Permutation Characters in GAP
THOMAS BREUER
Lehrstuhl D für Mathematik
RWTH, 52056 Aachen, Germany
April 17th, 1999
(Sections 14 and 15 were added in October 2001.)
This is a loose collection of examples of computations with
permutation characters and possible permutation characters in
the GAP system [GAP04].
We mainly use the GAP implementation of the algorithms to compute
possible permutation characters that are described in [BP98],
and information from the Atlas of Finite Groups [CCN+85].
A possible permutation character of a finite group G is a character
satisfying the conditions listed in Section
"Possible Permutation Characters" of the
GAP Reference Manual.
Contents
1 Some Computations with M24
2 All Possible Permutation Characters of M11
3 The Action of U6(2) on the Cosets of M22
4 Degree 20 736 Permutation Characters of U6(2)
5 Degree 57 572 775 Permutation Characters of O8+(3)
6 The Action of O7(3).2 on the Cosets of 27.S7
7 The Action of O8+(3).21 on the Cosets of 27.A8
8 The Action of S4(4).4 on the Cosets of 52.[25]
9 The Action of Co1 on the Cosets of Involution Centralizers
10 The Multiplicity Free Permutation Characters of G2(3)
11 Degree 11 200 Permutation Characters of O8+(2)
12 A Proof of Nonexistence of a Certain Subgroup
13 A Permutation Character of the Lyons group
14 Identifying two subgroups of Aut(U3(5))
15 A Permutation Character of Aut(O8+(2))
In the following, the GAP Character Table Library [Bre04]
will be used frequently.
gap> LoadPackage( "ctbllib" );
true
1 Some Computations with M24
We start with the sporadic simple Mathieu group G = M24
in its natural action on 24 points.
gap> g:= MathieuGroup( 24 );;
gap> SetName( g, "m24" );
gap> Size( g ); IsSimple( g ); NrMovedPoints( g );
244823040
true
24
The permutation character pi of G corresponding to the action on
the moved points is constructed.
This action is 5-transitive.
gap> NrConjugacyClasses( g );
26
gap> pi:= NaturalCharacter( g );
Character( CharacterTable( m24 ), [ 24, 2, 1, 1, 4, 6, 0, 0, 3, 3, 0, 1, 1,
8, 0, 2, 0, 0, 4, 2, 0, 0, 0, 1, 1, 0 ] )
gap> IsTransitive( pi ); Transitivity( pi );
true
5
gap> Display( pi );
CT1
2 10 . . . 2 3 . . 1 1 3 1 1 10 2 3 7 9 7 4 2
3 3 . 1 1 1 3 1 1 1 1 2 . . 1 1 1 1 1 . . 1
5 1 . 1 1 1 1 . . . . . . . . . . . 1 . . .
7 1 . . . . . 1 1 1 1 1 1 1 1 . . . . . . .
11 1 1 . . . . . . . . . . . . . . . . . . .
23 1 . . . . . . . . . . . . . . . . . . . .
1a 11a 15a 15b 5a 3a 21a 21b 7a 7b 3b 14a 14b 2a 12a 6a 4a 2b 4b 8a 12b
Y.1 24 2 1 1 4 6 . . 3 3 . 1 1 8 . 2 . . 4 2 .
2 3 5 . . 2
3 1 1 . . .
5 . . . . 1
7 . . . . .
11 . . . . .
23 . . 1 1 .
6b 4c 23a 23b 10a
Y.1 . . 1 1 .
pi determines the permutation characters of the G-actions on
related sets,
for example piop on the set of ordered and piup on the set of
unordered pairs of points.
gap> piop:= pi * pi;
Character( CharacterTable( m24 ), [ 576, 4, 1, 1, 16, 36, 0, 0, 9, 9, 0, 1,
1, 64, 0, 4, 0, 0, 16, 4, 0, 0, 0, 1, 1, 0 ] )
gap> IsTransitive( piop );
false
gap> piup:= SymmetricParts( UnderlyingCharacterTable(pi), [ pi ], 2 )[1];
Character( CharacterTable( m24 ), [ 300, 3, 1, 1, 10, 21, 0, 0, 6, 6, 0, 2,
2, 44, 1, 5, 4, 12, 12, 4, 0, 0, 0, 1, 1, 2 ] )
gap> IsTransitive( piup );
false
Clearly the action on unordered pairs is not transitive, since the pairs
[ i, i ] form an orbit of their own.
There are exactly two G-orbits on the unordered pairs,
hence the G-action on 2-sets of points is transitive.
gap> ScalarProduct( piup, TrivialCharacter( g ) );
2
gap> comb:= Combinations( [ 1 .. 24 ], 2 );;
gap> hom:= ActionHomomorphism( g, comb, OnSets );;
gap> pihom:= NaturalCharacter( hom );
Character( CharacterTable( m24 ), [ 276, 1, 0, 0, 6, 15, 0, 0, 3, 3, 0, 1, 1,
36, 1, 3, 4, 12, 8, 2, 0, 0, 0, 0, 0, 2 ] )
gap> Transitivity( pihom );
1
In terms of characters, the permutation character pihom is the difference
of piup and pi .
Note that GAP does not know that this difference is in fact a character;
in general this question is not easy to decide without knowing the
irreducible characters of G,
and up to now GAP has not computed the irreducibles.
gap> pi2s:= piup - pi;
VirtualCharacter( CharacterTable( m24 ), [ 276, 1, 0, 0, 6, 15, 0, 0, 3, 3,
0, 1, 1, 36, 1, 3, 4, 12, 8, 2, 0, 0, 0, 0, 0, 2 ] )
gap> pi2s = pihom;
true
gap> HasIrr( g ); HasIrr( CharacterTable( g ) );
false
false
The point stabilizer in the action on 2-sets is in fact a maximal
subgroup of G, which is isomorphic to the automorphism group
M22:2 of the Mathieu group M22.
Thus this permutation action is primitive.
But we cannot apply IsPrimitive to the character pihom for getting
this answer because primitivity of characters is defined in a different
way, cf. IsPrimitiveCharacter in the GAP Reference Manual.
gap> IsPrimitive( g, comb, OnSets );
true
We could also have computed the transitive permutation character of
degree 276 using the GAP Character Table Library instead of
the group G,
since the character tables of G and all its maximal subgroups are
available, together with the class fusions of the maximal subgroups
into G.
gap> tbl:= CharacterTable( "M24" );
CharacterTable( "M24" )
gap> maxes:= Maxes( tbl );
[ "M23", "M22.2", "2^4:a8", "M12.2", "2^6:3.s6", "L3(4).3.2_2",
"2^6:(psl(3,2)xs3)", "L2(23)", "L3(2)" ]
gap> s:= CharacterTable( maxes[2] );
CharacterTable( "M22.2" )
gap> TrivialCharacter( s )^tbl;
Character( CharacterTable( "M24" ), [ 276, 36, 12, 15, 0, 4, 8, 0, 6, 3, 0,
3, 3, 2, 2, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0 ] )
Note that the sequence of conjugacy classes in the library table of
G does in general not agree with the succession computed for the
group.
2 All Possible Permutation Characters of M11
We compute all possible permutation characters of the Mathieu group
M11, using the three different strategies available in GAP.
First we try the algorithm that enumerates all candidates via solving
a system of inequalities, which is described in [BP98,Section 3.2].
gap> m11:= CharacterTable( "M11" );;
gap> SetName( m11, "m11" );
gap> perms:= PermChars( m11 );
[ Character( m11, [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ),
Character( m11, [ 11, 3, 2, 3, 1, 0, 1, 1, 0, 0 ] ),
Character( m11, [ 12, 4, 3, 0, 2, 1, 0, 0, 1, 1 ] ),
Character( m11, [ 22, 6, 4, 2, 2, 0, 0, 0, 0, 0 ] ),
Character( m11, [ 55, 7, 1, 3, 0, 1, 1, 1, 0, 0 ] ),
Character( m11, [ 66, 10, 3, 2, 1, 1, 0, 0, 0, 0 ] ),
Character( m11, [ 110, 6, 2, 2, 0, 0, 2, 2, 0, 0 ] ),
Character( m11, [ 110, 6, 2, 6, 0, 0, 0, 0, 0, 0 ] ),
Character( m11, [ 110, 14, 2, 2, 0, 2, 0, 0, 0, 0 ] ),
Character( m11, [ 132, 12, 6, 0, 2, 0, 0, 0, 0, 0 ] ),
Character( m11, [ 144, 0, 0, 0, 4, 0, 0, 0, 1, 1 ] ),
Character( m11, [ 165, 13, 3, 1, 0, 1, 1, 1, 0, 0 ] ),
Character( m11, [ 220, 4, 4, 0, 0, 4, 0, 0, 0, 0 ] ),
Character( m11, [ 220, 12, 4, 4, 0, 0, 0, 0, 0, 0 ] ),
Character( m11, [ 220, 20, 4, 0, 0, 2, 0, 0, 0, 0 ] ),
Character( m11, [ 330, 2, 6, 2, 0, 2, 0, 0, 0, 0 ] ),
Character( m11, [ 330, 18, 6, 2, 0, 0, 0, 0, 0, 0 ] ),
Character( m11, [ 396, 12, 0, 4, 1, 0, 0, 0, 0, 0 ] ),
Character( m11, [ 440, 8, 8, 0, 0, 2, 0, 0, 0, 0 ] ),
Character( m11, [ 440, 24, 8, 0, 0, 0, 0, 0, 0, 0 ] ),
Character( m11, [ 495, 15, 0, 3, 0, 0, 1, 1, 0, 0 ] ),
Character( m11, [ 660, 4, 3, 4, 0, 1, 0, 0, 0, 0 ] ),
Character( m11, [ 660, 12, 3, 0, 0, 3, 0, 0, 0, 0 ] ),
Character( m11, [ 660, 12, 12, 0, 0, 0, 0, 0, 0, 0 ] ),
Character( m11, [ 660, 28, 3, 0, 0, 1, 0, 0, 0, 0 ] ),
Character( m11, [ 720, 0, 0, 0, 0, 0, 0, 0, 5, 5 ] ),
Character( m11, [ 792, 24, 0, 0, 2, 0, 0, 0, 0, 0 ] ),
Character( m11, [ 880, 0, 16, 0, 0, 0, 0, 0, 0, 0 ] ),
Character( m11, [ 990, 6, 0, 2, 0, 0, 2, 2, 0, 0 ] ),
Character( m11, [ 990, 6, 0, 6, 0, 0, 0, 0, 0, 0 ] ),
Character( m11, [ 990, 30, 0, 2, 0, 0, 0, 0, 0, 0 ] ),
Character( m11, [ 1320, 8, 6, 0, 0, 2, 0, 0, 0, 0 ] ),
Character( m11, [ 1320, 24, 6, 0, 0, 0, 0, 0, 0, 0 ] ),
Character( m11, [ 1584, 0, 0, 0, 4, 0, 0, 0, 0, 0 ] ),
Character( m11, [ 1980, 12, 0, 4, 0, 0, 0, 0, 0, 0 ] ),
Character( m11, [ 1980, 36, 0, 0, 0, 0, 0, 0, 0, 0 ] ),
Character( m11, [ 2640, 0, 12, 0, 0, 0, 0, 0, 0, 0 ] ),
Character( m11, [ 3960, 24, 0, 0, 0, 0, 0, 0, 0, 0 ] ),
Character( m11, [ 7920, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ]
gap> Length( perms );
39
Next we try the improved combinatorial approach that is sketched at the
end of Section 3.2 in [BP98].
We get the same characters, except that they may be ordered in a different
way; thus we compare the ordered lists.
gap> degrees:= DivisorsInt( Size( m11 ) );;
gap> perms2:= [];;
gap> for d in degrees do
> Append( perms2, PermChars( m11, d ) );
> od;
gap> Set( perms ) = Set( perms2 );
true
Finally, we try the algorithm that is based on Gaussian elimination
and that is described in [BP98,Section 3.3].
gap> perms3:= [];;
gap> for d in degrees do
> Append( perms3, PermChars( m11, rec( torso:= [ d ] ) ) );
> od;
gap> Set( perms ) = Set( perms3 );
true
GAP provides two more functions to test properties of permutation
characters.
The first one yields no new information in our case,
but the second excludes one possible permutation character;
note that TestPerm5 needs a p-modular Brauer table,
and the GAP character table library contains all Brauer tables
of M11.
gap> newperms:= TestPerm4( m11, perms );;
gap> newperms = perms;
true
gap> newperms:= TestPerm5( m11, perms, m11 mod 11 );;
gap> newperms = perms;
false
gap> Difference( perms, newperms );
[ Character( m11, [ 220, 4, 4, 0, 0, 4, 0, 0, 0, 0 ] ) ]
GAP knows the table of marks of M11,
from which the permutation characters can be extracted.
It turns out that M11 has 39 conjugacy classes of subgroups
but only 36 different permutation characters,
so three candidates computed above are in fact not permutation characters.
gap> tom:= TableOfMarks( "M11" );
TableOfMarks( "M11" )
gap> trueperms:= PermCharsTom( m11, tom );;
gap> Length( trueperms ); Length( Set( trueperms ) );
39
36
gap> Difference( perms, trueperms );
[ Character( m11, [ 220, 4, 4, 0, 0, 4, 0, 0, 0, 0 ] ),
Character( m11, [ 660, 4, 3, 4, 0, 1, 0, 0, 0, 0 ] ),
Character( m11, [ 660, 12, 3, 0, 0, 3, 0, 0, 0, 0 ] ) ]
3 The Action of U6(2) on the Cosets of M22
We are interested in the permutation character of U6(2)
(see [CCN+85,p. 115]) that corresponds to the action on the cosets
of a M22 subgroup (see [CCN+85,p. 39]).
The character tables of both the group and the point stabilizer
are available in the GAP character table library,
so we can compute class fusion and permutation character directly;
note that if the class fusion is not stored on the table of the subgroup,
in general one will not get a unique fusion but only a list of candidates
for the fusion.
gap> u62:= CharacterTable( "U6(2)" );;
gap> m22:= CharacterTable( "M22" );;
gap> fus:= PossibleClassFusions( m22, u62 );
[ [ 1, 3, 7, 10, 14, 15, 22, 24, 24, 26, 33, 34 ],
[ 1, 3, 7, 10, 14, 15, 22, 24, 24, 26, 34, 33 ],
[ 1, 3, 7, 11, 14, 15, 22, 24, 24, 27, 33, 34 ],
[ 1, 3, 7, 11, 14, 15, 22, 24, 24, 27, 34, 33 ],
[ 1, 3, 7, 12, 14, 15, 22, 24, 24, 28, 33, 34 ],
[ 1, 3, 7, 12, 14, 15, 22, 24, 24, 28, 34, 33 ] ]
gap> RepresentativesFusions( m22, fus, u62 );
[ [ 1, 3, 7, 10, 14, 15, 22, 24, 24, 26, 33, 34 ] ]
We see that there are six possible class fusions that are equivalent
under table automorphisms of U6(2) and M22.
gap> cand:= Set( List( fus,
> x -> Induced( m22, u62, [ TrivialCharacter( m22 ) ], x )[1] ) );
[ Character( CharacterTable( "U6(2)" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0,
0, 0, 48, 0, 16, 6, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0,
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ),
Character( CharacterTable( "U6(2)" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0,
0, 48, 0, 0, 16, 6, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0,
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ),
Character( CharacterTable( "U6(2)" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0,
48, 0, 0, 0, 16, 6, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0,
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ]
gap> PermCharInfo( u62, cand ).ATLAS;
[ "1a+22a+252a+616a+1155c+1386a+8064a+9240c",
"1a+22a+252a+616a+1155b+1386a+8064a+9240b",
"1a+22a+252a+616a+1155a+1386a+8064a+9240a" ]
gap> aut:= AutomorphismsOfTable( u62 );; Size( aut );
24
gap> elms:= Filtered( Elements( aut ), x -> Order( x ) = 3 );
[ (10,11,12)(26,27,28)(40,41,42), (10,12,11)(26,28,27)(40,42,41) ]
gap> Position( cand, Permuted( cand[1], elms[1] ) );
3
gap> Position( cand, Permuted( cand[3], elms[1] ) );
2
The six fusions induce three different characters,
they are conjugate under the action of the unique subgroup of order 3
in the group of table automorphisms of U6(2).
The table automorphisms of order 3 are induced by group automorphisms
of U6(2) (see [CCN+85,p. 120]).
As can be seen from the list of maximal subgroups of U6(2)
in [CCN+85,p. 115],
the three induced characters are in fact permutation characters
which belong to the three classes of maximal subgroups of type M22
in U6(2), which are permuted by an outer automorphism of order 3.
Now we want to compute the extension of the above permutation character
to the group U6(2).2,
which corresponds to the action of this group on the cosets of a M22.2
subgroup.
gap> u622:= CharacterTable( "U6(2).2" );;
gap> m222:= CharacterTable( "M22.2" );;
gap> fus:= PossibleClassFusions( m222, u622 );
[ [ 1, 3, 7, 10, 13, 14, 20, 22, 22, 24, 29, 38, 39, 42, 41, 46, 50, 53, 58,
59, 59 ] ]
gap> cand:= Induced( m222, u622, [ TrivialCharacter( m222 ) ], fus[1] );
[ Character( CharacterTable( "U6(2).2" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0,
48, 0, 0, 16, 6, 0, 0, 0, 0, 0, 6, 0, 2, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0,
0, 0, 0, 0, 0, 1080, 72, 0, 48, 8, 0, 0, 0, 18, 0, 0, 0, 8, 0, 0, 2, 0,
0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0 ] ) ]
gap> PermCharInfo( u622, cand ).ATLAS;
[ "1a+22a+252a+616a+1155a+1386a+8064a+9240a" ]
We see that for the embedding of M22.2 into U6(2).2,
the class fusion is unique,
so we get a unique extension of one of the above permutation characters.
This implies that exactly one class of maximal subgroups of type M22
extends to M22.2 in a given group U6(2).2.
4 Degree 20 736 Permutation Characters of U6(2)
Now we show an alternative way to compute the characters dealt with
in the previous example.
This works also if the character table of the point stabilizer is not
available.
In this situation we can compute all those characters that have certain
properties of permutation characters.
Of course this may take much longer than the above computations,
which needed only a few seconds.
(The following calculations may need several hours,
depending on the computer used.)
gap> cand:= PermChars( u62, rec( torso := [ 20736 ] ) );
[ Character( CharacterTable( "U6(2)" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0,
0, 0, 48, 0, 16, 6, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0,
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ),
Character( CharacterTable( "U6(2)" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0,
0, 48, 0, 0, 16, 6, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0,
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ),
Character( CharacterTable( "U6(2)" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0,
48, 0, 0, 0, 16, 6, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0,
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ]
For the next step, that is, the computation of the extension of the
permutation character to U6(2).2, we may use the above information,
since the values on the inner classes are prescribed.
The question which of the three candidates for U6(2) extends to
U6(2).2 depends on the choice of the class fusion of U6(2) into
U6(2).2.
With respect to the class fusion that is stored on the GAP library table,
the third candidate extends,
as can be seen from the fact that this one is invariant under the
permutation of conjugacy classes of U6(2) that is induced by the
action of the chosen supergroup U6(2).2.
gap> u622:= CharacterTable( "U6(2).2" );;
gap> inv:= InverseMap( GetFusionMap( u62, u622 ) );
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, [ 11, 12 ], 13, 14, 15, [ 16, 17 ], 18, 19,
20, 21, 22, 23, 24, 25, 26, [ 27, 28 ], [ 29, 30 ], 31, 32, [ 33, 34 ],
[ 35, 36 ], 37, [ 38, 39 ], 40, [ 41, 42 ], 43, 44, [ 45, 46 ] ]
gap> ext:= List( cand, x -> CompositionMaps( x, inv ) );
[ [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, [ 0, 48 ], 0, 16, 6, 0, 0, 0, 0, 0,
6, 0, 2, 0, 0, [ 0, 4 ], 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, [ 0, 48 ], 0, 16, 6, 0, 0, 0, 0, 0,
6, 0, 2, 0, 0, [ 0, 4 ], 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 48, 0, 0, 16, 6, 0, 0, 0, 0, 0, 6, 0,
2, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ] ]
gap> cand:= PermChars( u622, rec( torso:= ext[3] ) );
[ Character( CharacterTable( "U6(2).2" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0,
48, 0, 0, 16, 6, 0, 0, 0, 0, 0, 6, 0, 2, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0,
0, 0, 0, 0, 0, 1080, 72, 0, 48, 8, 0, 0, 0, 18, 0, 0, 0, 8, 0, 0, 2, 0,
0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0 ] ) ]
5 Degree 57 572 775 Permutation Characters of O8+(3)
The group O8+(3) (see [CCN+85,p. 140]) contains a subgroup of
type 23+6.L3(2),
which extends to a maximal subgroup U in O8+(3).3.
For the computation of the permutation character,
we cannot use explicit induction since the table of U is not available
in the GAP table library.
Since U ∩O8+(3) is contained in a O8+(2) subgroup
of O8+(3), we can try to find the permutation character of O8+(2)
corresponding to the action on the cosets of U ∩O8+(3),
and then induce this character to O8+(3).
This kind of computations becomes more difficult with increasing degree,
so we try to reduce the problem further.
In fact, the 23+6.L3(2) group is contained in a 26:A8 subgroup
of O8+(2), in which the index is only 15;
the unique possible permutation character of this degree can be read off
immediately.
Induction to O8+(3) through the chain of subgroups is possible
provided the class fusions are available.
There are 24 possible fusions from O8+(2) into O8+(3),
which are all equivalent w.r.t. table automorphisms of O8+(3).
If we later want to consider the extension of the permutation character
in question to O8+(3).3 then we have to choose a fusion of an
O8+(2) subgroup that does not extend to O8+(2).3.
But if for example our question is just whether the resulting permutation
character is multiplicity-free then this can be decided already from the
permutation character of O8+(3).
gap> o8p3:= CharacterTable("O8+(3)");;
gap> Size( o8p3 ) / (2^9*168);
57572775
gap> o8p2:= CharacterTable( "O8+(2)" );;
gap> fus:= PossibleClassFusions( o8p2, o8p3 );;
gap> Length( fus );
24
gap> rep:= RepresentativesFusions( o8p2, fus, o8p3 );
[ [ 1, 5, 2, 3, 4, 5, 7, 8, 12, 16, 17, 19, 23, 20, 21, 22, 23, 24, 25, 26,
37, 38, 42, 31, 32, 36, 49, 52, 51, 50, 43, 44, 45, 53, 55, 56, 57, 71,
71, 71, 72, 73, 74, 78, 79, 83, 88, 89, 90, 94, 100, 101, 105 ] ]
gap> fus:= rep[1];;
gap> Size( o8p2 ) / (2^9*168);
2025
gap> sub:= CharacterTable( "2^6:A8" );;
gap> subfus:= GetFusionMap( sub, o8p2 );
[ 1, 3, 2, 2, 4, 5, 6, 13, 3, 6, 12, 13, 14, 7, 21, 24, 11, 30, 29, 31, 13,
17, 15, 16, 14, 17, 36, 37, 18, 41, 24, 44, 48, 28, 33, 32, 34, 35, 35, 51,
51 ]
gap> fus:= CompositionMaps( fus, subfus );
[ 1, 2, 5, 5, 3, 4, 5, 23, 2, 5, 19, 23, 20, 7, 37, 31, 17, 50, 51, 43, 23,
23, 21, 22, 20, 23, 56, 57, 24, 72, 31, 78, 89, 52, 45, 44, 53, 55, 55,
100, 100 ]
gap> Size( sub ) / (2^9*168);
15
gap> List( Irr( sub ), Degree );
[ 1, 7, 14, 20, 21, 21, 21, 28, 35, 45, 45, 56, 64, 70, 28, 28, 35, 35, 35,
35, 70, 70, 70, 70, 140, 140, 140, 140, 140, 210, 210, 252, 252, 280, 280,
315, 315, 315, 315, 420, 448 ]
gap> cand:= PermChars( sub, 15 );
[ Character( CharacterTable( "2^6:A8" ), [ 15, 15, 15, 7, 7, 7, 7, 7, 3, 3,
3, 3, 3, 0, 0, 0, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1,
1, 1, 1, 1, 1, 0, 0 ] ) ]
gap> ind:= Induced( sub, o8p3, cand, fus );
[ Character( CharacterTable( "O8+(3)" ), [ 57572775, 59535, 59535, 59535,
3591, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2187, 0, 27, 135, 135, 135, 243,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 27, 27, 27, 0,
0, 0, 0, 27, 27, 27, 27, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ]
gap> o8p33:= CharacterTable( "O8+(3).3" );;
gap> inv:= InverseMap( GetFusionMap( o8p3, o8p33 ) );
[ 1, [ 2, 3, 4 ], 5, 6, [ 7, 8, 9 ], [ 10, 11, 12 ], 13, [ 14, 15, 16 ], 17,
18, 19, [ 20, 21, 22 ], 23, [ 24, 25, 26 ], [ 27, 28, 29 ], 30,
[ 31, 32, 33 ], [ 34, 35, 36 ], [ 37, 38, 39 ], [ 40, 41, 42 ],
[ 43, 44, 45 ], 46, [ 47, 48, 49 ], 50, [ 51, 52, 53 ], 54, 55, 56, 57,
[ 58, 59, 60 ], [ 61, 62, 63 ], 64, [ 65, 66, 67 ], 68, [ 69, 70, 71 ],
[ 72, 73, 74 ], [ 75, 76, 77 ], [ 78, 79, 80 ], [ 81, 82, 83 ], 84, 85,
[ 86, 87, 88 ], [ 89, 90, 91 ], [ 92, 93, 94 ], 95, 96, [ 97, 98, 99 ],
[ 100, 101, 102 ], [ 103, 104, 105 ], [ 106, 107, 108 ], [ 109, 110, 111 ],
[ 112, 113, 114 ] ]
gap> ext:= CompositionMaps( ind[1], inv );
[ 57572775, 59535, 3591, 0, 0, 0, 0, 0, 2187, 0, 27, 135, 243, 0, 0, 0, 0, 0,
0, 0, 27, 0, 0, 27, 27, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
gap> perms:= PermChars( o8p33, rec( torso:= ext ) );
[ Character( CharacterTable( "O8+(3).3" ), [ 57572775, 59535, 3591, 0, 0, 0,
0, 0, 2187, 0, 27, 135, 243, 0, 0, 0, 0, 0, 0, 0, 27, 0, 0, 27, 27, 0,
8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 3159, 3159, 243, 243, 39, 39, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3,
3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0
] ) ]
gap> PermCharInfo( o8p33, perms ).ATLAS;
[ "1a+780aabb+2457a+2808abc+9450aaabbcc+18200abcdddef+24192a+54600a^{5}b+70200\
aabb+87360ab+139776a^{5}+147420a^{4}b^{4}+163800ab+184275aabc+199017aa+218700a\
+245700a+291200aef+332800a^{4}b^{5}c^{5}+491400aaabcd+531441a^{5}b^{4}c^{4}+55\
2825a^{4}+568620aabb+698880a^{4}b^{4}+716800aaabbccdddeeff+786240aabb+873600aa\
+998400aa+1257984a^{6}+1397760aa" ]
6 The Action of O7(3).2 on the Cosets of 27.S7
We want to know whether the permutation character of O7(3).2
(see [CCN+85,p. 108]) on the cosets of its maximal subgroup U
of type 27.S7 is multiplicity-free.
As in the previous examples, first we try to compute the permutation
character of the simple group O7(3).
It turns out that the direct computation of all candidates from the
degree is very time consuming.
But we can use for example the additional information provided by the fact
that U contains an A7 subgroup.
We compute the possible class fusions.
gap> o73:= CharacterTable( "O7(3)" );;
gap> a7:= CharacterTable( "A7" );;
gap> fus:= PossibleClassFusions( a7, o73 );
[ [ 1, 3, 6, 10, 15, 16, 24, 33, 33 ], [ 1, 3, 7, 10, 15, 16, 22, 33, 33 ] ]
We cannot decide easily which fusion is the right one,
but already the fact that no other fusions are possible
gives us some information about impossible constituents of the
permutation character we want to compute.
gap> ind:= List( fus,
> x -> Induced( a7, o73, [ TrivialCharacter( a7 ) ], x )[1] );;
gap> mat:= MatScalarProducts( o73, Irr( o73 ), ind );;
gap> sum:= Sum( mat );
[ 2, 6, 2, 0, 8, 6, 2, 4, 4, 8, 3, 0, 4, 4, 9, 3, 5, 0, 0, 9, 0, 10, 5, 6,
15, 1, 12, 1, 15, 7, 2, 4, 14, 16, 0, 12, 12, 7, 8, 8, 14, 12, 12, 14, 6,
6, 20, 16, 12, 12, 12, 10, 10, 12, 12, 8, 12, 6 ]
gap> const:= Filtered( [ 1 .. Length( sum ) ], x -> sum[x] <> 0 );
[ 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 20, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58 ]
gap> Length( const );
52
gap> const:= Irr( o73 ){ const };;
gap> rat:= RationalizedMat( const );;
But much more can be deduced from the fact that certain zeros
of the permutation character can be predicted.
gap> names:= ClassNames( o73 );
[ "1a", "2a", "2b", "2c", "3a", "3b", "3c", "3d", "3e", "3f", "3g", "4a",
"4b", "4c", "4d", "5a", "6a", "6b", "6c", "6d", "6e", "6f", "6g", "6h",
"6i", "6j", "6k", "6l", "6m", "6n", "6o", "6p", "7a", "8a", "8b", "9a",
"9b", "9c", "9d", "10a", "10b", "12a", "12b", "12c", "12d", "12e", "12f",
"12g", "12h", "13a", "13b", "14a", "15a", "18a", "18b", "18c", "18d", "20a"
]
gap> List( fus, x -> names{ x } );
[ [ "1a", "2b", "3b", "3f", "4d", "5a", "6h", "7a", "7a" ],
[ "1a", "2b", "3c", "3f", "4d", "5a", "6f", "7a", "7a" ] ]
gap> torso:= [ 28431 ];;
gap> zeros:= [ 5, 8, 9, 11, 17, 20, 23, 28, 29, 32, 36, 37, 38,
> 43, 46, 47, 48, 53, 54, 55, 56, 57, 58 ];;
gap> names{ zeros };
[ "3a", "3d", "3e", "3g", "6a", "6d", "6g", "6l", "6m", "6p", "9a", "9b",
"9c", "12b", "12e", "12f", "12g", "15a", "18a", "18b", "18c", "18d", "20a" ]
Every order 3 element of U lies in an A7 subgroup of U,
so among the classes of element order 3, at most the classes 3B, 3C,
and 3F can have nonzero permutation character values.
The excluded classes of element order 6 are the square roots of the
excluded order 3 elements,
likewise the given classes of element orders 9, 12, and 18 are
excluded.
The character value on 20A must be zero because U does not contain
elements of this order.
So we enter the additional information about these zeros.
gap> for i in zeros do
> torso[i]:= 0;
> od;
gap> torso;
[ 28431,,,, 0,,, 0, 0,, 0,,,,,, 0,,, 0,,, 0,,,,, 0, 0,,, 0,,,, 0, 0, 0,,,,, 0,
,, 0, 0, 0,,,,, 0, 0, 0, 0, 0, 0 ]
gap> perms:= PermChars( o73, rec( torso:= torso, chars:= rat ) );
[ Character( CharacterTable( "O7(3)" ), [ 28431, 567, 567, 111, 0, 0, 243, 0,
0, 81, 0, 15, 3, 27, 15, 6, 0, 0, 27, 0, 3, 27, 0, 0, 0, 3, 9, 0, 0, 3,
3, 0, 4, 1, 1, 0, 0, 0, 0, 2, 2, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0 ] ) ]
gap> PermCharInfo( o73, perms ).ATLAS;
[ "1a+78a+168a+182a+260ab+1092a+2457a+2730a+4095b+5460a+11648a" ]
We see that this character is already multiplicity free,
so this holds also for its extension to O7(3).2,
and we need not compute this extension.
(Of course we could compute it in the same way as in the examples above.)
7 The Action of O8+(3).21 on the Cosets of 27.A8
We are interested in the permutation character of O8+(3).21
that corresponds to the action on the cosets of a subgroup of type
27.A8.
The intersection of the point stabilizer with the simple group O8+(3)
is of type 26.A8.
First we compute the class fusion of these groups,
modulo problems with ambiguities due to table automorphisms.
gap> o8p3:= CharacterTable( "O8+(3)" );;
gap> o8p2:= CharacterTable( "O8+(2)" );;
gap> fus:= PossibleClassFusions( o8p2, o8p3 );;
gap> NamesOfFusionSources( o8p2 );
[ "A9", "2^8:O8+(2)", "2^(1+8)+.O8+(2)", "2^6:A8", "2.O8+(2)", "2^2.O8+(2)",
"S6(2)" ]
gap> sub:= CharacterTable( "2^6:A8" );;
gap> subfus:= GetFusionMap( sub, o8p2 );
[ 1, 3, 2, 2, 4, 5, 6, 13, 3, 6, 12, 13, 14, 7, 21, 24, 11, 30, 29, 31, 13,
17, 15, 16, 14, 17, 36, 37, 18, 41, 24, 44, 48, 28, 33, 32, 34, 35, 35, 51,
51 ]
gap> fus:= List( fus, x -> CompositionMaps( x, subfus ) );;
gap> fus:= Set( fus );;
gap> Length( fus );
24
The ambiguities due to Galois automorphisms disappear when we are
looking for the permutation characters induced by the fusions.
gap> ind:= List( fus, x -> Induced( sub, o8p3,
> [ TrivialCharacter( sub ) ], x )[1] );;
gap> ind:= Set( ind );;
gap> Length( ind );
6
Now we try to extend the candidates to O8+(3).21;
the choice of the fusion of O8+(3) into O8+(3).21 determines
which of the candidates may extend.
gap> o8p32:= CharacterTable( "O8+(3).2_1" );;
gap> fus:= GetFusionMap( o8p3, o8p32 );;
gap> ext:= List( ind, x -> CompositionMaps( x, InverseMap( fus ) ) );;
gap> ext:= Filtered( ext, x -> ForAll( x, IsInt ) );
[ [ 3838185, 17577, 8505, 8505, 873, 0, 0, 0, 0, 6561, 0, 0, 729, 0, 9, 105,
45, 45, 105, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 189, 0, 0, 0, 9, 9, 27, 27,
0, 0, 27, 9, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0,
0, 0, 9, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0,
0, 0 ], [ 3838185, 17577, 8505, 8505, 873, 0, 6561, 0, 0, 0, 0, 0, 729,
0, 9, 105, 45, 45, 105, 30, 0, 0, 0, 0, 0, 0, 189, 0, 0, 0, 9, 0, 0, 0,
9, 27, 27, 0, 0, 9, 27, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0,
0, 0, 9, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0 ] ]
We compute the extensions of the first candidate;
the other belongs to another class of subgroups,
which is the image under an outer automorphism.
(These calculations may need about one hour,
depending on the computer used.)
gap> perms:= PermChars( o8p32, rec( torso:= ext[1] ) );
[ Character( CharacterTable( "O8+(3).2_1" ),
[ 3838185, 17577, 8505, 8505, 873, 0, 0, 0, 0, 6561, 0, 0, 729, 0, 9,
105, 45, 45, 105, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 189, 0, 0, 0, 9, 9,
27, 27, 0, 0, 27, 9, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0,
0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0,
0, 0, 0, 0, 3159, 1575, 567, 63, 87, 15, 0, 0, 45, 0, 81, 9, 27, 0, 0,
3, 3, 3, 3, 5, 5, 0, 0, 0, 4, 0, 0, 27, 0, 9, 0, 0, 15, 0, 3, 0, 0, 2,
0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ]
gap> PermCharInfo( o8p32, perms ).ATLAS;
[ "1a+260abc+520ab+819a+2808b+9450aab+18200a+23400ac+29120b+36400aab+46592abce\
+49140d+66339a+98280ab+163800a+189540d+232960d+332800ab+368550a+419328a+531441\
ab" ]
Now we repeat the calculations for O8+(3).22 instead of
O8+(3).21.
gap> o8p32:= CharacterTable( "O8+(3).2_2" );;
gap> fus:= GetFusionMap( o8p3, o8p32 );;
gap> ext:= List( ind, x -> CompositionMaps( x, InverseMap( fus ) ) );;
gap> ext:= Filtered( ext, x -> ForAll( x, IsInt ) );;
gap> perms:= PermChars( o8p32, rec( torso:= ext[1] ) );
[ Character( CharacterTable( "O8+(3).2_2" ), [ 3838185, 17577, 8505, 873, 0,
0, 0, 6561, 0, 0, 0, 0, 729, 0, 9, 105, 45, 105, 30, 0, 0, 0, 0, 0, 0,
189, 0, 0, 0, 9, 0, 9, 27, 0, 0, 0, 27, 27, 9, 0, 8, 1, 1, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0,
0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 199017, 2025, 297, 441, 73, 9, 0,
1215, 0, 0, 0, 0, 0, 81, 0, 0, 0, 0, 27, 27, 0, 1, 9, 12, 0, 0, 45, 0,
0, 1, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0
] ) ]
gap> PermCharInfo( o8p32, perms ).ATLAS;
[ "1a+260aac+520ab+819a+2808a+9450aaa+18200accee+23400ac+29120a+36400a+46592aa\
+49140c+66339a+93184a+98280ab+163800a+184275ac+189540c+232960c+332800aa+419328\
a+531441aa" ]
8 The Action of S4(4).4 on the Cosets of 52.[25]
We want to know whether the permutation character corresponding to the
action of S4(4).4 (see [CCN+85,p. 44]) on the cosets of its
maximal subgroup of type 52:[25] is multiplicity free.
The library names of subgroups for which the class fusions are stored
are listed as value of the attribute NamesOfFusionSources,
and for groups whose isomorphism type is not determined by the name
this is the recommended way to find out whether the table of the subgroup
is contained in the GAP library and known to belong to this group.
(It might be that a table with such a name is contained in the library
but belongs to another group,
and it may also be that the table of the group is contained in the
library -with any name- but it is not known that this group is
isomorphic to a subgroup of S4(4).4.
gap> s444:= CharacterTable( "S4(4).4" );;
gap> NamesOfFusionSources( s444 );
[ "S4(4)", "S4(4).2" ]
So we cannot simply fetch the table of the subgroup.
As in the previous examples, we compute the possible permutation
characters.
gap> perms:= PermChars( s444, rec( torso:= [ Size( s444 ) / ( 5^2*2^5 ) ] ) );
[ Character( CharacterTable( "S4(4).4" ), [ 4896, 384, 96, 0, 16, 32, 36, 16,
0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ),
Character( CharacterTable( "S4(4).4" ), [ 4896, 192, 32, 0, 0, 8, 6, 1, 0,
2, 0, 0, 36, 0, 12, 0, 0, 0, 1, 0, 6, 6, 2, 2, 0, 0, 0, 0, 1, 1 ] ),
Character( CharacterTable( "S4(4).4" ), [ 4896, 240, 64, 0, 8, 8, 36, 16,
0, 0, 0, 0, 0, 12, 8, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ]
So there are three candidates.
None of them is multiplicity free,
so we need not decide which of the candidates actually belongs
to the group 52:[25] we have in mind.
gap> PermCharInfo( s444, perms ).ATLAS;
[ "1abcd+50abcd+153abcd+170a^{4}b^{4}+680aabb",
"1a+50ac+153a+170aab+256a+680abb+816a+1020a",
"1ac+50ac+68a+153abcd+170aabbb+204a+680abb+1020a" ]
(If we would be interested which candidate is the right one,
we could for example look at the intersection with S4(4),
and hope for a contradiction to the fact that the group must lie
in a (A5 ×A5):2 subgroup.)
9 The Action of Co1 on the Cosets of Involution Centralizers
We compute the permutation characters of the sporadic simple Conway group
Co1 (see [CCN+85,p. 180]) corresponding to the actions on
the cosets of involution centralizers.
Equivalently, we are interested in the action of Co1 on conjugacy
classes of involutions.
These characters can be computed as follows.
First we take the table of Co1.
gap> t:= CharacterTable( "Co1" );
CharacterTable( "Co1" )
The centralizer of each 2A element is a maximal subgroup of Co1.
This group is also contained in the table library.
So we can compute the permutation character by explicit induction,
and the decomposition in irreducibles is computed with the command
PermCharInfo.
gap> s:= CharacterTable( Maxes( t )[5] );
CharacterTable( "2^(1+8)+.O8+(2)" )
gap> ind:= Induced( s, t, [ TrivialCharacter( s ) ] );;
gap> PermCharInfo( t, ind ).ATLAS;
[ "1a+299a+17250a+27300a+80730a+313950a+644644a+2816856a+5494125a+12432420a+24\
794000a" ]
The centralizer of a 2B element is not maximal.
First we compute which maximal subgroup can contain it.
The character tables of all maximal subgroups of Co1 are contained
in the GAP's table library,
so we may take these tables and look at the group orders.
gap> centorder:= SizesCentralizers( t )[3];;
gap> maxes:= List( Maxes( t ), CharacterTable );;
gap> cand:= Filtered( maxes, x -> Size( x ) mod centorder = 0 );
[ CharacterTable( "(A4xG2(4)):2" ) ]
gap> u:= cand[1];;
gap> index:= Size( u ) / centorder;
3
So there is a unique class of maximal subgroups containing the centralizer
of a 2B element, as a subgroup of index 3.
We compute the unique permutation character of degree 3 of this group,
and induce this character to G.
gap> subperm:= PermChars( u, rec( degree := index, bounds := false ) );
[ Character( CharacterTable( "(A4xG2(4)):2" ),
[ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3,
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ) ]
gap> subperm = PermChars( u, rec( torso := [ 3 ] ) );
true
gap> ind:= Induced( u, t, subperm );
[ Character( CharacterTable( "Co1" ), [ 2065694400, 181440, 119408, 38016,
2779920, 0, 0, 378, 30240, 864, 0, 720, 316, 80, 2520, 30, 0, 6480,
1508, 0, 0, 0, 0, 0, 38, 18, 105, 0, 600, 120, 56, 24, 0, 12, 0, 0, 0,
120, 48, 18, 0, 0, 6, 0, 360, 144, 108, 0, 0, 10, 0, 0, 0, 0, 0, 4, 2,
3, 9, 0, 0, 15, 3, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 12,
8, 0, 6, 0, 0, 3, 0, 1, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0 ] ) ]
gap> PermCharInfo( t, ind ).ATLAS;
[ "1a+1771a+8855a+27300aa+313950a+345345a+644644aa+871884aaa+1771000a+2055625a\
+4100096a+7628985a+9669660a+12432420aa+21528000aa+23244375a+24174150aa+2479400\
0a+31574400aa+40370176a+60435375a+85250880aa+100725625a+106142400a+150732800a+\
184184000a+185912496a+207491625a+299710125a+302176875a" ]
Finally, we try the same for the centralizer of a 2C element.
gap> centorder:= SizesCentralizers( t )[4];;
gap> cand:= Filtered( maxes, x -> Size( x ) mod centorder = 0 );
[ CharacterTable( "Co2" ), CharacterTable( "2^11:M24" ) ]
The group order excludes all except two classes of maximal subgroups.
But the 2C centralizer cannot lie in Co2 because the involution
centralizers in Co2 are too small.
gap> u:= cand[1];;
gap> GetFusionMap( u, t );
[ 1, 2, 2, 4, 7, 6, 9, 11, 11, 10, 11, 12, 14, 17, 16, 21, 23, 20, 22, 22,
24, 28, 30, 33, 31, 32, 33, 33, 37, 42, 41, 43, 44, 48, 52, 49, 53, 55, 53,
52, 54, 60, 60, 60, 64, 65, 65, 67, 66, 70, 73, 72, 78, 79, 84, 85, 87, 92,
93, 93 ]
gap> centorder;
389283840
gap> SizesCentralizers( u )[4];
1474560
So we try the second candidate.
gap> u:= cand[2];
CharacterTable( "2^11:M24" )
gap> index:= Size( u ) / centorder;
1288
gap> subperm:= PermChars( u, rec( torso := [ index ] ) );
[ Character( CharacterTable( "2^11:M24" ), [ 1288, 1288, 1288, 56, 56, 56,
56, 56, 56, 48, 48, 48, 48, 48, 10, 10, 10, 10, 7, 7, 8, 8, 8, 8, 8, 8,
4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 0,
0, 0, 0, 2, 2, 2, 2, 3, 3, 3, 1, 1, 2, 2, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0 ] ) ]
gap> subperm = PermChars( u, rec( degree:= index, bounds := false ) );
true
gap> ind:= Induced( u, t, subperm );
[ Character( CharacterTable( "Co1" ), [ 10680579000, 1988280, 196560, 94744,
0, 17010, 0, 945, 7560, 3432, 2280, 1728, 252, 308, 0, 225, 0, 0, 0,
270, 0, 306, 0, 46, 45, 25, 0, 0, 120, 32, 12, 52, 36, 36, 0, 0, 0, 0,
0, 45, 15, 0, 9, 3, 0, 0, 0, 0, 18, 0, 30, 0, 6, 18, 0, 3, 5, 0, 0, 0,
0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 1, 0, 0, 0, 0, 6, 0, 2,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ]
gap> PermCharInfo( t, last ).ATLAS;
[ "1a+17250aa+27300a+80730aa+644644aaa+871884a+1821600a+2055625aaa+2816856a+54\
94125a^{4}+12432420aa+16347825aa+23244375a+24174150aa+24667500aa+24794000aaa+3\
1574400a+40370176a+55255200a+66602250a^{4}+83720000aa+85250880aaa+91547820aa+1\
06142400a+150732800a+184184000aaa+185912496aaa+185955000aaa+207491625aaa+21554\
7904aa+241741500aaa+247235625a+257857600aa+259008750a+280280000a+302176875a+32\
6956500a+387317700a+402902500a+464257024a+469945476b+502078500a+503513010a+504\
627200a+522161640a" ]
10 The Multiplicity Free Permutation Characters of G2(3)
We compute the multiplicity free possible permutation characters of
G2(3) (see [CCN+85,p. 60]).
For each divisor d of the group order,
we compute all those possible permutation
characters of degree d of G for which each irreducible constituent
occurs with multiplicity at most 1;
this is done by prescribing the maxmult component of the second argument
of PermChars to be the list with 1 at each position.
gap> t:= CharacterTable( "G2(3)" );
CharacterTable( "G2(3)" )
gap> t:= CharacterTable( "G2(3)" );;
gap> n:= Length( RationalizedMat( Irr( t ) ) );;
gap> maxmult:= List( [ 1 .. n ], i -> 1 );;
gap> perms:= [];;
gap> divs:= DivisorsInt( Size( t ) );;
gap> for d in divs do
> Append( perms,
> PermChars( t, rec( bounds := false,
> degree := d,
> maxmult := maxmult ) ) );
> od;
gap> Length( perms );
42
gap> List( perms, Degree );
[ 1, 351, 351, 364, 364, 378, 378, 546, 546, 546, 546, 546, 702, 702, 728,
728, 1092, 1092, 1092, 1092, 1092, 1092, 1092, 1092, 1456, 1456, 1638,
1638, 2184, 2184, 2457, 2457, 2457, 2457, 3159, 3276, 3276, 3276, 3276,
4368, 6552, 6552 ]
For finding out which of these candidates are really permutation
characters, we could inspect them piece by piece, using the information
in [CCN+85].
For example, the candidates of degrees 351, 364, and 378 are
induced from the trivial characters of maximal subgroups of G,
whereas the candidates of degree 546 are not permutation characters.
Since the table of marks of G is available in GAP,
we can extract all permutation characters from the table of marks,
and then filter out the multiplicity free ones.
gap> tom:= TableOfMarks( "G2(3)" );
TableOfMarks( "G2(3)" )
gap> tbl:= CharacterTable( "G2(3)" );
CharacterTable( "G2(3)" )
gap> permstom:= PermCharsTom( tbl, tom );;
gap> Length( permstom );
433
gap> multfree:= Intersection( perms, permstom );;
gap> Length( multfree );
15
gap> List( multfree, Degree );
[ 1, 351, 351, 364, 364, 378, 378, 702, 702, 728, 728, 1092, 1092, 2184, 2184
]
11 Degree 11 200 Permutation Characters of O8+(2)
We compute the primitive permutation characters of degree 11 200 of
O8+(2) and O8+(2).2 (see [CCN+85,p. 85]).
The character table of the maximal subgroup of type 34:23.S4 in
O8+(2) is not available in the GAP table library.
But the group extends to a wreath product of S3 and S4 in the
group O8+(2).2, and the table of this wreath product can be
constructed easily.
gap> tbl2:= CharacterTable("O8+(2).2");;
gap> s3:= CharacterTable( "Symmetric", 3 );;
gap> s:= CharacterTableWreathSymmetric( s3, 4 );
CharacterTable( "Sym(3)wrS4" )
The permutation character pi of O8+(2).2 can thus be computed by
explicit induction, and the character of O8+(2) is obtained by
restriction of pi.
gap> fus:= PossibleClassFusions( s, tbl2 );
[ [ 1, 41, 6, 3, 48, 9, 42, 19, 51, 8, 5, 50, 24, 49, 7, 2, 44, 22, 42, 12,
53, 17, 58, 21, 5, 47, 26, 50, 37, 52, 23, 60, 18, 4, 46, 25, 14, 61,
20, 9, 53, 30, 51, 26, 64, 8, 52, 31, 13, 56, 38 ] ]
gap> pi:= Induced( s, tbl2, [ TrivialCharacter( s ) ], fus[1] )[1];
Character( CharacterTable( "O8+(2).2" ), [ 11200, 256, 160, 160, 80, 40, 40,
76, 13, 0, 0, 8, 8, 4, 0, 0, 16, 16, 4, 4, 4, 1, 1, 1, 1, 5, 0, 0, 0, 1, 1,
0, 0, 0, 0, 0, 2, 2, 0, 0, 1120, 96, 0, 16, 0, 16, 8, 10, 4, 6, 7, 12, 3,
0, 0, 2, 0, 4, 0, 1, 1, 0, 0, 1, 0, 0, 0 ] )
gap> PermCharInfo( tbl2, pi ).ATLAS;
[ "1a+84a+168a+175a+300a+700c+972a+1400a+3200a+4200b" ]
gap> tbl:= CharacterTable( "O8+(2)" );
CharacterTable( "O8+(2)" )
gap> rest:= RestrictedClassFunction( pi, tbl );
Character( CharacterTable( "O8+(2)" ), [ 11200, 256, 160, 160, 160, 80, 40,
40, 40, 76, 13, 0, 0, 8, 8, 8, 4, 0, 0, 0, 16, 16, 16, 4, 4, 4, 4, 1, 1, 1,
1, 1, 1, 5, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0 ] )
gap> PermCharInfo( tbl, rest ).ATLAS;
[ "1a+84abc+175a+300a+700bcd+972a+3200a+4200a" ]
12 A Proof of Nonexistence of a Certain Subgroup
We prove that the sporadic simple Mathieu group G = M22
(see [CCN+85,p. 39]) has no subgroup of index 56.
In [Isa76], remark after Theorem 5.18, this is stated as an example
of the case that a character may be a possible permutation character but
not a permutation character.
Let us consider the possible permutation character of degree 56 of
G.
gap> tbl:= CharacterTable( "M22" );
CharacterTable( "M22" )
gap> perms:= PermChars( tbl, rec( torso:= [ 56 ] ) );
[ Character( CharacterTable( "M22" ), [ 56, 8, 2, 4, 0, 1, 2, 0, 0, 2, 1, 1
] ) ]
gap> pi:= perms[1];;
gap> Norm( pi );
2
gap> Display( tbl, rec( chars:= perms ) );
M22
2 7 7 2 5 4 . 2 . . 3 . .
3 2 1 2 . . . 1 . . . . .
5 1 . . . . 1 . . . . . .
7 1 . . . . . . 1 1 . . .
11 1 . . . . . . . . . 1 1
1a 2a 3a 4a 4b 5a 6a 7a 7b 8a 11a 11b
2P 1a 1a 3a 2a 2a 5a 3a 7a 7b 4a 11b 11a
3P 1a 2a 1a 4a 4b 5a 2a 7b 7a 8a 11a 11b
5P 1a 2a 3a 4a 4b 1a 6a 7b 7a 8a 11a 11b
7P 1a 2a 3a 4a 4b 5a 6a 1a 1a 8a 11b 11a
11P 1a 2a 3a 4a 4b 5a 6a 7a 7b 8a 1a 1a
Y.1 56 8 2 4 . 1 2 . . 2 1 1
Suppose that pi is a permutation character of G.
Since G is 2-transitive on the 56 cosets of the point stabilizer S,
this stabilizer is transitive on 55 points,
and thus G has a subgroup U of index 56 ·55 = 3080.
We compute the possible permutation character of this degree.
gap> perms:= PermChars( tbl, rec( torso:= [ 56 * 55 ] ) );;
gap> Length( perms );
16
U is contained in S, so only those candidates must be considered
that vanish on all classes where pi vanishes.
Furthermore, the index of U in S is odd, so the Sylow 2 subgroups
of U and S are isomorphic;
S contains elements of order 8, hence also U does.
gap> OrdersClassRepresentatives( tbl );
[ 1, 2, 3, 4, 4, 5, 6, 7, 7, 8, 11, 11 ]
gap> perms:= Filtered( perms, x -> x[5] = 0 and x[10] <> 0 );
[ Character( CharacterTable( "M22" ), [ 3080, 56, 2, 12, 0, 0, 2, 0, 0, 2, 0,
0 ] ), Character( CharacterTable( "M22" ),
[ 3080, 8, 2, 8, 0, 0, 2, 0, 0, 4, 0, 0 ] ),
Character( CharacterTable( "M22" ), [ 3080, 24, 11, 4, 0, 0, 3, 0, 0, 2, 0,
0 ] ), Character( CharacterTable( "M22" ),
[ 3080, 24, 20, 4, 0, 0, 0, 0, 0, 2, 0, 0 ] ) ]
For getting an overview of the distribution of the elements of U to the
conjugacy classes of G, we use the output of PermCharInfo.
gap> infoperms:= PermCharInfo( tbl, perms );;
gap> Display( tbl, infoperms.display );
M22
2 7 7 2 5 2 3
3 2 1 2 . 1 .
5 1 . . . . .
7 1 . . . . .
11 1 . . . . .
1a 2a 3a 4a 6a 8a
2P 1a 1a 3a 2a 3a 4a
3P 1a 2a 1a 4a 2a 8a
5P 1a 2a 3a 4a 6a 8a
7P 1a 2a 3a 4a 6a 8a
11P 1a 2a 3a 4a 6a 8a
I.1 3080 56 2 12 2 2
I.2 1 21 8 54 24 36
I.3 1 3 4 9 12 18
I.4 3080 8 2 8 2 4
I.5 1 3 8 36 24 72
I.6 1 3 4 9 12 18
I.7 3080 24 11 4 3 2
I.8 1 9 44 18 36 36
I.9 1 3 4 9 12 18
I.10 3080 24 20 4 . 2
I.11 1 9 80 18 . 36
I.12 1 3 4 9 12 18
We have four candidates.
For each the above list shows first the character values,
then the cardinality of the intersection of U with the classes,
and then lower bounds for the lengths of U-conjugacy classes of these
elements.
Only those classes of G are shown that contain elements of U
for at least one of the characters.
If the first two candidates are permutation characters corresponding to
U then U contains exactly 8 elements of order 3
and thus U has a normal Sylow 3 subgroup P.
But the order of NG(P) is bounded by 72,
which can be shown as follows.
The only elements in G with centralizer order divisible by 9
are of order 1 or 3, so P is self-centralizing in G.
The factor NG(P)/CG(P) is isomorphic with a subgroup of
Aut(G) ≅ GL(2,3) which has order divisible
by 16, hence the order of NG(P) divides 144.
Now note that [ G : NG(P) ] ≡ 1 mod 3 by Sylow's Theorem,
and |G|/144 = 3 080 ≡ −1 mod 3.
Thus the first two candidates are not permutation characters.
If the last two candidates are permutation characters corresponding to
U then U has self-normalizing Sylow subgroups.
This is because the index of a Sylow 2 normalizer
in G is odd and divides 9,
and if it is smaller than 9 then U contains
at most 3 ·15 + 1 elements of 2 power order;
the index of a Sylow 3 normalizer
in G is congruent to 1 modulo 3 and divides 16,
and if it is smaller than 16 then U contains
at most 4 ·8 elements of order 3.
But since U is solvable and not a p-group,
not all its Sylow subgroups can be self-normalizing;
note that U has a proper normal subgroup N containing
a Sylow p subgroup P of U for a prime divisor p of |U|,
and U = N ·NU(P) holds by the Frattini argument
(see [Hup67,Satz I.7.8]).
13 A Permutation Character of the Lyons group
Let G be a maximal subgroup with structure 32+4:2A5.D8
in the sporadic simple Lyons group Ly.
We want to compute the permutation character 1GLy.
(This construction has been explained in [BP98,Section 4.2],
without showing explicit GAP code.)
In the representation of Ly as automorphism group of the rank 5
graph B with 9 606 125 points (see [CCN+85,p. 174]),
G is the stabilizer of an edge.
A group S with structure 3.McL.2 is the point stabilizer.
So the two point stabilizer U = S ∩G is a subgroup of index 2
in G.
The index of U in S is 15 400, and according to the list of
maximal subgroups of McL.2 (see [CCN+85,p. 100]),
the group U is isomorphic to the preimage in 3.McL.2 of a subgroup H
of McL.2 with structure 3+1+4:4S5.
Using the improved combinatorial method described
in [BP98,Section 3.2],
all possible permutation characters of degree 15 400 for the group McL
are computed.
(The method of [BP98,Section 3.3] is slower but also needs only
a few seconds.)
gap> ly:= CharacterTable( "Ly" );;
gap> mcl:= CharacterTable( "McL" );;
gap> mcl2:= CharacterTable( "McL.2" );;
gap> 3mcl2:= CharacterTable( "3.McL.2" );;
gap> perms:= PermChars( mcl, rec( degree:= 15400 ) );
[ Character( CharacterTable( "McL" ), [ 15400, 56, 91, 10, 12, 25, 0, 11, 2,
0, 0, 2, 1, 1, 1, 0, 0, 3, 0, 0, 1, 1, 1, 1 ] ),
Character( CharacterTable( "McL" ), [ 15400, 280, 10, 37, 20, 0, 5, 10, 1,
0, 0, 2, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0 ] ) ]
We get two characters, corresponding to the two classes of maximal
subgroups of index 15 400 in McL.
The permutation character π = 1H ∩McLMcL is the one with
nonzero value on the class 10A, since the subgroup of structure
2S5 in H ∩McL contains elements of order 10.
gap> ord10:= Filtered( [ 1 .. NrConjugacyClasses( mcl ) ],
> i -> OrdersClassRepresentatives( mcl )[i] = 10 );
[ 15 ]
gap> List( perms, pi -> pi[ ord10[1] ] );
[ 1, 0 ]
gap> pi:= perms[1];
Character( CharacterTable( "McL" ), [ 15400, 56, 91, 10, 12, 25, 0, 11, 2, 0,
0, 2, 1, 1, 1, 0, 0, 3, 0, 0, 1, 1, 1, 1 ] )
The character 1HMcL.2 is an extension of π,
so we can use the method of [BP98,Section 3.3] to compute all
possible permutation characters for the group McL.2 that have
the values of π on the classes of McL.
We find that the extension of π to a permutation character of McL.2
is unique.
Regarded as a character of 3.McL.2, this character is equal to 1US.
gap> map:= InverseMap( GetFusionMap( mcl, mcl2 ) );
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, [ 10, 11 ], 12, [ 13, 14 ], 15, 16, 17, 18,
[ 19, 20 ], [ 21, 22 ], [ 23, 24 ] ]
gap> torso:= CompositionMaps( pi, map );
[ 15400, 56, 91, 10, 12, 25, 0, 11, 2, 0, 2, 1, 1, 0, 0, 3, 0, 1, 1 ]
gap> perms:= PermChars( mcl2, rec( torso:= torso ) );
[ Character( CharacterTable( "McL.2" ), [ 15400, 56, 91, 10, 12, 25, 0, 11,
2, 0, 2, 1, 1, 0, 0, 3, 0, 1, 1, 110, 26, 2, 4, 0, 0, 5, 2, 1, 1, 0, 0,
1, 1 ] ) ]
gap> pi:= Inflated( perms[1], 3mcl2 );
Character( CharacterTable( "3.McL.2" ), [ 15400, 15400, 56, 56, 91, 91, 10,
12, 12, 25, 25, 0, 0, 11, 11, 2, 2, 0, 0, 0, 2, 2, 1, 1, 1, 0, 0, 0, 0, 3,
3, 0, 0, 0, 1, 1, 1, 1, 1, 1, 110, 26, 2, 4, 0, 0, 5, 2, 1, 1, 0, 0, 1, 1
] )
The fusion of conjugacy classes of S in Ly can be computed from
the character tables of S and Ly given in [CCN+85],
it is unique up to Galois automorphisms of the table of Ly.
gap> fus:= PossibleClassFusions( 3mcl2, ly );; Length( fus );
4
gap> g:= AutomorphismsOfTable( ly );;
gap> OrbitLengths( g, fus, OnTuples );
[ 4 ]
Now we can induce 1US to Ly, which yields (1US)Ly = 1ULy.
gap> pi:= Induced( 3mcl2, ly, [ pi ], fus[1] )[1];
Character( CharacterTable( "Ly" ), [ 147934325000, 286440, 1416800, 1082,
784, 12500, 0, 672, 42, 24, 0, 40, 0, 2, 20, 0, 0, 0, 64, 10, 0, 50, 2, 0,
0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0 ] )
All elements of odd order in G are contained in U,
for such an element g we have
1GLy(g) = |
|CLy(g)|
|G|
|
·|G ∩ClLy(g)| = |
|CLy(g)|
2 ·|U|
|
·|U ∩ClLy(g)| = |
1
2
|
·1ULy(g) , |
|
so we can prescribe the values of 1GLy on all classes of odd
element order.
For elements g of even order we have the weaker condition
U∩ClLy(g) ⊆ G ∩ClLy(g)
and thus 1GLy(g) ≥ [1/2] ·1ULy(g),
which gives lower bounds for the value of 1GLy on the
remaining classes.
gap> orders:= OrdersClassRepresentatives( ly );
[ 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 7, 8, 8, 9, 10, 10, 11, 11, 12, 12, 14, 15,
15, 15, 18, 20, 21, 21, 22, 22, 24, 24, 24, 25, 28, 30, 30, 31, 31, 31, 31,
31, 33, 33, 37, 37, 40, 40, 42, 42, 67, 67, 67 ]
gap> torso:= [];;
gap> for i in [ 1 .. Length( orders ) ] do
> if orders[i] mod 2 = 1 then
> torso[i]:= pi[i]/2;
> fi;
> od;
gap> torso;
[ 73967162500,, 708400, 541,, 6250, 0,,,, 0,,, 1,,, 0, 0,,,, 25, 1, 0,,, 0, 0,
,,,,, 0,,,, 0, 0, 0, 0, 0, 0, 0, 0, 0,,,,, 0, 0, 0 ]
Exactly one possible permutation character of Ly satisfies these
conditions.
gap> perms:= PermChars( ly, rec( torso:= torso ) );;
gap> Length( perms );
43
gap> perms:= Filtered( perms, cand -> ForAll( [ 1 .. Length( orders ) ],
> i -> cand[i] >= pi[i] / 2 ) );
[ Character( CharacterTable( "Ly" ), [ 73967162500, 204820, 708400, 541, 392,
6250, 0, 1456, 61, 25, 0, 22, 10, 1, 10, 0, 0, 0, 32, 5, 0, 25, 1, 0,
1, 2, 0, 0, 0, 0, 4, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2,
0, 0, 0, 0, 0 ] ) ]
(The permutation character 1GLy was used in the proof that the
character χ37 of Ly (see [CCN+85,p. 175])
occurs with multiplicity at least 2 in each character of Ly
that is induced from a proper subgroup of Ly.)
14 Identifying two subgroups of Aut(U3(5))
According to the Atlas of Finite Groups [CCN+85,p. 34],
the group Aut(U3(5)) has two classes of maximal subgroups of
order 24 ·33, which have the structures 32 \colon 2S4 and
62 \colon D12, respectively.
gap> tbl:= CharacterTable( "U3(5).3.2" );
CharacterTable( "U3(5).3.2" )
gap> deg:= Size( tbl ) / ( 2^4*3^3 );
1750
gap> pi:= PermChars( tbl, rec( torso:= [ deg ] ) );
[ Character( CharacterTable( "U3(5).3.2" ), [ 1750, 70, 13, 2, 0, 0, 1, 0, 0,
0, 10, 7, 10, 4, 2, 0, 0, 0, 0, 0, 0, 30, 10, 3, 0, 0, 1, 0, 0 ] ),
Character( CharacterTable( "U3(5).3.2" ), [ 1750, 30, 4, 6, 0, 0, 0, 0, 0,
0, 40, 7, 0, 6, 0, 0, 0, 0, 0, 0, 0, 20, 0, 2, 2, 0, 0, 0, 0 ] ) ]
Now the question is which character belongs to which subgroup.
We see that the first character vanishes on the classes of element order
8 and the second does not, so only the first one can be the permutation
character induced from 62 \colon D12.
gap> ord8:= Filtered( [ 1 .. NrConjugacyClasses( tbl ) ],
> i -> OrdersClassRepresentatives( tbl )[i] = 8 );
[ 9, 25 ]
gap> List( pi, x -> x{ ord8 } );
[ [ 0, 0 ], [ 0, 2 ] ]
Thus the question is whether the second candidate is really a permutation
character.
Since none of the two candidates vanishes on any outer coset of
U3(5) in Aut(U3(5)), the point stabilizers are extensions
of groups of order 23 ·32 in U3(5).
The restrictions of the candidates to U3(5) are different,
so we can try to answer the question using information about this group.
gap> subtbl:= CharacterTable( "U3(5)" );
CharacterTable( "U3(5)" )
gap> rest:= RestrictedClassFunctions( pi, subtbl );
[ Character( CharacterTable( "U3(5)" ), [ 1750, 70, 13, 2, 0, 0, 0, 0, 1, 0,
0, 0, 0, 0 ] ), Character( CharacterTable( "U3(5)" ),
[ 1750, 30, 4, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ]
The intersection of the 32 \colon 2S4 subgroup with U3(5) lies
inside the maximal subgroup of type M10,
which does not contain elements of order6.
Only the second character has this property.
gap> ord6:= Filtered( [ 1 .. NrConjugacyClasses( subtbl ) ],
> i -> OrdersClassRepresentatives( subtbl )[i] = 6 );
[ 9 ]
gap> List( rest, x -> x{ ord6 } );
[ [ 1 ], [ 0 ] ]
In order to establish the two characters as permutation characters,
we could also compute the permutation characters of the degree in question
directly from the table of marks of U3(5), which is contained in the
GAP library of tables of marks.
gap> tom:= TableOfMarks( "U3(5)" );
TableOfMarks( "U3(5)" )
gap> perms:= PermCharsTom( subtbl, tom );;
gap> Set( Filtered( perms, x -> x[1] = deg ) ) = Set( rest );
true
We were mainly interested in the multiplicities of irreducible characters
in these characters.
The action of Aut(U3(5) on the cosets of 32 \colon 2S4
turns out to be multiplicity-free whereas that on the cosets of
62 \colon D12 is not.
gap> PermCharInfo( tbl, pi ).ATLAS;
[ "1a+21a+42a+84aac+105a+125a+126a+250a+252a+288bc",
"1a+42a+84ac+105ab+125a+126a+250a+252b+288bc" ]
It should be noted that the restrictions of the multiplicity-free character
to the subgroups U3(5).2 and U3(5).3 of Aut(U3(5) are not
multiplicity-free.
gap> subtbl2:= CharacterTable( "U3(5).2" );;
gap> rest2:= RestrictedClassFunctions( pi, subtbl2 );;
gap> PermCharInfo( subtbl2, rest2 ).ATLAS;
[ "1a+21aab+28aa+56aa+84a+105a+125aab+126aab+288aa",
"1a+21ab+28a+56a+84a+105ab+125aab+126a+252a+288aa" ]
gap> subtbl3:= CharacterTable( "U3(5).3" );;
gap> rest3:= RestrictedClassFunctions( pi, subtbl3 );;
gap> PermCharInfo( subtbl3, rest3 ).ATLAS;
[ "1a+21abc+84aab+105a+125abc+126abc+144bcef",
"1a+21bc+84ab+105aa+125abc+126adg+144bcef" ]
15 A Permutation Character of Aut(O8+(2))
According to the Atlas of Finite Groups [CCN+85,p. 85],
the group G = Aut(O8+(2)) has a class of maximal subgroups of
order 213 ·32, thus the index of these subgroups in G is
34 ·52 ·7.
The intersection of these subgroups with H = O8+(2) lie inside maximal
subgroups of type 26 \colon A8.
We want to show that the permutation character of the action of G on
the cosets of these subgroups is not multiplicity-free.
Since the table of marks for H is available in GAP, but not that for
G, we first compute the H-permutation characters of the intersections
with H of index 34 ·52 ·7 = 14 175 subgroups in G.
(Note that these intersections have order 212 ·3 because
subgroups of order 212 ·32 are contained in O8+(2).2
and hence are not maximal in G.)
gap> t:= CharacterTable( "O8+(2).3.2" );;
gap> s:= CharacterTable( "O8+(2)" );;
gap> tom:= TableOfMarks( s );;
gap> perms:= PermCharsTom( s, tom );;
gap> deg:= 3^4*5^2*7;
14175
gap> perms:= Filtered( perms, x -> x[1] = deg );;
gap> Length( perms );
4
gap> Length( Set( perms ) );
1
We see that there are four classes of subgroups S in H that may
belong to maximal subgroups of the desired index in G,
and that the permutation characters are equal.
They lead to such groups if they extend to G,
so we compute the possible permutation characters of G that extend
these characters.
gap> fus:= PossibleClassFusions( s, t );
[ [ 1, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 8, 9, 10, 10, 10, 11, 12, 12, 12, 13,
13, 13, 14, 14, 14, 15, 16, 16, 16, 17, 17, 17, 18, 19, 20, 21, 22, 22,
22, 23, 23, 23, 24, 24, 24, 25, 26, 26, 26, 27, 27, 27 ] ]
gap> fus:= fus[1];;
gap> inv:= InverseMap( fus );;
gap> comp:= CompositionMaps( perms[1], inv );
[ 14175, 1215, 375, 79, 0, 0, 27, 27, 99, 15, 7, 0, 0, 0, 0, 9, 3, 1, 0, 1,
1, 0, 0, 0, 0, 0, 0 ]
gap> ext:= PermChars( t, rec( torso:= comp ) );
[ Character( CharacterTable( "O8+(2).3.2" ),
[ 14175, 1215, 375, 79, 0, 0, 27, 27, 99, 15, 7, 0, 0, 0, 0, 9, 3, 1, 0,
1, 1, 0, 0, 0, 0, 0, 0, 63, 9, 15, 7, 1, 0, 3, 3, 3, 1, 0, 0, 1, 1,
945, 129, 45, 69, 21, 25, 13, 0, 0, 0, 9, 0, 3, 3, 7, 1, 0, 0, 0, 3, 1,
0, 0, 0, 0, 0, 0 ] ) ]
gap> PermCharInfo( t, ext[1] ).ATLAS;
[ "1a+50b+100a+252bb+300b+700b+972bb+1400a+1944a+3200b+4032b" ]
Thus we get one permutation character of G which is not multiplicity-free.
References
- [BP98]
-
Thomas Breuer and Götz Pfeiffer, Finding Possible
Permutation Characters, J. Symbolic Comput. 26 (1998),
343-354.
- [Bre04]
-
Thomas Breuer, Manual for the GAP Character Table Library, Version
1.1, Lehrstuhl D für Mathematik, Rheinisch
Westfälische Technische Hochschule, Aachen, Germany,
2004.
- [CCN+85]
-
J[ohn] H. Conway, R[obert] T. Curtis, S[imon] P. Norton, R[ichard] A. Parker,
and R[obert] A. Wilson, Atlas of finite groups, Oxford University
Press, 1985.
- [GAP04]
-
The GAP Group, GAP - Groups, Algorithms, and Programming, Version
4.4, 2004, http://www.gap-system.org.
- [Hup67]
-
B[ertram] Huppert, Endliche Gruppen I, Grundlehren Math. Wiss., vol.
134, Springer, Berlin, Heidelberg and New York, 1967.
- [Isa76]
-
I. M. Isaacs, Character theory of finite groups, Pure and applied
mathematics, vol. 69, Academic Press, New York, 1976, xii+303 pp., ISBN
0-12-374550-0.
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