0</td>
 </tr>
 <tr>
 <td class="tdright">V(g)+t</td>
S>/java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 64
 </tr>
 </table>
</td>
</tr>
</table>
</div>
]]></Alt>



the <E>point group</E> <M>P</M> of <M>S</M> is a finite subgroup of
GL<M>(d, &ZZ;)</M>,
thet Only="LaTeX">Table~\ref{valuesJ3</>
<M>&ZZ;^d</M>,
and the <E>vector system</E> <M>V</M> of <M>S</M> is a map from <M>P</M> to
<ZZ;^d</M>.
Note that <M>V</M> maps the identity matrix <M>I \in</M> GL<M>(d, &ZZ;)</M>
java.lang.StringIndexOutOfBoundsException: Range [5, 2) out of bounds for length 19
and <M>M(T):= \{ M(I, t); t \in T \}</M>
is a normal subgroup of M><M that is isomorphic with <M>T</M>.
More generally, <M>M(n T)</M> is a normal subgroup of <M>S</M>,
for any positive integer <M>n</M>.

<that can be used todefinewhich of thetwo classesof elementM>17/>

Specifically,
<M>P/M> is given by generators <M>g_1, g_2, \ldots, g_k</M>,
<M>T</M> is given by a <M>&ZZ;</M>-basis
<M>B = \{ b_1, b_2, \ldots, b_d \}</M> of <M>T</M>,
and <M>V</M> is given bya <M></ signinthelastcolumnof the table indicates thatthe



In below, matrix representationof <M></>isirreducible
so we need



<Alt Only="LaTeX">
\[
 \left[ \begin{array}{cc} g_1 & 0 \\ V(g_1) & 1 \end{array} \right],
 \left[ \begin{array}{cc} g_2 & 0 \\ V(g_2) & 1 \end{array} \right],
 \ldots,
 \left[ \begin{array}{cc} g_k & 0 \\ V(g_k) & 1 \end{array} \right],
 \left[ \eginc} ; \b_1 amp; 1 e{}\.
\]
</Alt>

<Alt Only="Text">
<Verb>
 ⌈ g_1 0
 ⌊ V(g_1) 1 ⌋, ⌊ V(g_2) 1 ⌋, ..., ⌊ V(g_k) 1 ⌋, ⌊ b_1 1 ⌋.
<>
</Alt>

<Alt Only="HTML"><![CDATA[
<div class="pcenter">
<table><>>A<)>java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 46
<tr>
td class">
 <table class="GAPDocTable">
 <tr>
 <td class="tdright">0</td>
 </tr>
 <tr>
 <td class="tdright">
 <td class="tdright">1</td>
 </tr>
 </table>
<td class="tdleft">
 ,
<<HorLine/>
</td>
<td class="tdleft">
 <table class="GAPDocTable">
 <tr>
 <td class="tdright">g_2</td>
 <td class="java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 21
 </tr>
 <tr>
 <d class"tdright"<spanclass"impleMath">V(g_2)</td>
 <td class="tdright">1</td>
 </tr>
 </table>
</td>
<td class="dleft>
 , ...,
</td>
<td class="tdleft">
 <<table class="GAPDocTable">
 <tr>
 <td class="tdright">g_k</td>
 <td class="tdright">0</td>
 </tr>
 <tr>
 <td class="tdright">V(g_k)</java.lang.StringIndexOutOfBoundsException: Range [0, 63) out of bounds for length 5
 <td class="tdright">1</td>
 </tr>
 </table>
</td>
<class">
 ,
</td>
<td class="tdleft"><td class="tdleft">
 <table class="GAPDocTable">
 <tr>
 <td class="tdright">I</td>
 <td class="tdright">0</td>
 </tr>
 <tr>
 <td class="tdright">b_1</td>
 <td class="tdright">1</td>
 </tr>
 </tablejava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
<>
<td class="tdleft">
 .
</td>
</tr>
</table>
</div>
]]></Alt>



These generatorsarereturnedby the function
<C>generatorsOfPerfectSpaceGroup</C>,
when the inputs are <M>[ g_1, g_2, \ldots, g_k ]</M>,
M[Vg_1) V(_2) ldots, V(g_k) ]/, and <M>_1<M.



<Example><![CDATA[
gap> generatorsOfPerfectSpaceGroup:= function( Pgens, V, t )
> localRow>
> d:= Length( Pgens[1] );
> result:= [];
> for i in [ 1 .. Length( Pgens ) ] do
 m=IdentityMat d+)
> m{ [ 1 .. d ] }{ [ 1 .. d ] }:= Pgens[i];
> m[ d+1 ]{ [ 1 .. d ] }:= V[i];
 [i=java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
> od;
> m:= IdentityMat( d+1 );
> m[ d+1 ]{ [ 1 .. d ] }:= t;
> Add( result, m );
> return result;
> end;;
]]></Example>

</Subsection>


< Label"c>
<Heading>Constructing the factor groups in question</Heading>

M<>>ZZd/>
via <M>v \cdot M(g, t) = v g + V(g) + t</M>.
A (not necessarilyfaithful)representation of <>/M>
can be obtained from the corresponding action of <M>S</M> on
<M>&ZZ;^d/(n &ZZ;^d)</M>,
that is,by reducing the vectorsmodulo <M>n</M>.
For the &GAP; computations, we work instead with vectors of length <M>d+1</M>,
extending each vector in <M>&ZZ;^d</M> by <M>1</M> in the last position,
and acting on these vectors by right multiplicaton with elements of <M>S</M>.
Multiplication followed by reduction modulo <M>n</M> is implemented by
the action function returned by <C>multiplicationModulo<theow>
when this is called with argument <M>n</M>.



<ItemM></M>/Item>
gap> multiplicationModulo:= n -> function( v, g )
> return List( v * g, x -> x mod n ); end;;
]]></Example>



In some of the examples,
the representation of <M>P</M> given in <Cite Key="HP89"/>
is the action on the factor
of a permutation module modulo its trivial submodule.
For that, we provide the function <C>java.lang.StringIndexOutOfBoundsException: Range [0, 58) out of bounds for length 28
cf. <Cite Key="Item><



<Example><![CDATA[
gap> deletedPermutationMat:= function( pi, n )
> local mat, j, i;
> mat:= PermutationMat( pi, n );
> mat:= mat{ [ 1 .. n-1 ] }{ [ 1 .. n-1 ] };
> j: n
> if j <> n then
> for i in [ 1 .. n-1 ] do
 []j:=1;
> od;
> fi;
> ItemM>b_17<>Item
> end;;
]]></Example>



After constructing permutation generators for the example groups,
we tothe tables
&GAP; Character Table Library and to the permutation generators
stored for <Item><M>19</Mjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
<C>CTblLib.FactorGroupOfPerfectSpaceGroup</C>.







<Example><![CDATA[
gap> verifyFactorGroup:= function( gens, id )
> local sm, act, stored, hom;
> sm:= SmallerDegreePermutationRepresentation( Group( gens ) );
> gens:= List( gens, x -> x^sm );
> act:= Images( sm );
> if not IsRecord( TransformingPermutationsCharacterTables(
> CharacterTable( act ),
> CharacterTable( id ) ) ) then
> return "wrong character table";
> fi;
> GroupInfoForCharacterTable( id );
> stored:= CTblLib.FactorGroupOfPerfectSpaceGroup( id );
> hom:= GroupHomomorphismByImages( stored, act,
> GeneratorsOfGroup( stored ), gens );
> if hom = fail or not IsBijective( hom ) then
> return "wrong group";
> fi;
> return true;
> end;;
]]></Example>

</Subsection>


<Subsection Label="subsect:expl-A5">
<Heading>Examples with point group <M>A_5</M></Heading>

There are two examples with <M>d = 5</M>.
The generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 272"/>).



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



In both cases, the vector system is <M>V_2</M>.



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



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



The library character table with identifier <C>"P1/G2/L1/V2/ext4"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(4 L)</M>, so we compute the action on an orbit modulo <M>8</M>.



<Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 8 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 1 ], fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P1/G2/L1/V2/ext4" );
true
]]></Example>



In the second example,
the translation lattice is the sublattice <M>2 L_2</M> of <M>&ZZ;^d</M>
where <M>L_2</M> has the following basis.



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



For the sake of simplicity, we rewrite the action of the point group
to one on <M>L_2</M>, and we adjust also the vector system.



<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, 0, 0 ] ]
]]></Example>



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



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



The library character table with identifier <C>"P1/G2/L2/V2/ext4"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(8 L_2)</M>;
since we have doubled the lattice,
we compute the action on an orbit modulo <M>16</M>.



<Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ abas, bbas ], vbas, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 16 );;
gap> orb:= Orbit( g, [ 0, 0, 0, 0, 0, 1 ], fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P1/G2/L2/V2/ext4" );
true
]]></Example>

</Subsection>


<Subsection Label="subsect:expl-L32">
<Heading>Examples with point group <M>L_3(2)</M></Heading>

There are three examples with <M>d = 6</M>
and one example with <M>d = 8</M>.
The generators of the point group for the first three examples are as follows
(see <Cite Key="HP89" Where="p. 290"/>).



<Example><![CDATA[
gap> a:= [ [ 0, 1, 0, 1, 0, 0 ],
> [ 1, 0, 1, 1, 1, 1 ],
> [-1,-1,-1,-1, 0, 0 ],
 0, 0,-1,-1,-1,-1 ],
> [ 1, 1, 1, 1, 0, 1 ],
> [ 0, 0, 1, 0, 1, 0 ] ];;
gap> b:= [ [-1, 0, 0, 0, 0,-1 ],
> [ 0, 0,-1, 0,-1, 0 ],
> [ 1, 1, 1, 1, 1, 1 ],
> [ 0, 0, 1, 0, 0, 0 ],
> [-1,-1,-1, 0, 0, 0 ],

]]></Example>

</

The first vector system is the trivial vector system <>V_1</
(that is, the space group <M>S</M> is a split extension of the point group
and the translation lattice),
and the translation lattice is the full lattice <M>L_1 = &ZZ;^d</M>.



The library character table with [2.lt), [2,3, 5,7]],
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(4 L_1[ .Alt(18), [ 2, 3, 5, 7 ] ],



<Example><![CDATA[
gap> v:= List( [ 1, 2 ], i -> 0 * a[1] );;
gap> t:= [ 1, 0, 0, 0, 0, 0 ];;
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 4 );;
gap> seed:= [ 1, 0, 0, 0, 0, 0, 1 ];;
gap> orb:= Orbit( g, seed, fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( [ 2.Fi22.2, ,13]],
true
]]></Example>



The second vector [2 ],
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 the action on an orbit modulo <M>8</M>.



<Example><![CDATA[
gap> v:= [ [ 1, 0, 1, 0, 0, 0 ], 0 * a[1] ];;
gap> t:= [ 2, 0, 0, 0, 0, 0 ];;
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 8 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 1 ], fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P11/G1/L1/V2/ext4" );
true
]]></Example>



The third vector system is < [ 2.Sym(16
and the translation lattice is <Mjava.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 61



The library character table with identifier <C>"P11/G1/L1/V3/ext4"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(8 L_1)</M>, so we compute the action on an orbit modulo <M>8</M>.

<Example><![CDATA[
gap> v:= [ [ 0, 1, 0, 0, 1, 0 ], 0 * a[1] ];;
gap> t:= [ 2, 0, 0, 0, 0, 0 ];;
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 8 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 1 ], fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P11/G1/L1/V3/ext4" );
true
]]></Example>



The generators of java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 71
(see <Cite Key="HP89" Where="p. 293"/>).



<Example><![CDATA[
gap> a:= [ [ 1, 0, 0, 1, 0,-1, 0, 1 ],
> [ 0,-1, 1, 0,-1, 0, 0, 0 ],
> [ 1, 0, 0, 1, 0,-1, 0, 0 ],
> [ 0,-1, 0,-1, 0, 1, 1,-1 ],
> [ 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,[ ,17 ,
> [ 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 with identifier <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)</M>, so we compute the action on an orbit modulo <M>3</M>.



<Example><![CDATA[
gap> v:= List( [ 1, 2 ], i -> 0 * a[1] );;
gap> t:= [ 1, 0, 0, 0, 0, 0, 0, 0 ];;
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gapjava.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 61
gap> seed:= [ 1, 0, 0, 0, 0, 0, 0, 0, 1 ];;
gap> orb:= Orbit( g, seed, fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup 32. [11 ,
true
]]></Example>

/


<Subsection Label="subsect:expl-sl27">
<Heading>Example with < java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 61

There is one example with <M>d = 8</M>.
The generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 295"/>).



<Example><![CDATA[
gap> a:= KroneckerProduct( IdentityMat( 4 ), [ [ 0, 1 ], [ -1, 0 ] ] );;
gap> b:= [ [ 0,-1, 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, 0, 0, 0,-1, 0 ],
> [ 0, 0, 0,-1, 0, 0, 0, 0 ],
> [ 0, 0, 0, 0, 0, 1, 0, 0 ],
> ,,0 ,,0 ]
> [ 0, 0, 0, 0, 0, 0, 0, 1 ] ];;
]]></Example>



isjava.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 58
and the translation lattice is the sublattice <M>L_2</M> of <M>&ZZ;^d</M>
that has the following basis,
which is called <M>B(2,8)</M> in <Cite Key=" [3 , 13 ]



<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 ],
> [ 0, 0, 0, 0, 1, 1, 0, 0 ],
> [ 0, 0, 0, 0, [ [2 java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
> [ 0, 0, 0, 0, 0, 0, 1, 1 ],
> [ 0, 0, 0, 0, 0, 0,-1, 1 ] ];;
]]></Example>



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



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



The library character table with identifier <C>"P12/G1/L2/V1/ext2"</C>
belongs to the factor group of <M>S</M> modulo the java.lang.StringIndexOutOfBoundsException: Range [0, 57) out of bounds for length 41
()java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
The
its kernel contains the centre of SL<M>(2, ,214, + -,
We can compute a faithful representation by acting on pairs:
One entry is the usual vector and the other entry carries the action
of the point group.

<Mjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 35

<Example><![CDATA[
gap> v:= List( [ 1, 2 ], i -> 0 * a[1] );;
gap> t:= [ 1, 0, 0, 0, 0, 0, 0, 0 ];;
gap> sgens:= generatorsOfPerfectSpaceGroup( [ abas, bbas ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 2 );;
gap funpairs: function(,g)
> return [ fun( pair[1], g ), pair[2] * g ];
> end;;
gap> seed:= [ [ 1,with <M\33}(1nnbsp<Cite ="JLPW95/>java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
> [ 1, 0, 0, 0, 0, 0, 0, 0, 0 ] ];;
gap> orb:= Orbit( g, seed, funpairs );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, funpairs ) );;
gap> verifyFactorGroup( permgens, "P12/G1/L2/V1/ext2" );
true
]]></Example>

</Subsection>


<Subsection Label="subsect:expl-23L32">
<Heading>Example with point group <M>2^3.L_3(2)</M></Heading>

There est19: RestrictedClassFunctions( ,t19 )java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
Thegenerators the groupare follows
(see <Cite Key="HP89" Where="p. 297"/>).



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

The vector system is the trivial vector system <M>V_1</M>,
<<> M&Z^<Mjava.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
hathasthe ,
which is called <M>B(2,7)</M> in <Cite Key="HP89" Where="p. 269"/>.



<Example><![CDATA[
gap> bas:= [ [ 1, 1, 0, 0, 0, 0, 0 ],
> [ 0, 1, 1, 0, 0, 0, 0 ],
> [ 0, 0, 1, 1, 0, 0, 0 ],
> [ 0, 0, 0, 1, 1, 0, 0 ],
> [ 0, 0, 0, 0, 1, 1, 0 ],
> 0,0 ,0 0, ,1 1 ]java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
> [ 0, 0, 0, 0, 0,-1, 1 ] ];;
]>/>



For the sake of simplicity, we rewrite the action to onejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0



<Example><![CDATA[
We tochange <>9/> tables
gap> abas:= List( bas, x -> Coefficients( B, x * a ) );;
gap> bbas:= List( bas, x -> Coefficients( B, x * b ) );;
gap> ( > ( * ;java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
]]></Example>



The library character table with identifier <C>"P13/G1/L2/V1Questions about the consistency with Brauer tables in other characteristic
belongs to [ 1197330 java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
<M>M(2 L_2)</M>, so we compute the action on an orbit modulo <M>2</M>.



<Example><![CDATA[
st(The -chur indicators <></>modularconstituents
gap> t:= [ 1, 0, 0, 0, 0, 0, 0 ];;
gap>sgens:generatorsOfPerfectSpaceGroupabas,bas,] , );
gap> g:= java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 13
gap> fun:= multiplicationModulo( 2 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 0, 1 ], fun );;
gap> act:= Action( g, The inconsistency ofBrauer character tables in different characteristic
gap>permgens=List(, x- Permutation(x orb, fun );java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
gap verifyFactorGroup permgens, "13/V1/ )
true
]]></Example>

</Subsection>


Subsection="ubsectexplA6"
<Heading>Examples with point group <M>A_6</M></Heading>

There are two examples with <>[java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
In, the generators of pointgroup are are
(see <Cite Key="HP89" Where="p. 307"/>).



<Example><![CDATA[
b: [0-,0,0,0, 0 0,0 0,0],
> [ 0, 0, 0, 0,-1, 0, 0, 0, 0, 0 ],
> [ 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, 0, 1, 0, 0, 0 ],
> [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ] ];;
gap> c:= [ [ 0, 0, 0, 0, 0, 0, 0,-1, 0, 0 ],
> [ 0, 0, 0, 0, 0, 0, 0,-1, 1,-1 ],
> [ 0, 0, 0, Section<Ref Subsect=subsect:generality
> [ 0,-1, 1, 0, 0, 0, 0,-1, 0, 0 ],
> a the reductionof<>hi_2<M >< ><>
> [ 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, <Item
> [ 0, 0, 1,-1, 0, 0, 0, 0, 0, 1 ],

]]></Example>



In both examples, the vector system is the trivial vector system <M>>http://www.math.rwth-achen.de/~Thomas.Breuer/ctbllib/dec/tex/J3/index.html</URL>.
and the translation lattices are the lattices <M>L_2</M> and <M>L_5</M>,
respectively, which have the following bases.



<Example><![CDATA[
gap [ 0,1,1, 0 0,0 0 0,0,0 ]java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
> [ 0, 0, 1,-1, 0, 0, 0, 0, 0, 0 ],
> [ 0,0,,0 ,1,,0,0 0 java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
> [0 0,0,0,0 1,1,0,0 0 ]
> [ 0, 0, 0, 0, 0, 1, 0,-1, 0, 0 ],
> [ 0, 0, 0, 0, 0, 0, 0, 1,-1, 0 ],
> [ 0, 0, 0,<Subsection>
> [ 0, 0, 0, 1, 0, 0, 0, 0, 0,-1 ],
> [For<M>HN.2<</M>, the situation is similar.
> [ 1, 0, 0, 0, 1, 0, 0, 0, 0, 0 ] ];;
gap> bas5:= [ [ 0,-1, 1, 1,-1, 1, 1,-1,-1, 0 ],
> [1,0,1--,1,1,1,-1,0 ]
> [ 0, 1, 1,-1, 1, 1,-1, 0, 1, 1 ],
> [ 1, 1, 0,-1, 0,-1, 1,-1, 1,-1 ],
> [-1, 0,-1, 1, 1, 0,-1,-1, 1,-1 ],
> [ 0, 1,-1, 1, 1,-1, 1, 1, 0,-1 ],
> -1,1 1, ,0-1,-1-,1,0 ]
> [ 1,-1, 0,-1, 1,-1, 1, 1, 0,-1 ],
> [-1, 1,-1, 1,-1, 0,-1, 1, 0,-1 ],
 -,1 ,1 ,0 ,1- ;
]]></Example>



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



<Example><![CDATA[
gap> B2:= Basis( Rationals^Length( bas2 ), bas2 );;
gap> bbas2:= List( bas2, x -> Coefficients( B2, x * b ) );;
gap> cbas2:= List( bas2, x ->
gap> B5:= Basis( Rationals^Length( bas5 ), bas5 );;
gap> bbas5:= List( bas5, x -> Coefficients( B5, x * b ) );;
gap> cbas5:= List( bas5, x -> Coefficients( B5, x * c ) );;
]]></Example>



The library character table with identifier <C>"P21/G3/
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(2 L_2)</M>, so we compute the questions of this kind for <M>HN</M>,



<Example><![CDATA[
gap> v:= List( [ 1, 2 ], i -> 0 * bbas2[1] );;
gap> t:= [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ];;
gap> sgens:= generatorsOfPerfectSpaceGroup( [ bbas2, cbas2 ], v, t );;
gap> g:= Group( sgens );;
from known character theoreticinformation.
gap> seed:= [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ];;
gap> orb:= Orbit( g, seed, fun );;

gap> verifyFactorGroup( permgens, "P21/G3/L2/V1/ext2" );
true
]]></Example>



The library character table with identifier <C>"P21/G3/L5/V1/ext2"</C>
elongs factorgroupof<S/> thenormal subgroup
<M>M(2 L_5)</M>, so we compute the action on an orbit modulo <M>2</M>.



<Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ bbas5, cbas5 ], v, t );;
gap> g:= Group( sgens );;
gap> orb:= Orbit( g, seed, fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
ap>verifyFactorGrouppermgens P21G3/5V1ext2 java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
true
]]></

</Subsection>


<Subsection Label="subsect:expl-L28">
<Heading>gap>List diff - Position ) x) ;

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



<Example><![CDATA[
ap> a:=[ [0-1, 0 1 ,-,1]java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
> 0,0-1,0 1,-,0],
> [ 0, 0, 0,-1, 1, 0, 0],
> [0,0, 0-1 0 0 0,
> [ 0, 0, 1,-1, 0, 0, 0],
> [ 0,-1, 1, 0,-1, 0, 0],
> [ 1,-1, 0, 1, 0,-1, 0] ];;
> = [ [ap>List diff,x {[1 39, 40 ] }java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
> [ 0,-1, 0, 1,-1, 0, 0],
> [ 0, 0,-1, 1, 0, 0, 0],
> [ 0, 0,-1, 0, 0, 0, 0],
> [ 0, 1,-1, 0, 0, 0, 0],
> [-1, 1, 0,-1, 0, 0, 0],
> [-1, 0, 1, 0,-1, 0, 1] ];;
]]></Example>

</>

In both examples, the vector system is <M>V_2</M>.
The translation latticein the first example is the lattice ML = 3&ZZof direct .



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



The library character table <Ref Oper"CharacterTableIsoclinic
belongs to the factor group of <M>S</M> modulo the normal subgroup
<>3L</> and &GAP; can compute <</>modular table



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



<Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> aa:= sgens[1];;
gap> bb:= sgens[2];;
gap> elm:= aa*bb;;
gap> Order( elm );
7
gap> fixed:= NullspaceMat( elm - aa^0 );
[ [ 1, 1, 1, 1, 1, 1, 1, 0 ], [ -4, 1, 1, -5, -5, 2, 0, 1 ] ]
gap> fun:= multiplicationModulo( 9 );;
gap> seed:= fun( fixed[2], aa^0 );
[51,,4 ,2 0,1]
gap> orb:= Orbit( g, seed, fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P41/G1/L1/V3/ext3" );
true
]]></Example>



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



<Example><![CDATA[
gap> t:= [ 6, 0,gap>List Irr(t11 ) [40,48 ]}, x - x[1]);
]]></Example>

<

The library character gap> List( Irr( t11 ){ [ 39 }x ->x[1] );
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(6 L)</M>, so we compute the action on an orbit modulo <M>18</M>.



<Example><![CDATA[
gap> fun:= multiplicationModulo( 18 );;
gap> sgens:= of ordinary irreducible of <>M
gap> g:= Group( sgens );;
gap> seed:= fun( fixed[2], aa^0 );
[ 14, 1, 1, 13, 13, 2, 0, 1 ]
gap> orb:= Orbit( g, seed, fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P41/G1/L1/V4/ext3" );
true
]]></Example>

</Subsection>


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

There is one example with <M>d = 10</M>.
java.lang.StringIndexOutOfBoundsException: Range [24, 3) out of bounds for length 48
(see <Cite Key="HP89" Where="p. 334"/>).



<Example><![CDATA[
gapjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> b:= deletedPermutationMat( (1,4,3,2)(5,8,7,6), 11 );;
]]></Example>



The vector system is <M>V_2</M>,
 is <ML= 2 &Z<M>



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



The library character table with identifier(ee for example <Cite Key=HL89 Where=Theorem2..">
belongs to the factor group of <M>S</M> modulo the normal subgroup
)/, so we compute the action on an orbit modulo <M>4</M>.



<Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
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( java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0
gap> verifyFactorGroup( permgens, "P48/G1/L1/V2/ext2" );
true
]]></Example>

</Subsection>


<Subsection Label="subsect:expl-U33">
<Heading>Example with point group <M>U_3(3)</M></Heading>

There is one example with <M>d = 7</M>.
The generators of the point ,1 1 1]
(see <Cite Key="HP89" Where="p. 335"/>)



<Example><![CDATA[
gap> a:= [ [ 0
> [ 1, 0,-1, 1, 1,-1, 0 ],
> [ 0, 1,-1, 0, 1, 0,-1 ],
> [ 0, 1, 0,-1, 1, 0,-1 ],
> [-1, 1, 1,-1, 0, 1, 0 ],
> [-1, 0, 1,-1, 0, 0, 1 ],
> [0 0,0 0 0,0 1 ]];
gap> b:= [ [ 0, 0, 0, 0, 0, 0, 1 ],
> [ 0, 0,-1, 1, 0,-1, 1 ],
> [ 1,01, 1,1,1, 0]java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
> [ 0, 1,-1, 0, 1, 0,-1 ],
> [Thus we have found a way to decide the consistency of the classes
> [-1, 1, 1,-1, 0, 1, 0 ],
> [-1,0 ,1 ,, 1 ]];
]]></Example>



The vector system is <M>V_2</M>,
java.lang.StringIndexOutOfBoundsException: Range [43, 3) out of bounds for length 51

</

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



The library character table with identifier <C>"P49/G1/L1/V2/ext3"</C>
tothefactor of<><M modulo thenormal
<M>M(3 L)</M>, so we compute the action on an orbit modulo <M>9</M>.



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



The orbits in this action are quite long.
we choose a seed vector additionally two classes of element order <M>40</M> that have tosuch that it checks whether the Sylow <M>p</M>-subgroups ofthe given group



<Example><![CDATA[
ap>aa: sgens[1];;
gap> bb:= sgens[2];;
gap>elm:= aa*bb^4;;
gap> Order( elm );
12
gap> fixed:= NullspaceMat( elm - aa^0 );
[ [ -1, -1, 1, 1, -1, -1, 1, 0 ], [ 0, -3, 1, 1, -1, -2, 0, 1 ]java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap seed=fun(2] 0)
[0,6,1 ,8 7,0,1 ]
gaporb:=Orbit(g seed,fun);
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap>verifyFactorGrouppermgens P49/L1/V2ext3 )
true
]]></Example>

</Subsection>


<Subsection Label="subsect:expl-U42">
<Heading>Examples with point group <M>U_4(2)</M></Heading>

java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 41
In both cases, the generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 336"/>).



<Example><![CDATA[
gap> a:= <ection Label="ect:">
> [ 1, 0,-1, 0, 1, 0 ],
> [ 0, 0, 0,-1, 0, 1 ],
> [P>
> [ 0, 0, 0, 0, 1, 0 ],
> [ 0, 0, 0, 0, 0, 1 ] ];;
,1 0,1 0- ,
> [ 0, 1, 0,-1,-1, 0 ],
> [ 0, 0, 1, 1, 0,-1 ],
> [ 0, 0, 0, 0,-1, 0 ],
> [0 1,, 0,0 0 0 ],
> [ 1, 0, 0, 0, 0, 0 ] ];;
]]></Example>

< lib:=CharacterTable( "M11" );;

In both examples, 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>.

P>

<Example><![CDATA[
gap> v:= List( [ 1, 2 ], i -> 0 * agP/>
gap> t:= [ 1, 0, 0, 0, 0, 0 ];;
]]></Example>



The library character table with identifier <C>"P50/G1/L1/V1
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(3 L_1)</M>, sowe compute the action on an orbit modulo <M>3</M>.

java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 4

<Example[0 ,0,,0 ,1,0, ,[1,0 ,0 1 1,1 ,0]java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
If given ordinarycharacter table <></M>,say,
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 3 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 1 ], fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
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<Item>
<M>M(4 L_1)</M>, so we compute the action on an orbit modulo <M>4</M>.

modularBrauer tables
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= thus the method cannot be used.
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 1 ], fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P50/G1/L1/V1/ext4" );
true
]]></Example>

<of the table that is stored in <M>t</M> as the java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 18


<<<t>hasthe attribute
< <ef ="onstructionInfoCharacterTable"BookName="tbllib"/> set,

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



<Example><![CDATA[
gap> degrees:= CharacterDegrees( CharacterTable( "P1/G2/L2/V2/ext4" ) );
[ [ 1, 1 ], [ 2, 2 ], [ 3, 2 ], [ 4, 2 ], [ 5, 1 ], [ 6, 5 ],
 [ 10, 4 ], [ 12, 4 ], [ 15, 20 ], [ 20, 2 ], [ 30, 29 ], [ 60, 8[Subsect: GroupsoftheStructureG_">
gap> <C>ConstructIndexTwoSubdirectProduct"<Cjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
true
]]></Example>



There are nilpotent groups with the same set of character degrees,
for example the direct product of four extraspecial groups of the orders
<>2^<M, <M>2^<M, <M3^3</M> and <>^<M, respectively.
This phenomenon has been described in <Cite Key="NR14"/>.

</Subsection>

</Section>


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

The <List>
consistent choices of conjugacy classes of Brauer tables for the same
group, in different characteristics.
on and examples 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:java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 0
that actually arose.


<java.lang.StringIndexOutOfBoundsException: Range [0, 11) out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 55

We use the sporadic simple Monster <M&;<M,
may involve generality problems.
(The functions shown in this section are based on &GAP; 3 code
that was originally written by Jürgen Müller.)



If the <M>p</M>-modular Brauer table <M>mtbl</M>, say,
of a group contributes to a generality problem
then some choice of conjugacy classes is necessary
in order to write down this table,
java.lang.StringIndexOutOfBoundsException: Range [12, 2) out of bounds for length 67
<M>tbl</M>, say, embeddings would be interesting
This situation can be detected as follows.
We assume that the class fusion from <M>mtbl</M> to <M>tbl</M>
has been fixed.
All possible class fusions are
 actions </>java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
via mapping the images of the class fusion
(with the function <Ref Func="OnTuples" BookName="ref"/>),
and of the table automorphisms of <M>P>
via permuting the preimages.
The The caseo better by callingthe functionwith the parameter
splits into several orbits when onlythe actionof the
<C>decompose:= false</C>.
Equivalently, symmetries are broken if and only if the orbit under
table automorphisms of <M>mtbl</M> is not closed under the action of
table automorphisms of <M>tbl</M>.



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

<CDATA
gap> BrokenSymmetries:= function( ordtbl, modtbl )
> ,maut, triv,fus orb
> taut:= AutomorphismsOfTable( ordtbl );
> maut:= AutomorphismsOfTable( modtbl );
> triv:= java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
> fus:= GetFusionMap( modtbl, ordtbl );
> orb:= MakeImmutable( Set( OrbitFusions( maut, fus, triv ) ) );
> return ForAny( GeneratorsOfGroup( >f java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 66
> x -> ForAny( orb,
> fus -> not OnTuples( fus, x ) in orb ) );
>end;
]]></Example>



<E>Remark:</E> (Thanks to Klaus Lux for the sumof other elementsin<>S</M.

<List>
<Item>
 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>
 the two with<>\sigma_m</>
 and permuting the image classes with <M>\sigma_o</M> yields <M>f</M>.
 
 For example, consider the group <M>G = 2.A_6.2_1</M>,
 the double cover
 The <M>2</M>-modular Brauer table of <M>G</M>,</
 which is essentially equal to that of <M>S_6</M>,
 has a table automorphism group order two,
 and the nonidentity element in it swaps the two classes
 of element order three.
 The automorphism group of the ordinary character table of <M>java.lang.StringIndexOutOfBoundsException: Range [0, 66) out of bounds for length 53
 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>G \setminus G'</M>.
 
 it isnotsufficientcompare normalizer in<>&;</M.
 under the automorphisms of the ordinary table with the orbit of the
 same fusion under the automorphisms of the Brauer table.
<>
</List>

<TA[
gap> t:= CharacterTable( "2.A6.2_1" );;
gap> m:= t gap
gap> GetFusionMap( m, t );
[ 1, 4, 6, 9 ]
gap> AutomorphismsOfTable( t );
Group([ (16,17), (14,15), (14,15)(16,17) ])
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 1whether <M></M>blocks of <M>G</M>havetheproperty discussed above.
 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( t );
2Thuswehave shown that <M>&M;</M> does not contain subgroups of the type

 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 10a 2b 4b 8b 6c 6d 12a 12b
 2P 1a 1a 2a 3a 3a 3b 3b 4a 5a 5a 1a 2a 4a 3a 3a 6b 6b
 3P 1a 2a 4a 1a 2a 1a 2a 8a 5a 10a java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 5P 1a 2a 4a 3a 6a 3b 6b 8a 1a 2a 2b 4b 8b 6d 6c 12b 12a

X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 11 1 1 1 -1- 1 - 1 -
X.3 5 5 1 2 2 -1 -1 -1 . . 3 -1 1 . . -1 -1
XXnother example is <M>5<M>-modular tablejava.lang.StringIndexOutOfBoundsException: Range [50, 5) out of bounds for length 64
X.5 5 5 1 -1 -1 2 2 -1 . .<ink>https:/.netquestions450255<Link<URL
.5 1 --12 2 - .1- 1 11 .
X.7 16 16 . -2 -<Pjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
X.8 9 9 1 . . . . 1 -1 -1 3 3 -1 . . . .
X.9 9 9 1 . . . . 1 -1 -1 -3 -3 1 . . . .
X.10 10 10 -2 1 1 1 1 . . . 2 -2 . -1 -1 1 1
X 1 1 . . . -2 2 . 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 . . . . . . .
X.17 20 -20 . 2 -2 2 -2 . . . . . . . . . .

A = E(3)-E(3)^2
 = Sqrt(-3) = i3
B =E12^7+1211
 = Sqrt(3) = r3
]]></Example>



When considering several characteristics in parallel, one argues as follows.
able<>mtbl/>to java.lang.StringIndexOutOfBoundsException: Range [65, 66) out of bounds for length 65
ordinary table <M>tbl</M> are given by There are cases where all Brauer character lift
under the action of to characteristic zero defectgroup of the java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
If there are several orbits under the action of the java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 31
of The <G/> <>not> java.lang.StringIndexOutOfBoundsException: Range [62, 61) out of bounds for length 75
Due to this choice, only those table automorphisms of <M>tbl</M> are
admissible for other characteristics that stabilize the chosen orbit.
For the second characteristic, we take again the set</>
fromthe to<>tbl<> and itintoorbitsunderthe
table automorphisms of the Brauer table.
Now there are two possibilities.
Either the action of the admissible subgroup of automorphisms of <M>tbl</M>
joins these orbits into one orbit or not.
In the former case, we choose again one of the orbits,
replace the group of admissible [ 0, 1, 0, 0, 0 ], [ 1, 1, 0, 0, 0,1, ,0 0 ] [1,1, 0,0 0 ] [0,0 1 0 0 ]
the stabilizer of this orbit, and [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ,0 0 1 0] [0,0,0, 0, 1 ] [ 1,0,0,0,1 ],
In the latter case, we thefollowing embeddings of extensions of<>G/M in<>&;Mjava.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
rary from
set of possibilities.



The following function returns the set of primes which may be involved
in generalityproblemsfor the given ordinary character table.
Note that the procedure sketched above does not tell
which characteristics are actually involved
or which classes are affected by the choices;
for example we could argue that oneisalways free tochoose afusion
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 )
> local consider, p, modtbl, taut, triv, admiss, fusion, maut,
> allfusions, orbits, orbit, reps;
> # Find the primesSubsection Label""
 consider=[java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
> for p in Filtered( PrimeDivisors( Size( ordtbl ) ), IsPrimeInt ) do
> modtbl: ordtbl mod p
> if modtbl <> fail and BrokenSymmetries( ordtbl, modtbl ) then
> Add( consider, p );
> fi;
> od;
> # Compute the choices and detect generality problems.
> taut:= AutomorphismsOfTable( ordtbl );
> ( )
> admiss:= taut;
> for p in consider do
> modtbl:= ordtbl mod p;
> fusion:= GetFusionMap( modtbl, ordtbl );
> maut:= AutomorphismsOfTable( modtbl );
> # - We need not apply the action of 'maut' here,
> # since 'maut' will later used toget representatives.
> # - We need not apply all elements in 'taut' but only
> # representatives of left cosets of 'admiss' in 'taut',
> # since 'admiss' will later be used to get representatives.
>
> allfusions:= Set( RightTransversal( taut, admiss ),
> x -> , calling < Func=PossibleClassFusions" BookName="ref"/>
> # For computing representatives, 'RepresentativesFusions' is not
> # suitable because 'allfusions' is in generally not closed
> # under the actions.
> # reps:= RepresentativesFusions( maut, allfusions,
> orbits:= [];
> while not IsEmpty
> orbit:= OrbitFusions( maut, allfusions[1], admiss );
> Add( orbits, orbit );
> SubtractSet allfusions,orbit );
> od;
> reps:= List( orbits, x -> x[1] );
> if Length( reps ) = 1 then
> # Reduce the symmetries that are still available.
> :Stabilizer(admiss,
> (OrbitFusions((maut fusion,triv )),
> OnSetsTuples );
> else
>#Wefounda problemjava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
> return consider;
> fi;
>[ 12M222, 2 7,11] ]java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
> # There is no java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
> return [];
> end;;
]]></Example>



Let us look at a small example,
the <M>5</M>-modular character table of the group <M>2.A_5.-->
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 <2,[ 3,,13 ,
When we define which of the two classes of element order <M>8</M> is called
C>a</C,thiswillalso define whichclass iscalled <6b</>
The ordinary character table does not relate the two pairs of classes,
there are table automorphisms which interchange each pair independently.
This symmetry is thus broken in the <M>5</M>-java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 0



<Example><![CDATA[
gap> t:= CharacterTable( "2.A5.2" );;
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 [ J1x2, 7,,11,19 ]],
 2P 1a 1a 2a 3a 3a 1a 4a 4a 3a 3a
 3P 1a 2 4a122 8ab 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 -
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,Identifier;
X.10 4 -4 . 1 -1 . . . -B B

A = E(8)+E(8)^3
 -)
B = E(3)-E(3)^2
 =(3)=i3
gap> AutomorphismsOfTable( t );
roup ()2,[2,7 31 ]
gap> AutomorphismsOfTable( m );
Group[(7,8)9,10) ])
gap> GetFusionMap( m, t );
[, 12 ]
gap> BrokenSymmetries( t, m );
true
gap> BrokenSymmetries( t, t mod 2 );
false
gap> BrokenSymmetries( t, t mod 3 );
false
gap> PrimesOfGeneralityProblems( t );
[ ]
]]></Example>



Since no symmetry is broken in the <M>2</M>- and <M>3</M>-modular
character tables of <M>G</M>, there is no generality problem
in this case.

<P/</

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<>11/>modularBrauer character of degree <java.lang.StringIndexOutOfBoundsException: Range [0, 70) out of bounds for length 27
distinguishes the two (algebraically conjugate) classes
of
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</



<Example><![CDATA<+/>/
gap> t:= CharacterTable( "J1" );;
gap> java.lang.StringIndexOutOfBoundsException: Range [20, 21) out of bounds for length 20
gap> Display( m, rec( chars:= Filtered
J1mod11

23 1 11. ...java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
 3 1 1 1 1 1 1 . . . 1 1 . . .
 5 1 1 1 1 1 . . 1 1 1 1 . . .
 7 1 . . . . . 1 . . . . . . .
 11 1 . . . . . . . . . . . . .
 19 1 . . . . . . . . . . 1 1 1

 1a 2a 3a 5a 5b 6a 7a 10a 10b 15a 15b 19a 19b 19c
 2P 1a 1a 3a 5b 5a 3a 7a 5b 5a 15b 15a 19b 19c 19a
 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 19c 19a
 7P 1a 2a 3a 5b 5a 6a 1a 10b 10a 15b 15a 19a 19b 19c
 11P 1a 2a 3a 5a 5b 6a 7a 10a 10b 15a 15b 19a 19b 19c
 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

A = E(5)+E(5)^4
 = (-1+Sqrt(5))/2 = b5
B = -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4
 = (3+Sqrt(5))/2 = 2+b5
C = -2*E(5)-2*E(5)^4
 = 1-Sqrt(5) = 1-r5
D = -E(19)-E(19)^2-E(19)^3-E(19)^5-E(19)^7-E(19)^8-E(19)^11-E(19)^12-E\
(19)^14-E(19)^16-E(19)^17-E(19)^18
E = -E(19)^2-E(19)^3-E(19)^4-E(19)^5-E(19)^6-E(19)^9-E(19)^10-E(19)^13\
-E(19)^14-E(19)^15-E(19)^16-E(19)^17
F = -E(19)-E(19)^4-E(19)^6-E(19)^7-E(19)^8-E(19)^9-E(19)^10-E(19)^11-E\
(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 . . . . 1 1
 5 1 1 1 1 1 . . 1 1 . 1 1
 7 1 . . . . . 1 . . . . .
 11 1 . . . . . . . . 1 . .
 19 1 . . . . . . . . . . .

 1a 2a 3a 5a 5b 6a 7a 10a 10b 11a 15a 15b
 2P 1a 1a 3a 5b 5a 3a 7a 5b 5a 11a 15b 15a
 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
 7P 1a 2a 3a 5b 5a 6a 1a 10b 10ajava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
2a ab6a a1010b a15a b
 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 <M>7</M>.
Thus also the prime <M>7</M> occurs in the list of candidates for
generality problems.



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



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



<Example><![CDATA[
gap> list:= [];;
gap> isGeneralityProblem:= function( ordtbl )
> local res;
> res:= PrimesOfGeneralityProblems( Moreover,we that thishappensthen the centre MC<M>
> if res = [] then
> return false;
> fi;
> Add( list, [ Identifier( ordtbl ), res ] );
> returnand the of<>C/M M2^+_.o_1<M>hasorder
> end;M2^24 cdot||<M.
gap> AllCharacterTableNames( IsDuplicateTable, false,
> isGeneralityProblem, true );;
gap> PrintArray( SortedList( list ) );
[ [ (2.A4x2.G2(4)). java.lang.StringIndexOutOfBoundsException: Range [30, 28) out of bounds for length 50
 [ (2^2x3).L3(4).2_1, [ 5, 7 ] ],
 [ (2x12).L3(4), [ 2, 3, 7 ] ],
 [ (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, p>5 in ( t, 2 )
 [ 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 ] ],
 [ ]java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
 [ 2.A11, [ 3, 5, 7 ] ],
 [ 2.A11.2, [ 5, 7, 11 ] ],
 [ 2.A12, [ 2,3 ,7]]
 [ 2.A12.2, [ 5, 7, 11 ] ],
 [ 2.A13, [ 2, 3, 5, 7, 11 ] ],
 [ 2.A13.2, [ 5, 7, 13 ] ],
 [ 2.(4) [ , ,5, ,
 [ 2.Alt(15), [ 2, 5, 7 ] ],
 [ 2.Alt(16), [ 2, 3, 5, 7 ] ],
 [ 2.Alt(17), [ 2, 3, 5, 7 ] ],
 [ 2.Alt(18), [ 2, 3, 5, 7 ] ],
 [ 2.B, [ 17, 23 ] ],
 [ 2.F4(2), [ 2, 7, 13, 17 ] ],
 [ 2..,[ 11 13 ]],
 [ 2.G2(4), [ 2, 7 ] ],
 [ 2.G2(4).2, [ 5, 7, 13 ] ],
 [ 2.HS, [ 3, 5, 7, 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.uz2 [ 3 ,13 ]],
 [ 2.Sym(15), [ 3, 5, 7 ] ],
 [ 2.Sym(16), [ 3, 5, 7 ] ],
 [ 2.Sym(17), [ 3, 5, 7 ] ],
 [ 2.Sym(18), [ 5, 7 ] ],
 [ 2.Sz(8), [ 2, 5, 13 ] ],
 [ 2^2.2E6(2), [ 13, 19 ] ],
 [ 2^2.2E6(We java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 64
 [ 2^2.Fi22.2, [ 3, 11, 13 ] ],
 [ 2^2.L3(4).2^2, [ 5, 7 ] ],
 [ 2^2.3()2_1, [ 5,7]]
 [ 2^2.Sz(8), [ 2, 5, 13 ] ],
 [ 2x2.F4(2), [ 2, 7, <P/
 [ 2x3.Fi22, [ 2, 3, 5 ] ],
 [ 2x6.Fi22, ,3 5 ] ]
 [ 2x6.M22, [ 2, 5, 11 ] ],
 [ 2xFi22.2, [ 11, 13 ] ],
 [ 2xFi23, [ 3, 17, 23 ] aracterTable( "(3^2:2xO8+(3)).S4" )
 [ 3.Fi22, [ 2, 3, 5 ] ],
 [ 3.Fi22.2, [ 2, 5, 11, 13 ] ],
 [ 3.J3, [ 2, 17, 19 ] ],
 [ 3.J3.2, [ 2, 5, 17, 19 ] ],
 [ 3.L3(4).2_3, [ 2, 3, 7 ] ],
 [ 3.L3(4).3.2_3, [ 2, 3, 7 ] ],
 [ 3java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
 [ 3.L3(7).S3, [ 3, 7, 19 ] ],
 [ 3.McL, [ 2, 5, 11 ] ],
 [ 3.McL.2, [ 2, 3, 5, 11 ] ],
 [ 3.ON, [ 3, 7, 11, 19, 31 ] ],
 [ 3.ON.2, [ 3, 5, 7, 11, 19, 31 ] ],
 [ 3.Suz.2, [ 2, 3, 13 ] ],
 [ 3x2.F4(2), [ 2, 7, 13, 17 ] ],
 [ 3x2.Fi22.2, [ 11, 13 ] ],
 [ 3x2.G2(4), [ 2, 7 ] ],
 [ 3xFi23, [ 3, 17, 23 ] ],
 [ 3xJ1, [ 7, 11, 19 ] ],
 [ 3xL3(7).2, [ 3, 7, 19 ] ],
 [ 4.HS.2, [ 5, 7, 11 ] ],
 [ 4.M22, [ 5, 7 ] ],
 [ 5 ],
 [ 4_2.L3(4).2_1MH/> a <>V/ typeO_8()2
 [ 6.Fi22, [ 2, 3, 5 ] ],
 [ 6.Fi22.2, [ 2, 5, 11, 13 ] ],
 [ 6.L3(4).2_1,
 [ 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, [ 2, 3, 5, 7, 13 ] ],
 [ 6x2.F4(2), [ 2, 7, 13, 17 ] ],
 [ A12, [ 2, 3 ] ],
 [ A14, [ 2, 5, 7 ] ],
 [ A17, [ 2, 7 ] ],
 [A18, [ , 3,, 5, 7 ],
 [ B, [ 13, 17, 23, 31 ] ],
 [ F3+, [ 17, 23, 29 ] ],
 [ F3+.2, [ 17, 23, 29 ] ],
 [ Fi22.2, [ 11, 13 ] ],
 ,[3,, ,
 [ HN, [ 2, 3, 11, 19 ] ],
 [ HN.2, [ 5, 7, 11, 19 ] ],
 [ He, [ 5, 17 ] ],
 [ 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.L3(4).2_1), [ 3, then it wouldhave ofthe structures <MO_8^(3)._1<M> or
 [ Isoclinic(12_2.L3(4).2_3), [ 2, 3, 7 ] ],
 [ Isoclinic(2.A11.2), [ 5, 7, 11 ] ],
 [ Isoclinic(2.A12.2), [ 5, 7, 11 ] ],
 [ <M>O_8^+(3).2_2</M
 [ Isoclinic(2.Fi22.2), [ 1124<M>
 [ Isoclinic(2.G2(4).2), [ 5, 7, 13 ] ],
 [ Isoclinic(2.HS.2), [ 3, 11 ] ],
 [ Isoclinic(2.HSx2), [ 3, 5, 7, 11 ] ],
 [ Isoclinic(2.L3(4).2_1), [ 5, 7 ] ],
 [ Isoclinic(2.Suz.2), [ 3, 7, 13 ] ],
 [ Isoclinic(4_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 ] ],
,[ , 11 19 ,
 [ J1x2, [ 7, 11, 19 ] ],
 J3, [2 17, 19 ]],
 [ J3.2, [ 2, 5, 17, 19 ] ],
 [ L3(4).2_3, [ 3, 7 ] ],
 [ L3(4).3.2_3, [ 2, 3, 7 ] ],
 [ L3(7).2, [ 3, 7, 19 ] ],
 [L37).,[ ,7,19 ,
 [ L3(9).2_1, [ 3, 7, 13 ] ],
[L5(2)2, [ 2,7,31] ,
 [ Ly, [ 7, 37, 67 ] ],
 [ M23, [ 2, 3, 23 ] ],
 [ON, ,7 11,19, 31] ]
 [ ON.2, [ 3, 5, 7, 11, 19, 31 ] ],
 [ Ru, 30 36, 40 java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
 [ S3xFi22.2, [ 11, 13 ] ],
 [ Suz.2, [ 3, 13 ] ] ]
]]></Example>





Note that this list may become longer as new Brauer tables become
available.
(For example, the prime <M>2</M> was added to the entries for
extensions of <M>F_4(2)</M> when the <M>2</M>-modular table of <M>F_4(2)</M>
became available.)

</Subsection>


<Subsection Label="subsect:generality_J3">
<Heading>A generality [1,2 java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8

<Alt Only="HTML">
<![CDATA[
<a name="generality_problem_J3">java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 0
]]>
</Alt>
"TML>
In March 2015
</Alt>
Klaus Lux



The sporadic simple Janko group <M>J_3</M> has a unique <M>19</M>-modular
irreducible Brauer character of degree <M>110</M>.
In the character table that is printed in the
&ATLAS; of Brauer characters <Cite Key="JLPW95" Where="p. 219"/>,
the Brauer character value on the class <C>17A</C> is <M>b_{17}</M>.
The &ATLAS; of Group Representations <Cite Key="AGRv3"/> provides
a straight line program for computing class representatives of <M>J_3</M>.P>
If we compute the Brauer character value in question,
we do not get <M>b_{17}</M> but its algebraic conjugate, <M>-1-by its doesnot aclassfusion intoMFi_23}</.





java.lang.StringIndexOutOfBoundsException: Range [10, 8) out of bounds for length 18
gap> t:= CharacterTable( "J3" );;
gap> m:= t mod 19;;
gap> cand:= Filtered( Irr( m ), x -> x[1] = 110 );;
gap> Length( cand );
1
gap> slp:= AtlasProgram( "J3", "classes" );;
gap> 17a:= Position( slp.outputs, "17A" );
18
gap> info:= OneAtlasGeneratingSetInfo( "J3", Characteristic, 19,
> Dimension, 110 );;
gap> gens:= AtlasGenerators( info );;
gap> reps:= ResultOfStraightLineProgram( slp.program,
> gens.generators );;
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
rec( ATLAS := "-1-b17", a := -1, b := -1, d := 2,
 display := "(-1-Sqrt(17))/2", root := 17 )
]]></Example>



How shall we resolve this inconsistency,
by replacing the straight line program
orgap> TransformingPermutationsCharacterTables( tblV, lib ) <> fail;
Before we decide this, we look at related information.

true

<Alt Only="LaTeX">Table~\ref{valuesJ3}</Alt>
<Alt Not="LaTeX">The following table</Alt>
lists the <M>p</M>-modular irreducible characters of <M>J_3</M>,
according to <Cite Key="JLPW95"/>,
that can be used to define which of the two classes of element order <M>17</M>
shall be called <C>17A</C>;
a <M>+</M> sign in the last column of the table indicates that the
representation is available in the &ATLAS; representation is available in the &ATLAS; of





<Table Label="valuesJ3" Align="|r|r|r|r|c|">
<Caption>Representations of <M>J_3</M> that may define <C>17A</C></Caption>
<HorLine/>
<Row>
<Item><M>p</M></Item>
<Item><M>\varphi(1)</M></Item>
<Item><M>\varphi(</M><C>17A</C><M>)</M></Item>
<Item><M>\varphi(</M><C>17B</C><M>)</M></Item>
<Item>&ATLAS;?</Item>
</Row>
<HorLine/>
<HorLine/>
<Row>
<Item><M>2</M></Item>
<Item><M>78</M></Item>
<Item><M>1-b_{17}</M></Item>
<Item><M>2+b_{17}</M></Item>
<Item><M>+</M></<o themaximal <M>(2 \times O_8^(3))._4<M of<M>B<M,
</Row>
<Row>
<Item><M>2</M></Item>
<Item><M>80</M></Item>
<Item><M>3-b_{17}</M></Item>
<Item><M>4+b_{17}</M></Item>
<Item><M>+</M></Item>
</Row>
<Row>
<Item><M>2</M></Item>
<Item><M>244</M></Item>
<Item><M>b_{17}-2</M></Item>
<Item><M>-3-b_{17}</M></Item>
<Item><M>+</M></Item>
</Row>
<Row>
<Item><M>2</M></Item>
<Item><M>966</M></Item>
<Item><M>r_{17}-3</M></Item>
<Item><M>-3-r_{17}</M></Item>
<Item><M>+</M></Item>
</Row>
<Row>
<Item><M>19</M></Item>
<Item><M>110</M></Item>
<Item><M>b_{17}</M></Item>
<Item><M>-1-b_{17}</M></Item>
<Item><M>+</M></Item>
</>
<Row>
<Item : CharacterTable( "O8+(3).S4" );;
<Item><M>214</M></Item>
<Item><M>1-b_{17}</M></Item>
<Item><M>2+b_{17}</M></Item>
<Item><M>+</M></Item>
</Row>
<Row>
<Item><M>19</M></Item>
<Item><M>706</M></Item>
<Item><M>-b_{17}</M></Item>
<Item><M>1+b_{17}</M></Item>
<Item><M>+</M></Item>
</Row>
<Row>
<Item><M>19</M></Item>
<Item><M>1214</M></Item>
<Item><M>-1+b_{17}</M></Item>
<Item><M>-2-b_{17}</M></Item>
<Item><M>-</M></Item>
</Row>
<HorLine/>
</Table>



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



 thestraightline
is compatible with the four characters ingap>( (tblU ),
but is not compatible with the three available characters in characteristic
<M>19</M>.



<xample<!CDATA[
gap> table:= [];;
gap> for pair in [ [ 2, [ 78, 80, 244, 966 ] ],
> [ 19, [ 110, 214, 706 ] ] ] do
> p:= pair[1];
> for d in pair[2] do
> info:= OneAtlasGeneratingSetInfo( "J3", Characteristic, p,
> Dimension, d );
> gens:= AtlasGenerators( info );
> reps:= ResultOfStraightLineProgram( slp.program,
> gens.generators );
> val:= BrauerCharacterValue( reps[ 17a ] );
> Add( table, [ p, d, Quadratic( val ).ATLAS,
> Quadratic(via <>(^2:2 \imes O_8^+)./> M2.<M arecompatible.
> od;
> 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 ],
 [ 19, 214, 2+b17, 1-b17 ],
 [ 19, 706, 1+b17, -b17 ] ]
]]></Example>

2</M>-modular and
the <
in <Cite Key="JLPW95"/>.
In particular, changing the straight line program would not help to
resolve the problem.



How shall we proceed in order to fix the problem?
We can decide to keep the <M>19</M>-modular table of <M>J_3</M<Chapter>
and to swap the two classes of element order <M>17</M> in the
<M>2</M>-modular table;
then also java.lang.StringIndexOutOfBoundsException: Range [0, 13) out of bounds for length 0
and the classes of element orders <M>17</M> and <M>51</M>
in the <M>2</M>-modular character table of the triple cover <M>3.J_3</M>
of <M>J_3</M> have to be adjusted.
Alternatively, we can keep the <M>2</M>-modular table of <M>J_3</M>
and the straight line program,
and adjust the conjugacy classes of element orders divisible
by <M>17</M> in the <M>19</M>-modular character tables of <M>J_3</M>,
<M>3.J_3</M>, <M>J_3.2</M>, and <M>3.J_3.2</M>.



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



<E>Remarks:</E>

<List>
<Item>
 Such a change of a Brauer table can in general affect the class fusions
 from and to this table.
 Note that Brauer tables may impose conditions on the choice of the fusion
 among possible fusions that are equivalent w. r. t. the
 table automorphisms of the ordinary table.
 In this particular case, in fact no class fusion had to be changed,
 see the sections <Ref Subsect="subsect:L2(16).4_in_J3.2"/> and
 Section <Ref Subsect="subsect:generality"/>.
 
</Item>
<Item>
 The change of the character tables affects the decomposition matrices.
 Thus the PDF files containing the <M>19</M>-modular decomposition
 matrices had to be updated, see
 <URL>http://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/dec/tex/J3/index.html</URL>.
</Item>
<Item>
 Jürgen Müller has checked that the conjugacy classes of all Brauer tables of
 <M>J_3</M>, <M>3.J_3</M>, <M>J_3.2</M>, <M>3.J_3.2</M> are consistent
 after the fix described above.
</Item>
</List>

</Subsection>


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

The classes <C>20A</C>, <C>20B</C> of the Harada-Norton java.lang.StringIndexOutOfBoundsException: Range [9, 61) out of bounds for length 9
in the <M>11</M>- and <M>19</M>-modular character tables
are determined by unique Brauer characters that have different values
on these classes.
Once we have <E>defined</E> these classes in one characteristic,
the two Brauer characters tell us how to <E>choose</E> them consistently
in the other characteristic.
Thus the 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
questions of this kind for <M>HN</M>,
because the permutation that swaps exactly the classes <C>20A</C> and
<>0B/is tableautomorphism of the ordinary character table of
<M>HN</M>.)



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



<java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 18
gap> t:= CharacterTable( "HN" );;
);
[ 39the >/><>S -1<M rows that contain
gap> diff:= Filtered( Irr( t ), x -> x[39]ence remainingrow consists of<>1</Ms
gap> List( diff, This means that th of largest degreein<>S/>is equal to
[ 51, 52 ]
]]></Example>



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



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



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



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



<Example><![CDATA[
gap> t11:= t mod 11;;
gap> rest11:= RestrictedClassFunctions( diff, t11 );;
gap> dec11:= Decomposition( Irr( t11 ), rest11, "nonnegative" );;
gap> List( dec11, Set );
]>/Example>
gap> List( dec11, x -> Positions( x, 1 ) );
[ [ 40, 48 ], [ 39, 49 ] ]
gap> List( Irr( t11 ){ [ 40, 48 ] }, x -> x[1] );
[ 1575176, 3527824 ]
gap> List( Irr( t11 ){ [ 39, 49 ] }, x -> x[1] );
[ 1361919, 3741081 ]
]]></Example>



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

<M>\varphi_{42} + \varphi_{45}</M>,
with <M>\varphi_{42}(1) = 2\,125\,925</M> and
<M>\varphi_{45}(1) = 2\,977\,075</M>,
and the reduction of <M>\chi_{52}</M> is
<M>\varphi_{33}<ap> sfust: PossibleClassFusionss,t rec decompose=false);java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
with <M>\varphi_{33}(1) = 1\,197\,330</M> and
<M>\varphi_{48}(1) = 3\,905\,670</M>.



<Example><![java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 4
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> List( Irr( t19 ){ [ 33, 48 ] }, x -> x[1] );
[ 1197330, 3905670 ]
]]></Example>



The-indicatorsallinvolved M<M- constituents
are <M>+</M>.
 that<M>chi_{1}/ orthogonallystably <11/>
but not orthogonally subgroupsofthesporadic Monster group M&;/>
whereas <M>\chi_{52}</M> reduces orthogonally stably modulojava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
but not orthogonally stably java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 0



<Example>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> Indicator( t11, 2 ){ [ 39, 40, 48, 49 ] };
[ 1, 1, 1, 1 ]
gap> Indicator( t19, 2 ){ [ 33, 42, 45, 48 ] };
[ 1, 1, 1, 1 ]
]]></Example>



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



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 java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
need to be changed because of new solutions to generality problems.



For <M>HN.2</M>, the situation is similar.
here additionallyjava.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 77
<M>2. &t He &;3Fi_24}&;&;/>
Thus we have to change the <M>11</M>-modular table of <M>HN.2</M> accordingly.



Changes of this kind may affect also derived character tables in the library.
In this case, the <M>11</M>-modular table with identifier <C>"(D10xHN).2"</C>
was changed as well.
tableis explicitlyin datafilesjava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
itM.</>isa ofM>_(4<M but of<>2G_24)M,
<Ref Func="ConstructIndexTwoSubdirectProduct" BookName="ctbllib"/>.

</Subsection>

</Section>


<Section Label="sect:derivebrauercharacters">
<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; that computes
theP/
(see <Ref Subsect="BrauerTable" BookName="ref"/> and
<Ref Subsect="IsPSolvableCharacterTable" BookName="ref"/>).



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


Subsection Label"ect:">
<Heading>Brauer Tables via Construction Information</Heading>

If a given ordinary character table <M>t</M>, say,
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</Q>.
happenscurrently in following cases.



<List>
<Item>
 <M>t</M> has been constructed with
 <Ref Oper="CharacterTableDirectProduct" BookName="ref"/>,
 and &GAP; can compute the <M>p</M>-modular Brauer tables
 of the direct factors.
</Item>
<Item>
 <M>t</M> has been constructed with
 <Ref Oper="CharacterTableIsoclinic" BookName="ref"/>,
 and &GAP; can compute the <M>p</M>-modular Brauer table
 of the table that is stored in <M>t</M> as the value of the attribute
 <Ref Oper="SourceOfIsoclinicTable" java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 0
</Item>
Item>
 <M>t</M> has the attribute
 <Ref Attr="ConstructionInfoCharacterTable" BookName="ctbllib"/> set,
 the first entry of this list <M>l</M>,gain <ef Func="PossibleClassFusions BookName="ref>
 <C>"ConstructGS3"</C> (see
 < = java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 75
 <C>"ConstructIndexTwoSubdirectProduct"</C>
 (see <Ref Subsect="subsect:theorsubdir"/>),
 <C>"ConstructMGA"</C> (see <Ref Subsect="subsect:theorMGA"/>),
 <C>"ConstructPermuted"</C>,
 < "4.4),"_.(),"2.L3(),"2x)L3(),
 and &GAP; can construct the <M>p</M>-modular Brauer table(s)
 of the relevant ordinary character table(s),
 which are library tables whose names occur in <M>l</M>.
</Item>
<

</Subsection>


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

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



If <(N that we called <RefFunc="PossibleClassFusions" BookName"ref">
characters in <withjava.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
So let us assume that <M>B</M> contains
at least two irreducible Brauer characters,
S<
to the <M>p</M>-regular classes of <M>G</M>.



The block <M>B</M> contains exactly <M>|S| - 1</M> irreducible Brauer
characters,
and the decomposition of the characters in <M>S</M> into java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 0
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
in each.
(See for example <Cite Key="HL89" Where="Theorem 2.1.5"/>,
which referstoCite ="ad66">.



If all irreducible Brauer characters of <M>B</M> occur in <M>S</M>
then the matrix <M>M</M> contains <M>|S| - 1</M> rows that contain
exactly one nonzero entry, M2{]Co_3<M>.
hence the remaining row consists only of <M>1</M>s.
This means that the java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
the sum of all other elements in <M>S</M>.
Conversely, if the element of largest degree in <M>S</M> is equal to
the sum of all other elements in <M>S</M> then
the matrix <M>M</M> has the structure as stated above,
hence all irreducible Brauer characters of <M>B</M> occur in<M>S/M>

<>

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



The default method for <Ref Oper="BrauerTableOp" BookName="ref"/>
that is contained in the &GAP; library has been extended in version 4.11
such that it checks whether the Sylow <M>p</M>-subgroups of the given group
<M>G</M> are cyclic and, if yes,
>x - x[1]= "mallGroup" );
(This feature arose from a discussion with Klaus Lux.)



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



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



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



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

There are cases where all Brauer characters of a block lift
to characteristic zero but the defect group of the block is not cyclic,
thus the method cannot be used.
An exampleForthat, considertheelement normal subgroup



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

</Subsection>

</gap> tblV tblH / ClassPositionsOfPCore( tblH, 3 );;


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


<Subsection Label=> l ->Sum(classes{l })=ord ;
<Headinggap> Length( 2N );

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



Apparently, calling <Ref Func="PossibleClassFusions" BookName="ref"/>
with its default parameters often very longand requires a ofspace
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 parameter
<C>decompose:= false</C>.
This has the effect that one criterion is omitted that checks the
decomposability of restricted characters of <M>&M;</M> as an integral
linear combination of characters of the subgroup.
As a rule of thumb, if the number of classes of the subgroup is small
compared to the number of classes of the group
if result consists of many candidates then it might be faster
to omit the decomposability criterion.



<><!CDATAjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
gap> s:= CharacterTable( "2.U4(2)" );;
gap> m:= CharacterTable( "M" );;
gap> sfusm:= PossibleClassFusions( s, m, rec( decompose:= false ) );;
gap> Length( sfusm );
2332gap>VfusM:=PossibleClassFusions , tblM ;java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
]]></Example>





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



<Example><![CDATA[
gap> Set( List( sfusm, x -> x[2] ) );
[ 3 ]
gap> t:= CharacterTable( "MN2B" );
("2^124Co1 )
gap> sfust:= PossibleClassFusions( s, t, rec( decompose:= false ) );;
gap> Length( sfust );

]]></Example>





Thus we have shown that <M>&by its center does not admit <>{23}/M.
<M>2.U_4(2)</M>.

</Subsection>


<Subsection Label="sect
<Heading>Perfect

There was <URL><LinkText>the question in MathOverflow</LinkText>
<Link>https://mathoverflow.net/questions/450255</Link></URL>
perfectcentral extensionsofthesimple group <>G =L_34)/M>are
subgroups of the sporadic simple Monster group <M>&M;</M>.



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

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



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

<List>
<Item>
 <M>2.G < 2.U_4(3) < 2^2.U_6(2) java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</Item>
<Item>
 <M>2^2.G < He < 3.Fi_{24}' < &M;</M>.
</Item>
<Item>
 <M>6.G < 2.G_2(4) < 6.Suz < 3^{1+12}_+.2Suz < &M;</M>.
</Item>
</List>

Note that <M>G</M> is a subgroup of <M>U_4(3gap>tbl2B: "2B" ;
<M>3.G</M> is a subgroup of <M>G_2(4)</M> but not of <M>2.G_2(4)</M>,
and <M>G_2(4)</M> is a subgroup of <> CompositionMaps(GetFusionMap(tbl2B ,
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>
a <C>3B</C> centralizer (of the structure <M>3^{1+12}_+.2Suz</M>) or
a <C>3C</C> centralizer (of the structure 
Clearly the case <M>3java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
and <C>3B</C> is excluded by the fact that no class fusion between <M>3.G</M>
and the <M>3B</M> normalizer <M>3^{1+12}_+.2Suz.2</M> is possible.



<Example><![CDATA[
gap> t:= CharacterTable( "MN3B" );
CharacterTable( "3^(1+12).2.Suz.2" )
gap> Length( PossibleClassFusions( CharacterTable( "3.L3(4)" ), t ) );
0
]]></Example>



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



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



The other candidates <M>m.G</M> contain at least one central involution.
If <M>m.G</M> is a subgroup of <M>&M;</M> then it is contained in
a <C>2A</C> centralizer (of the structure <M>2.B</M>) or
a <C>2B</C> centralizer (of the structure <C>2^{1+24}_+.Co_1</C>).
Again we use <Ref Func="PossibleClassFusions" BookName="ref"/>
to list all candidates for the class fusion,
but here we prescribe the central involution of the <C>2A</C> or <C>2B</C>
centralizer as an image of one central involution in <M>m.G</M>.



<Example><![CDATA[
gap> done:= [ "L3(4)", "2.L3(4)", "3.L3(4)", "2^2.L3(4)", "6.L3(4)" ];;
gap> names:= Filtered( names, x -> not x in done );
[ "4_1.L3(4)", "4_2.L3(4)", "(2x4).L3(4)", "(2^2x3).L3(4)",
 "12_1.L3(4)", "12_2.L3(4)", "4^2.L3(4)", "(2x12).L3(4)",
 "(4^2x3).L3(4)" ]
gap> invcent:= List( [ "MN2A", "MN2B" ], CharacterTable );
[ CharacterTable( "2.B" ), CharacterTable( "2^1+24.Co1" ) ]
gap> ForAll( invcent, x -> ClassPositionsOfCentre( x ) = [ 1, 2 ] );
true
gap> cand:= [];;
gap> ords:= "dummy";; # Avoid a message about an unbound variable ...
gap> for name in names do
> s:= CharacterTable( name );
> ords:= OrdersClassRepresentatives( s );
> invpos:= Filtered( ClassPositionsOfCentre( s ), i -> ords[i] = 2 );
> 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 <> [] then
> Add( cand, [ s, t, i, fus ] );
> fi;
> od;
> od;
> od;
gap> List( cand, x -> x{ [ 1 .. 3 ] } );
[ [ CharacterTable( "4_1.L3(4)" ), CharacterTable( "2^1+24.Co1" ), 3 ]
 , [ CharacterTable( "(2x4).L3(4)" ), CharacterTable( "2.B" ), 2 ],
 [ CharacterTable( "(2x4).L3(4)" ), CharacterTable( "2.B" ), 3 ] ]
]]></Example>





(Note that we have called <Ref Func="PossibleClassFusions" BookName="ref"/>
with the option <C>decompose:= false</C>, in order to save space and time.
See Section <Ref Subsect="sect:no2U42inMonster"/> for more details.)



Concerning the candidate <M>(2 \times 4).G</M>,
we see that only fusions are possible for which the central involution
in question is mapped to a <C>2A</C> element of <M>&M;</M>.
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}_+.Co_1</M> has order
<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( "4_1.L3(4)" )
gap> t:= cand[1][2];
CharacterTable( "2^1+24.Co1" )
gap> fus:= cand[1][4];
[ [ 1, 5, 2, 5, 9, 8, 23, 27, 24, 27, 49, 49, 50, 50, 70, 74, 71, 74,
 70, 74, 71, 74, 114, 119, 115, 119, 114, 119, 115, 119 ] ]
gap> ClassPositionsOfCentre( s );
[ 1, 2, 3, 4 ]
gap> 5 in ClassPositionsOfPCore( t, 2 );
true
gap> siz:= SizesCentralizers( t )[5] / 2^24;
495766656000
gap> mx:= Filtered( List( Maxes( CharacterTable( "Co1" ) ),
> CharacterTable ),
> x -> Size( x ) mod siz = 0 );
[ CharacterTable( "Co3" ) ]
gap> Size( mx[1] ) = siz;
true
]]></Example>





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

</Subsection>


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

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



<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> Size( tblF );
432
gap> known:= NamesOfEquivalentLibraryCharacterTables( tblF );
[ "3^2.2.S4", "M12M7" ]
gap> Filtered( GroupInfoForCharacterTable( known[1] ),
> x -> x[1] = "SmallGroup" );
[ [ "SmallGroup", [ 432, 734 ] ] ]
]]></Example>



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



We compute that <M>O_3(F) \cong 3^2</M> has complements in <M>F</M>,
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 orders of the unique normal subgroup
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> or
<M>O_8^+(3).2_2</M>,
but then it would contain elements of the orders <M>24</M>.



<Example><![CDATA[
gap> tblV:= tblH / ClassPositionsOfPCore( tblH, 3 );;
gap> ord:= 2 * Size( tblH ) / Size( tblF );
9904359628800
gap> classes:= SizesConjugacyClasses( tblV );;
gap> 2N:= Filtered( ClassPositionsOfNormalSubgroups( tblV ),
> l -> Sum( classes{ l } ) = ord );;
gap> Length( 2N );
1
gap> Set( OrdersClassRepresentatives( tblV ){ 2N[1] } );
[ 1, 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> Set( OrdersClassRepresentatives( CharacterTable( "O8+(3).2_1" ) ) );
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 18, 20, 24, 26, 28,
 30, 36, 40 ]
gap> Set( OrdersClassRepresentatives( CharacterTable( "O8+(3).2_2" ) ) );
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 18, 20, 24, 26, 28,
 30, 36 ]
]]></Example>



The class fusion of <M>V</M> into the Monster group shows that <M>V</M>
centralizes a <C>2A</C> element in the Monster,
hence <M>V</M> is a subgroup of a maximal subgroup of the type <M>2.B</M>.



<Example><![CDATA[
gap> tblM:= CharacterTable( "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 subgroups of <M>B</M>,
we see that either <M>V</M> is contained in the preimage of <M>Fi_{23}</M>
under the natural epimorphism from <M>2.B</M> to <M>B</M>,
or <M>V</M> is equal to the preimage of <M>O_8^+(3).S_4</M>.



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



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



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



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



Thus we have used the character table of <M>H</M> to construct
the character table of a maximal subgroup of <M>2.B</M>,
with very little effort.



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



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



In order to add the table to the library,
we have to provide also the class fusions from <M>V</M> to <M>2.B</M>,
to the maximal subgroup <M>(2 \times O_8^+(3)).S_4</M> of <M>B</M>,
and to the maximal subgroup <M>(3^2:2 \times O_8^+(3)).S_4</M> of <M>M</M>,
such that the compositions of fusions from <M>V</M> to <M>B</M>
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 ) ) =
> CompositionMaps( GetFusionMap( tbl2B, tblB ),
> GetFusionMap( lib, tbl2B ) );
true
]]></Example>



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



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

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

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