[GAP Forum] Sporadic groups.

[Thomas Breuer]

Dear GAP Forum,

Fernando Fantino wrote

> I write you in order to do the following question.
> With a colleague,  Leandro Vendramin, I am working on some problem related
> to finite groups.
> In particular, we need to do some computations with the sporadic
> groups (G); for instance, we need to compute character
> tables of centralizers of (some) elements in G, etc. When the
> (sporadic) group is not so long these operation can be implemented
> without (serious) problems. But, for instance, in the monster
> cases the centralizers are so long that the program cannot do such
> a operation. Nevertheless, many of these centralizers are in the
> Library of GAP.
> THE QUESTION: is that true? In that case, how can I identify them?

Let us take the Fischer group Fi22 an example.

A look into the Atlas of Finite group shows that the centralizers
of involutions in the classes 2A and 2B are maximal subgroups of Fi22,
in the 1st and 7th class of maximal subgroups.
In GAP, one can get them as follows.

    gap> t:= CharacterTable( "Fi22" );
    CharacterTable( "Fi22" )
    gap> mx:= Maxes( t );
    [ "2.U6(2)", "O7(3)", "Fi22M3", "O8+(2).3.2", "2^10:m22", "2^6:s6f2", 
      "(2x2^(1+8)):U4(2):2", "S3xU4(3).2_2", "2F4(2)'", "2^(5+8):(S3xA6)", 
      "3^(1+6):2^(3+4):3^2:2", "A10.2", "A10.2", "M12" ]
    gap> CharacterTable( mx[1] );
    CharacterTable( "2.U6(2)" )
    gap> CharacterTable( mx[7] );
    CharacterTable( "(2x2^(1+8)):U4(2):2" )

Alternatively, also the following works.

    gap> CharacterTable( "Fi22C2A" );
    CharacterTable( "2.U6(2)" )
    gap> CharacterTable( "Fi22C2B" );
    CharacterTable( "(2x2^(1+8)):U4(2):2" )

Other element centralizers may be not maximal.
For example, the 3A centralizer has order 19595520,
and it has index two in the 3A normalizer, which is maximal and has
the structure $S_3 \times U_4(3).2_2$;
so the 3A centralizer has the structure $3 \times U_4(3).2_2$,
and its character table can be obtained as follows.

    gap> s:= CharacterTable( "C3" ) * CharacterTable( "U4(3).2_2" );
    CharacterTable( "C3xU4(3).2_2" )

Other element centralizers are very small, so it is no problem to compute
them from a representation of the group.

All the best,
Thomas