<!-- %W maintain.xml GAP 4 package CTblLib Thomas Breuer -->
<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 the &GAP; Character Table Library satisfy the usual tests.
However,
there are table candidates for which these tests are not good enough.
<!-- (mention that this should be run when a table is going to be added) --> <!-- (example: the candidate with nonintegral structure constants) -->
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.
(This example arose from a discussion with Jack Schmidt.)
<P/>
Up to version 1.1.3 of the &GAP; Character Table Library,
the table with identifier <C>"P41/G1/L1/V4/ext2"</C> was not correct.
The problem occurs already in the microfiches
that are attached to <Cite Key="HP89"/>.
<P/>
In the following, we show that this table is not the character table
of a finite group,
using the &GAP; library of perfect groups.
Currently we do not know how to prove this inconsistency
alone from the table.
<P/>
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.
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.
The &GAP; Library of Perfect Groups contains representatives of the
four isomorphism types of perfect groups of order <M>|G| = 64\,512</M>.
<P/>
<Example><![CDATA[
gap> n:= Size( tbl );
64512
gap> NumberPerfectGroups( n );
4
gap> grps:= List( [ 1 .. 4 ], i -> PerfectGroup( IsPermGroup, n, i ) );
[ L2(8) 2^6 E 2^1, L2(8) N 2^6 E 2^1 I, L2(8) N 2^6 E 2^1 II,
L2(8) N 2^6 E 2^1 III ]
]]></Example>
<P/>
If we believe that the classification of perfect groups of order <M>|G|</M>
is correct then all we have to do is to show that none of the
character tables of these four groups is equivalent to the given table.
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>, and <M>63</M>;
moreover, the value <M>7</M> occurs in exactly two classes.
<P/>
<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>
<P/>
(Another way to state this is that in each of the four tables <M>t</M> in
question,
there are ten preimage classes of the involution class in the simple
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>.)
<P/>
In the erroneous table, however,
there is only one class with the value <M>7</M> in this character.
<P/>
<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>
<P/>
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.
<P/>
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.)
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> corresponds to the group
<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>PerfectGroup( 64512, 2 )</C>,
have equivalent matrices of irreducibles.
<!-- % Let us suppose that we are not convinced yet, % perhaps because it might be that a perfect group of order <M>64\,512</M> % has been overlooked up to now. % Then we consider the factor group <M>F</M> of <M>G</M> modulo its centre. % % <Example><![CDATA[ % gap> cen:= ClassPositionsOfCentre( tbl ); % [ 1, 4 ] % gap> facttbl:= tbl / cen; % CharacterTable( "P41/G1/L1/V4/ext2/[ 1, 4 ]" ) % ]]></Example> % % The group <M>F</M> is a perfect group of order <M>32\,256</M>. % According to the &GAP; Library of Perfect Groups, % there are exactly two such groups, up to isomorphism, % and <M>F</M> must be isomorphic to the second of these groups. % % <Example><![CDATA[ % gap> factgrps:= List( [ 1 .. 2 ], i -> PerfectGroup( IsPermGroup, n/2, i ) ); % [ L2(8) 2^6, L2(8) N 2^6 ] % gap> facttbls:= List( factgrps, CharacterTable );; % gap> List( facttbls, x -> IsRecord( % > TransformingPermutationsCharacterTables( x, facttbl ) ) ); % [ false, true ] % ]]></Example> % % (In fact the situation is a bit more subtle: % The matrices of irreducibles of the two perfect groups of order <M>32\,256</M> % are equivalent, % the two character tables differ just by their second power maps and, % as a consequence, by their element orders. % However, the second power map of the given table of <M>G</M> % is uniquely determined by the matrix of irreducibles of this table.) % % No!! % % -> what about element orders? % % <Example><![CDATA[ % gap> IsRecord( TransformingPermutations( % > Irr( facttbls[1] ), Irr( facttbls[2] ) ) ); % true % gap> IsRecord( TransformingPermutationsCharacterTables( % > facttbls[1], facttbls[2] ) ); % false % % ]]></Example> % % % ... % % (And the power maps are uniquely det. by the matrix.) % % (and not equiv. to the other! % note that one is split, the other nonsplit, % and the matrices of irreds are equivalent) % % So if we believe that there is no other perfect group of order <M>32\,256</M> % that has the same character table % then <M>G</M> is a central extension of this group by a group of order two. % % We compute the possible central extensions of a maximal subgroup % of the structure <M>2^6.2^3:7</M>, % and show that none of them admits an embedding into the table. % % (Note that the extensions can be computed for solvable groups, % that's why we switch to a subgroup.) % % <Example><![CDATA[ % gap> sort; % CharacterTable( "P41/G1/L1/V4/ext2" ) % gap> g:= PerfectGroup( IsPermGroup, 32256, 2 );; % gap> nsg:= NormalSubgroups( g ); % ... % gap> epi:= NaturalHomomorphismByNormalSubgroup( g, nsg[2] );; % gap> img:= Image( epi );; % gap> mx:= MaximalSubgroupClassReps( img ); % [ Group([ (3,4,8,6,7,5,9), (1,2)(4,9)(5,8)(6,7) ]), Group([ (1,2,3)(4,7,8)(5,6,9), (1,4,9,2,7,5,3,8,6), (2,3)(4,6)(5,7)(8,9) ]), Group([ (2,3)(4,6)(5,7)(8,9), (2,4)(3,6)(5,9)(7,8), (2,5)(3,7)(4,9)(6,8), % (3,4,8,6,7,5,9) ]) ] % gap> List( mx, Size ); % [ 14, 18, 56 ] % gap> pre:= PreImages( epi, mx[3] );; % gap> iso:= IsomorphismPcGroup( pre );; % gap> G:= Image( iso );; % gap> mats:= List( Pcgs( G ), x -> IdentityMat( 1, GF(2) ) );; % gap> M:= GModuleByMats( mats, GF(2) );; % gap> ext:= Extensions( G, M );; % gap> Length( ext ); % 8 % gap> List( ext, Size ); % [ 7168, 7168, 7168, 7168, 7168, 7168, 7168, 7168 ] % gap> tbls:= List( ext, CharacterTable );; % gap> List( tbls, t -> Length( PossibleClassFusions( t, sort ) ) ); % [ 0, 0, 0, 0, 0, 0, 0, 0 ] % ]]></Example> % % % now: how to replace the wrong table by the correct one! % -> to which of the two? % -> note that V1/ext2 and V2/ext2 belong to the first and third group, % they have equiv. matrices; % V4/ext2 is wrong, and there are the second and fourth group, % again with equiv. matrices but different power maps! % (so add two tables? with which names?) % % <Example><![CDATA[ % ]]></Example> %
-->
</Subsection>
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<Subsection Label="subsect:errorE62">
<Heading>An Error in the Character Table of <M>E_6(2)</M> (March 2016)</Heading>
In March 2016, Bill Unger computed the character table of the simple group
<M>E_6(2)</M> with Magma (see <Cite Key="CP96"/>)
and compared it with the table that was contained in the
&GAP; Character Table Library since 2000.
It turned out that the two tables did not coincide.
<P/>
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 in both tables;
thus it may be that the assumption of a wrong power has implied the wrong
character values, or vice versa.)
Specifically, the <M>11</M>th power map in the &GAP; table
fixed all elements of order <M>91</M>.
Using the smallest matrix representation of <M>E_6(2)</M> over the field with
two elements, one can easily find an element <M>g</M> of order <M>91</M>,
and show that the characteristic polynomials of <M>g</M> and <M>g^{11}</M>
differ.
Hence these two elements cannot be conjugate in <M>E_6(2)</M>.
In other words, the &GAP; table was wrong.
<P/>
<Example><![CDATA[
gap> g:= AtlasGroup( "E6(2)" );;
gap> repeat x:= PseudoRandom( g ); until Order( x ) = 91;
gap> CharacteristicPolynomial( x ) = CharacteristicPolynomial( x^11 );
false
]]></Example>
<P/>
The wrong &GAP; table has been corrected in version 1.3.0 of the
&GAP; Character Table Library.
<!-- The wrong table involved irrationalities E(91)^6+E(91)^20+E(91)^31+E(91)^38+E(91)^48+E(91)^54+E(91)^66+E(91)^68 +E(91)^69+E(91)^73+E(91)^75+E(91)^89, and the new one contains irrationalities E(91)^17+E(91)^27+E(91)^34+E(91)^45+E(91)^54+E(91)^59+E(91)^68+E(91)^75 +E(91)^83+E(91)^87+E(91)^89+E(91)^90.
These values define different subfields of CF(91). -->
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:error2F422">
<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 group <M>2.F_4(2).2</M> was computed
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 <M>2.F_4(2).2</M> had been wrong.
<P/>
In fact, this was the one and only case of a power map for an &ATLAS; group
which was not determined by the character table,
and the <Ref Attr="InfoText" BookName="ref"/> value of the character table
had mentioned the two alternatives.
<P/>
Note that the ambiguity is not present in the table of the factor group
<M>F_4(2).2</M>, and only four faithful irreducible characters of
<M>2.F_4(2).2</M> distinguish the four relevant conjugacy classes.
I had not found a suitable subgroup of <M>2.F_4(2).2</M> whose
character table could be used to decide the question which of the
two alternatives is the correct one.
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:LyN2">
<Heading>A Character Table with a Wrong Name (May 2017)</Heading>
(This example is much older.)
<P/>
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;.
<P/>
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 centralizer order <M>2^6</M>,
and this does not occur in <M>Ly</M>.
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 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>P</M>.
Clearly the Sylow <M>2</M> subgroups in the factor group <M>McL.2</M>
are isomorphic with <M>P</M>.
Thus we can start with a rather small permutation representation.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:spacegroupfactors">
<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).
<P/>
Using the information from <Cite Key="HP89"/>,
it is straightforward to construct
such groups as factor groups of infinite groups.
Since version 1.3.0 of the &CTblLib; package,
calling
<Ref Func="GroupInfoForCharacterTable" BookName="ctbllib"/>
for the <M>16</M> library tables
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 undocumented auxiliary function that becomes available
automatically when
<Ref Func="GroupInfoForCharacterTable" BookName="ctbllib"/>
has been called for the first time.
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 groups in question</Heading>
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
the <E>point group</E> <M>P</M> of <M>S</M> is a finite subgroup of
GL<M>(d, &ZZ;)</M>,
the <E>translation lattice</E> <M>T</M> of <M>S</M> is a sublattice of
<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
<M>&ZZ;^d</M>.
Note that <M>V</M> maps the identity matrix <M>I \in</M> GL<M>(d, &ZZ;)</M>
to the zero vector,
and <M>M(T):= \{ M(I, t); t \in T \}</M>
is a normal subgroup of <M>S</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>.
<P/>
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 by the vectors <M>V(g_1), V(g_2), \ldots, V(g_k)</M>.
<P/>
In the examples below, the matrix representation of <M>P</M> is irreducible,
so we need just the following <M>k+1</M> elements to generate <M>S</M>:
These generators are returned by the function
<C>generatorsOfPerfectSpaceGroup</C>,
when the inputs are <M>[ g_1, g_2, \ldots, g_k ]</M>,
<M>[ V(g_1), V(g_2), \ldots, V(g_k) ]</M>, and <M>b_1</M>.
<P/>
<Example><![CDATA[
gap> generatorsOfPerfectSpaceGroup:= function( Pgens, V, t )
> local d, result, i, m;
> d:= Length( Pgens[1] );
> result:= [];
> for i in [ 1 .. Length( Pgens ) ] do
> m:= IdentityMat( d+1 );
> m{ [ 1 .. d ] }{ [ 1 .. d ] }:= Pgens[i];
> m[ d+1 ]{ [ 1 .. d ] }:= V[i];
> result[i]:= m;
> od;
> m:= IdentityMat( d+1 );
> m[ d+1 ]{ [ 1 .. d ] }:= t;
> Add( result, m );
> return result;
> end;;
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:constructfactors">
<Heading>Constructing the factor groups in question</Heading>
The space group <M>S</M> acts on <M>&ZZ;^d</M>,
via <M>v \cdot M(g, t) = v g + V(g) + t</M>.
A (not necessarily faithful) representation of <M>S/M(n T)</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 vectors modulo <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</C>
when this is called with argument <M>n</M>.
<P/>
<Example><![CDATA[
gap> multiplicationModulo:= n -> function( v, g )
> return List( v * g, x -> x mod n ); end;;
]]></Example>
<P/>
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>deletedPermutationMat</C>,
cf. <Cite Key="HP89" Where="p. 269"/>.
<P/>
<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 ^ pi;
> if j <> n then
> for i in [ 1 .. n-1 ] do
> mat[i][j]:= -1;
> od;
> fi;
> return mat;
> end;;
]]></Example>
<P/>
After constructing permutation generators for the example groups,
we verify that the groups fit to the character tables from the
&GAP; Character Table Library and to the permutation generators
stored for the construction of the group via
<C>CTblLib.FactorGroupOfPerfectSpaceGroup</C>.
<P/>
<!--
(In earlier versions of &GAP;, a call of the function <Ref Func="SmallerDegreePermutationRepresentation" BookName="ref"/> was sufficient for computing a reasonably small permutation representation of each of the example groups. Meanwhile, this function has become more sophisticated, with the effect that it requires much more space in some of our cases. Therefore, we reduce the number of points by hand.)
<Example><![CDATA[ gap> verifyFactorGroup:= function( gens, id ) > local g, s, sgens, act, sm, stored, hom; > # sm:= SmallerDegreePermutationRepresentation( Group( gens ) ); > g:= Group( gens ); > s:= NormalClosure( g, Subgroup( g, [ gens[ Length( gens ) ] ] ) ); > sgens:= MinimalGeneratingSet( s ); > s:= Subgroup( s, sgens{ [ 2 .. Length( sgens ) ] } ); > act:= Action( g, RightTransversal( g, s ), OnRight ); > if Size( act ) <> Size( g ) then > Error( "wrong group order!" ); > fi; > gens:= GeneratorsOfGroup( act ); > sm:= SmallerDegreePermutationRepresentation( act ); > 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>
-->
<P/>
<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"/>).
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>.
<P/>
<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>
<P/>
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.
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.
<P/>
<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>
<P/>
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.
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>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<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"/>).
The first vector system is the trivial vector system <M>V_1</M>
(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>.
<P/>
The library character table with identifier <C>"P11/G1/L1/V1/ext4"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(4 L_1)</M>, so we compute the action on an orbit modulo <M>4</M>.
The second vector system is <M>V_2</M>,
and the translation lattice is <M>2 L_1</M>.
<P/>
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>.
The third vector system is <M>V_3</M>,
and the translation lattice is <M>2 L_1</M>.
<P/>
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>.
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/>
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>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:expl-sl27">
<Heading>Example with point group SL<M>_2(7)</M></Heading>
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"/>).
The vector system is the trivial vector system <M>V_1</M>,
and the translation lattice is the sublattice <M>L_2</M> of <M>&ZZ;^d</M>
that has the following basis,
which is called <M>B(2,8)</M> in <Cite Key="HP89" Where="p. 269"/>.
For the sake of simplicity, we rewrite the action to one on <M>L_2</M>.
<P/>
<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>
<P/>
The library character table with identifier <C>"P12/G1/L2/V1/ext2"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(2 L_2)</M>.
The action on an orbit modulo <M>2</M> is not faithful,
its kernel contains the centre of SL<M>(2,7)</M>.
We can compute a faithful representation by acting on pairs:
One entry is the usual vector and the other entry carries the action
of the point group.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:expl-23L32">
<Heading>Example with point group <M>2^3.L_3(2)</M></Heading>
There is one example with <M>d = 7</M>.
The generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 297"/>).
The vector system is the trivial vector system <M>V_1</M>,
and the translation lattice is the sublattice <M>L_2</M> of <M>&ZZ;^d</M>
that has the following basis,
which is called <M>B(2,7)</M> in <Cite Key="HP89" Where="p. 269"/>.
For the sake of simplicity, we rewrite the action to one on <M>L_2</M>.
<P/>
<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> cbas:= List( bas, x -> Coefficients( B, x * c ) );;
]]></Example>
<P/>
The library character table with identifier <C>"P13/G1/L2/V1/ext2"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(2 L_2)</M>, so we compute the action on an orbit modulo <M>2</M>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:expl-A6">
<Heading>Examples with point group <M>A_6</M></Heading>
There are two examples with <M>d = 10</M>.
In both cases, the generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 307"/>).
In both examples, the vector system is the trivial vector system <M>V_1</M>,
and the translation lattices are the lattices <M>L_2</M> and <M>L_5</M>,
respectively, which have the following bases.
For the sake of simplicity, we rewrite the action to actions on <M>L_2</M>
and <M>L_5</M>, respectively.
<P/>
<Example><![CDATA[
gap> B2:= Basis( Rationals^Length( bas2 ), bas2 );;
gap> bbas2:= List( bas2, x -> Coefficients( B2, x * b ) );;
gap> cbas2:= List( bas2, x -> Coefficients( B2, x * c ) );;
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>
<P/>
The library character table with identifier <C>"P21/G3/L2/V1/ext2"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(2 L_2)</M>, so we compute the action on an orbit modulo <M>2</M>.
The library character table with identifier <C>"P21/G3/L5/V1/ext2"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(2 L_5)</M>, so we compute the action on an orbit modulo <M>2</M>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:expl-L28">
<Heading>Examples with point group <M>L_2(8)</M></Heading>
There are two examples with <M>d = 7</M>.
In both cases, the generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 327"/>).
The library character table with identifier <C>"P41/G1/L1/V3/ext3"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(3 L)</M>, so we compute the action on an orbit modulo <M>9</M>.
<P/>
The orbits in this action are quite long.
we choose a seed vector from the fixed space of an element of order <M>7</M>.
The library character table with identifier <C>"P41/G1/L1/V4/ext3"</C>
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>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:expl-M11">
<Heading>Example with point group <M>M_{11}</M></Heading>
There is one example with <M>d = 10</M>.
The generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 334"/>).
The library character table with identifier <C>"P48/G1/L1/V2/ext2"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(2 L)</M>, so we compute the action on an orbit modulo <M>4</M>.
<P/>
<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( sgens, x -> Permutation( x, orb, fun ) );;
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 group are as follows
(see <Cite Key="HP89" Where="p. 335"/>).
The library character table with identifier <C>"P49/G1/L1/V2/ext3"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(3 L)</M>, so we compute the action on an orbit modulo <M>9</M>.
<P/>
<Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 9 );;
]]></Example>
<P/>
The orbits in this action are quite long.
we choose a seed vector from the fixed space of an element of order <M>12</M>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:expl-U42">
<Heading>Examples with point group <M>U_4(2)</M></Heading>
There are two examples with <M>d = 6</M>.
In both cases, the generators of the point group are as follows
(see <Cite Key="HP89" Where="p. 336"/>).
The library character table with identifier <C>"P50/G1/L1/V1/ext3"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(3 L_1)</M>, so we compute the action on an orbit modulo <M>3</M>.
<P/>
<Example><![CDATA[
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
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>
<P/>
The library character table with identifier <C>"P50/G1/L1/V1/ext4"</C>
belongs to the factor group of <M>S</M> modulo the normal subgroup
<M>M(4 L_1)</M>, so we compute the action on an orbit modulo <M>4</M>.
<P/>
<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, 1 ], fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P50/G1/L1/V1/ext4" );
true
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:A-remark-on-one-of-the-example-groups">
<Heading>A remark on one of the example groups</Heading>
The (perfect) character table with identifier <C>"P1/G2/L2/V2/ext4"</C>
has the property that its character degrees are exactly
the divisors of <M>60</M>.
There are nilpotent groups with the same set of character degrees,
for example the direct product of four extraspecial groups of the orders
<M>2^3</M>, <M>2^3</M>, <M>3^3</M>, and <M>5^3</M>, respectively.
This phenomenon has been described in <Cite Key="NR14"/>.
The term <Q>generality problem</Q> is used for problems concerning
consistent choices of conjugacy classes of Brauer tables for the same
group, in different characteristics.
The definition and some examples are given
in <Cite Key="JLPW95" Where="p. x"/>.
<P/>
Section <Ref Subsect="subsect:generalityproblems_list"/>
shows how to detect generality problems
and lists the known generality problems,
and Section <Ref Subsect="subsect:generality_J3"/> gives an example
that actually arose.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:generalityproblems_list">
<Heading>Listing possible generality problems</Heading>
We use the following idea for finding character tables which
may involve generality problems.
(The functions shown in this section are based on &GAP; 3 code
that was originally written by Jürgen Müller.)
<P/>
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,
in the sense that some symmetry of the corresponding ordinary table
<M>tbl</M>, say, is broken in <M>mtbl</M>.
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 obtained as the orbit of this
class fusion under the actions of table automorphisms of <M>tbl</M>,
via mapping the images of the class fusion
(with the function <Ref Func="OnTuples" BookName="ref"/>),
and of the table automorphisms of <M>mtbl</M>,
via permuting the preimages.
The case of broken symmetries occurs if and only if this orbit
splits into several orbits when only the action of the
table automorphisms of <M>mtbl</M> is considered.
Equivalently, symmetries are broken if and only if the orbit under
table automorphisms of <M>mtbl</M> is not closed under the action of
table automorphisms of <M>tbl</M>.
<!-- It is sufficient to test generators of 'taut' because 'orb' is invariant under the action of 'taut'
if and only if everey generator leaves 'orb' invariant. -->
<E>Remark:</E> (Thanks to Klaus Lux for discussions on this topic.)
<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>
between the two tables with <M>\sigma_m</M>
and permuting the image classes with <M>\sigma_o</M> yields <M>f</M>.
<P/>
For example, consider the group <M>G = 2.A_6.2_1</M>,
the double cover of the symmetric group <M>S_6</M> on six points.
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>G</M>,
however, fixes the two classes of element order three;
note that exactly one of these classes possesses square roots in the
<Q>outer half</Q> <M>G \setminus G'</M>.
<P/>
Thus it is not sufficient to compare the orbit of the fixed class fusion
under the automorphisms of the ordinary table with the orbit of the
same fusion under the automorphisms of the Brauer table.
</Item>
</List>
<Example><![CDATA[
gap> t:= CharacterTable( "2.A6.2_1" );;
gap> m:= t mod 2;;
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
A = E(3)-E(3)^2
= Sqrt(-3) = i3
B = -E(12)^7+E(12)^11
= Sqrt(3) = r3
]]></Example>
<P/>
When considering several characteristics in parallel, one argues as follows.
The possible class fusions from a Brauer table <M>mtbl</M> to its
ordinary table <M>tbl</M> are given by the orbit of a fixed class fusion
under the action of the table automorphisms of <M>tbl</M>.
If there are several orbits under the action of the automorphisms
of <M>mtbl</M> then we choose one orbit.
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 of all class fusions
from the Brauer table to <M>tbl</M>, and split it into orbits under the
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 automorphisms of <M>tbl</M> by
the stabilizer of this orbit, and proceed with the next characteristic.
In the latter case, we have found a generality problem,
since we are not free to choose an arbitrary class fusion from the
set of possibilities.
<P/>
The following function returns the set of primes which may be involved
in generality problems for the given ordinary character table.
Note that the procedure sketched above does not tell
which characteristics are actually involved
or which classes are affected by the choices;
for example, we could argue that one is always free to choose a fusion
for the first characteristics, and that only the other ones cause problems.
We return <E>all</E> those primes <M>p</M> for which broken symmetries
between the <M>p</M>-modular table and the ordinary table have been detected.
<P/>
<Example><![CDATA[
gap> PrimesOfGeneralityProblems:= function( ordtbl )
> local consider, p, modtbl, taut, triv, admiss, fusion, maut,
> allfusions, orbits, orbit, reps;
> # Find the primes for which symmetries are broken.
> consider:= [];
> 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 );
> triv:= TrivialSubgroup( taut );
> admiss:= taut;
> for p in consider do
> modtbl:= ordtbl mod p;
> fusion:= GetFusionMap( modtbl, ordtbl );
> maut:= AutomorphismsOfTable( modtbl );
> # - We need not apply the action of 'maut' here,
> # since 'maut' will later be used to get representatives.
> # - We need not apply all elements in 'taut' but only
> # representatives of left cosets of 'admiss' in 'taut',
> # since 'admiss' will later be used to get representatives.
> # allfusions:= OrbitFusions( maut, fusion, taut );
> allfusions:= Set( RightTransversal( taut, admiss ),
> x -> OnTuples( fusion, x^-1 ) );
> # For computing representatives, 'RepresentativesFusions' is not
> # suitable because 'allfusions' is in generally not closed
> # under the actions.
> # reps:= RepresentativesFusions( maut, allfusions, admiss );
> orbits:= [];
> while not IsEmpty( allfusions ) do
> 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.
> admiss:= Stabilizer( admiss,
> Set( OrbitFusions( maut, fusion, triv ) ),
> OnSetsTuples );
> else
> # We have found a generality problem.
> return consider;
> fi;
> od;
> # There is no generality problem for this table.
> return [];
> end;;
]]></Example>
<P/>
Let us look at a small example,
the <M>5</M>-modular character table of the group <M>2.A_5.2</M>.
The irreducible characters of degree <M>2</M> have the values
<M>\pm \sqrt{{-2}}</M> on the classes <C>8a</C> and <C>8b</C>,
and the values <M>\pm \sqrt{{-3}}</M> on the classes <C>6b</C> and <C>6c</C>.
When we define which of the two classes of element order <M>8</M> is called
<C>8a</C>, this will also define which class is called <C>6b</C>.
The ordinary character table does not relate the two pairs of classes,
there are table automorphisms which interchange each pair independently.
This symmetry is thus broken in the <M>5</M>-modular character table.
<P/>
<Example><![CDATA[
gap> t:= CharacterTable( "2.A5.2" );;
gap> m:= t mod 5;;
gap> Display( m );
2.A5.2mod5
A = E(8)+E(8)^3
= Sqrt(-2) = i2
B = E(3)-E(3)^2
= Sqrt(-3) = i3
gap> AutomorphismsOfTable( t );
Group([ (11,12), (9,10) ])
gap> AutomorphismsOfTable( m );
Group([ (7,8)(9,10) ])
gap> GetFusionMap( m, t );
[ 1, 2, 3, 4, 5, 8, 9, 10, 11, 12 ]
gap> BrokenSymmetries( t, m );
true
gap> BrokenSymmetries( t, t mod 2 );
false
gap> BrokenSymmetries( t, t mod 3 );
false
gap> PrimesOfGeneralityProblems( t );
[ ]
]]></Example>
<P/>
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 <M>11</M>-modular Brauer character of degree <M>7</M>
distinguishes the two (algebraically conjugate) classes
of element order <M>5</M>.
Since also the unique irreducible <M>19</M>-modular Brauer character
of degree <M>22</M> distinguishes these classes,
we have to choose these classes consistently.
<P/>
<Example><![CDATA[
gap> t:= CharacterTable( "J1" );;
gap> m:= t mod 11;;
gap> Display( m, rec( chars:= Filtered( Irr( m ), x -> x[1] = 7 ) ) );
J1mod11
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>
<P/>
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.
<!-- the list was computed only from those tables which store table automorphisms; there are some huge Brauer tables that are
constructed as direct products, this takes quite long time. -->
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 problem concerning the group <M>J_3</M> (April 2015)</Heading>
<Alt Only="HTML">
<![CDATA[
<a name="generality_problem_J3">In March 2015,</a>
]]>
</Alt>
<Alt Not="HTML">
In March 2015,
</Alt>
Klaus Lux reported an inconsistency in the character data of &GAP;:
<P/>
The sporadic simple Janko group <M>J_3</M> has a unique <M>19</M>-modular
irreducible Brauer character of degree <M>110</M>.
In the character table that is printed in the
&ATLAS; of Brauer characters <Cite Key="JLPW95" Where="p. 219"/>,
the Brauer character value on the class <C>17A</C> is <M>b_{17}</M>.
The &ATLAS; of Group Representations <Cite Key="AGRv3"/> provides
a straight line program for computing class representatives of <M>J_3</M>.
If we compute the Brauer character value in question,
we do not get <M>b_{17}</M> but its algebraic conjugate, <M>-1-b_{17}</M>.
<P/>
<!-- We cannot show the ``old'' status,
since the table has been corrected. -->
How shall we resolve this inconsistency,
by replacing the straight line program
or by swapping the classes <C>17A</C> and <C>17B</C> in the character table?
Before we decide this, we look at related information.
<P/>
<Alt Only="LaTeX">Table~\ref{valuesJ3}</Alt>
<Alt Not="LaTeX">The following table</Alt>
lists the <M>p</M>-modular irreducible characters of <M>J_3</M>,
according to <Cite Key="JLPW95"/>,
that can be used to define which of the two classes of element order <M>17</M>
shall be called <C>17A</C>;
a <M>+</M> sign in the last column of the table indicates that the
representation is available in the &ATLAS; of Group Representations.
<!-- and fetching a representation with given char. and degree will have
the shown values -->
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.
<P/>
The following computations show that the given straight line program
is compatible with the four characters in characteristic <M>2</M>
but is not compatible with the three available characters in characteristic
<M>19</M>.
We see that the problem is an inconsistency between the <M>2</M>-modular and
the <M>19</M>-modular character table of <M>J_3</M>
in <Cite Key="JLPW95"/>.
In particular, changing the straight line program would not help to
resolve the problem.
<P/>
How shall we proceed in order to fix the problem?
We can decide to keep the <M>19</M>-modular table of <M>J_3</M>,
and to swap the two classes of element order <M>17</M> in the
<M>2</M>-modular table;
then also the straight line program has to be changed,
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>.
<P/>
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.
<P/>
<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"/>. <!-- the relevant sections in <Cite Key="AmbigFus"/>. -->
</Item>
<Item>
The change of the character tables affects the decomposition matrices.
Thus the PDF files containing the <M>19</M>-modular decomposition
matrices had to be updated, see
<URL>http://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/dec/tex/J3/index.html</URL>.
</Item>
<Item>
Jürgen Müller has checked that the conjugacy classes of all Brauer tables of
<M>J_3</M>, <M>3.J_3</M>, <M>J_3.2</M>, <M>3.J_3.2</M> are consistent
after the fix described above.
</Item>
</List>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:generality_HN">
<Heading>A generality problem concerning the group <M>HN</M> (August 2022)</Heading>
The classes <C>20A</C>, <C>20B</C> of the Harada-Norton group <M>HN</M>
in the <M>11</M>- and <M>19</M>-modular character tables
are determined by unique Brauer characters that have different values
on these classes.
Once we have <E>defined</E> these classes in one characteristic,
the two Brauer characters tell us how to <E>choose</E> them consistently
in the other characteristic.
Thus the question is whether the two Brauer tables are consistent
w.r.t. this property or not.
<P/>
(Note that this question can be answered independently of all other
questions of this kind for <M>HN</M>,
because the permutation that swaps exactly the classes <C>20A</C> and
<C>20B</C> is a table automorphism of the ordinary character table of
<M>HN</M>.)
<P/>
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>.
<P/>
<Example><![CDATA[
gap> t:= CharacterTable( "HN" );;
gap> pos20:= Positions( OrdersClassRepresentatives( t ), 20 );
[ 39, 40, 41, 42, 43 ]
gap> diff:= Filtered( Irr( t ), x -> x[39] <> x[40] );;
gap> List( diff, x -> Position( Irr( t ), x ) );
[ 51, 52 ]
]]></Example>
<P/>
These values are irrational and lie in the field that is generated by
the square root of <M>5</M>.
In each of the characteristics <M>p \in \{ 11, 19 \}</M>,
the <M>p</M>-modular reductions of <M>\chi_{51}</M> and <M>\chi_{52}</M>
decompose differently into irreducibles.
Note that the Galois automorphism of the ordinary character table
that maps <M>\sqrt{5}</M> to <M>-\sqrt{5}</M> does not live in the
<M>11</M>- and <M>19</M>-modular Brauer tables.
<P/>
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>.
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} + \varphi_{48}</M>,
with <M>\varphi_{33}(1) = 1\,197\,330</M> and
<M>\varphi_{48}(1) = 3\,905\,670</M>.
The Frobenius-Schur indicators of all involved <M>p</M>-modular constituents
are <M>+</M>.
This implies that <M>\chi_{51}</M> reduces orthogonally stably modulo <M>11</M>
but not orthogonally stably modulo <M>19</M>,
whereas <M>\chi_{52}</M> reduces orthogonally stably modulo <M>19</M>
but not orthogonally stably modulo <M>11</M>.
In version up to 1.3.4 of the character table library,
this condition was not satisfied:
The reduction of <M>\chi_{51}</M> modulo both <M>11</M> and <M>19</M> was
orthogonally stable,
and the reduction of <M>\chi_{52}</M> modulo both <M>11</M> and <M>19</M> was
not orthogonally stable.
However, this cannot happen, due to theoretical results about
orthogonal discriminants of the involved characters.
Thus we have found a way to decide the consistency of the classes
<C>20A</C> and <C>20B</C> in characteristics <M>11</M> and <M>19</M>:
Either the <M>11</M>-modular character table or the <M>19</M>-modular
character table of <M>HN</M> had to be changed for version 1.3.5,
by swapping the classes <C>20A</C> and <C>20B</C>.
<P/>
We decided to change the <M>11</M>-modular table,
because there are no other generality problems for <M>HN</M> involving the
<M>19</M>-modular table, and hence we are sure that this table will not
need to be changed because of new solutions to generality problems.
<P/>
For <M>HN.2</M>, the situation is similar.
There are additionally two classes of element order <M>40</M> that have to be
swapped if <C>20A</C> and <C>20B</C> get swapped.
Thus we have to change the <M>11</M>-modular table of <M>HN.2</M> accordingly.
<P/>
Changes of this kind may affect also derived character tables in the library.
In this case, the <M>11</M>-modular table with identifier <C>"(D10xHN).2"</C>
was changed as well.
Note that this table is not stored explicitly in the data files,
it gets constructed from ordinary and modular library tables via
<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.
<P/>
For quite some time, a method is available in &GAP; that computes
the Brauer characters of <M>p</M>-solvable groups
(see <Ref Subsect="BrauerTable" BookName="ref"/> and
<Ref Subsect="IsPSolvableCharacterTable" BookName="ref"/>).
<P/>
The following sections list other situations where Brauer tables
can be computed by &GAP;.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="sect:brauercharactersbyconstruction">
<Heading>Brauer Tables via Construction Information</Heading>
If 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>.
This happens currently in the following cases.
<P/>
<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" BookName="ref"/>.
</Item>
<Item>
<M>t</M> has the attribute
<Ref Attr="ConstructionInfoCharacterTable" BookName="ctbllib"/> set,
the first entry of this list <M>l</M>, say, is one of the strings
<C>"ConstructGS3"</C> (see
<Ref Subsect="sect:Character Tables of Groups of the Structure G.S_3"/>),
<C>"ConstructIndexTwoSubdirectProduct"</C>
(see <Ref Subsect="subsect:theorsubdir"/>),
<C>"ConstructMGA"</C> (see <Ref Subsect="subsect:theorMGA"/>),
<C>"ConstructPermuted"</C>,
<C>"ConstructV4G"</C> (see <Ref Subsect="subsect:theorV4G"/>),
and &GAP; can construct the <M>p</M>-modular Brauer table(s)
of the relevant ordinary character table(s),
which are library tables whose names occur in <M>l</M>.
</Item>
</List>
Let <M>B</M> be a <M>p</M>-block of cyclic defect for the finite group
<M>G</M>.
It can be read off from the set Irr<M>(B)</M>
of ordinary irreducible characters of <M>B</M>
whether all irreducible Brauer characters in <M>B</M> are restrictions
of ordinary characters to the <M>p</M>-regular classes of <M>G</M>,
as follows.
<P/>
If <M>B</M> has only one irreducible Brauer character then all ordinary
characters in <M>B</M> restrict to this Brauer character.
So let us assume that <M>B</M> contains
at least two irreducible Brauer characters,
and consider the set <M>S</M>, say, of restrictions of Irr<M>(B)</M>
to the <M>p</M>-regular classes of <M>G</M>.
<P/>
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 these Brauer
characters is described by
an <M>|S|</M> by <M>|S| - 1</M> matrix <M>M</M>, say,
whose entries are zero and one, such that exactly two nonzero entries
occur in each column.
(See for example <Cite Key="HL89" Where="Theorem 2.1.5"/>,
which refers to <Cite Key="Dad66"/>.)
<P/>
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,
hence the remaining row consists only of <M>1</M>s.
This means that the element of largest degree in <M>S</M> is equal to
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>.
<P/>
Alternatively,
one could state that all irreducible Brauer characters of <M>B</M>
are restricted ordinary characters if and only if the Brauer tree
of <M>B</M> is a <E>star</E> (see <Cite Key="HL89" Where="p. 2"/>.
If <M>B</M> contains at least two irreducible Brauer characters
then this happens if and only if one of the types <M>\times</M> or
<M>\circ</M> occurs for exactly one node in the Brauer graph of <M>B</M>,
see <Cite Key="HL89" Where="Lemma 2.1.13"/>,
and the distribution to types is determined by Irr<M>(B)</M>.
<P/>
The default method for <Ref Oper="BrauerTableOp" BookName="ref"/>
that is contained in the &GAP; library has been extended in version 4.11
such that it checks whether the Sylow <M>p</M>-subgroups of the given group
<M>G</M> are cyclic and, if yes,
whether all <M>p</M>-blocks of <M>G</M> have the property discussed above.
(This feature arose from a discussion with Klaus Lux.)
<P/>
Examples where this method is successful for all blocks
are the <M>p</M>-modular character tables of the groups PSL<M>(2, q)</M>,
where <M>p</M> is odd and does not divide <M>q</M>.
<P/>
<Example><![CDATA[
gap> t:= CharacterTable( PSL( 2, 11 ) );;
gap> modt:= t mod 5;;
gap> modt <> fail;
true
gap> InfoText( modt ); "computed using that all Brauer characters lift to char. zero"
]]></Example>
<P/>
Another such example is the <M>5</M>-modular table
of the Mathieu group <M>M_{11}</M>.
There are cases where all Brauer characters of a block lift
to characteristic zero but the defect group of the block is not cyclic,
thus the method cannot be used.
An example is the <M>2</M>-modular table of the Mathieu group <M>M_{11}</M>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:monstersubgroups">
<Heading>Information about certain subgroups of the Monster group</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="sect:no2U42inMonster">
<Heading>The Monster group does not contain subgroups of the type <M>2.U_4(2)</M> (August 2023)</Heading>
In the context of a question about decomposition numbers of the
sporadic simple Monster group <M>&M;</M>,
Benjamin Sambale was interested in possible embeddings of certain groups
<M>G</M> into <M>&M;</M> such that the decomposition matrices of <M>G</M>
are known.
For a given <M>G</M>, the first steps were to compute the possible class
fusions of <M>G</M> in <M>&M;</M> and then to check whether the corresponding
embeddings would be interesting.
<P/>
Apparently, calling <Ref Func="PossibleClassFusions" BookName="ref"/>
with its default parameters often runs very long and requires a lot of space
when <M>G</M> is a small group such as <M>2.U_4(2)</M>.
We 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
and if the result consists of many candidates then it might be faster
to omit the decomposability criterion.
<!-- The computation takes about 1928 seconds on my notebook. -->
<P/>
Looking at the (many) candidates,
we see that all map the central involution of
<M>2.U_4(2)</M> to the class <C>2B</C> of <M>&M;</M>,
thus any subgroup of the type <M>2.U_4(2)</M> lies inside the <C>2B</C>
normalizer in <M>&M;</M>.
We compute the possible class fusions into this subgroup.
<!-- The computation takes about 29 seconds on my notebook. -->
<P/>
Thus we have shown that <M>&M;</M> does not contain subgroups of the type
<M>2.U_4(2)</M>.
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="sect:ML34inMonster">
<Heading>Perfect central extensions of <M>L_3(4)</M> (August 2023)</Heading>
There was <URL><LinkText>the question in MathOverflow</LinkText>
<Link>https://mathoverflow.net/questions/450255</Link></URL>
which perfect central extensions of the simple group <M>G = L_3(4)</M> are
subgroups of the sporadic simple Monster group <M>&M;</M>.
<P/>
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).
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).
<!-- In <Cite Key="NW02"/>, $L_3(4)$ is listed as perhaps contained as a subgroup. (And it is stated in Section 4.2 which class fusion such a subgroup
would have.) -->
<P/>
And the following embeddings of central extensions of <M>G</M> in <M>&M;</M>
can be established using known subgroups of <M>&M;</M>.
Note that <M>G</M> is a subgroup of <M>U_4(3)</M> but not of <M>2.U_4(3)</M>,
<M>3.G</M> is a subgroup of <M>G_2(4)</M> but not of <M>2.G_2(4)</M>,
and <M>G_2(4)</M> is a subgroup of <M>Suz</M> but not of <M>2.Suz</M>.
The positive statements follow from <Cite Key="CCN85" Where="pp. 52, 97, 131"/>
and the negative ones from the following computations.
The group <M>3.G</M> centralizes an element of order three.
If <M>3.G</M> is a subgroup of <M>&M;</M> then it is contained in
a <C>3A</C> centralizer (of the structure <M>3.Fi_{24}'</M>),
a <C>3B</C> centralizer (of the structure <M>3^{1+12}_+.2Suz</M>) or
a <C>3C</C> centralizer (of the structure <M>3 \times Th</M>).
Clearly the case <M>3C</M> cannot occur,
and <C>3B</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.
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>..
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>.
<P/>
<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>
<!-- These computations take about 3445 seconds on my notebook. -->
<P/>
(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.)
<P/>
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>.
<P/>
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>.
<P/>
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>.
<!-- Inside Co_3, several maximal subgroups can contain the given L_3(4). -->
<P/>
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>.
(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>.)
<P/>
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>.
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>.
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>.
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>.
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>
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