[GAP Forum] checking subgroups and conjugacy classes

[Alexander Hulpke]

Dear Forum, Dear Hung Nguyen,

> \Let g:=M_{10}=A_6 . 2_3 - this is the stabilizer of a point in M_{11}. First I want to check whether g is isomorphic to a subgroup of GL(2,19). If the answer is yes, then list the structures of all groups G such that G/(C_{19} * C_{19}) = g where g acts naturally on C_{19} * C_{19}. Here C_{19} is the cyclic group of order 19.
> 
> For such a group G and a prime p dividing |G|, list the number of p-regular conjugacy classes of G (a class is p-regular if its element order is not divisible by p). 

There is an old theorem (by Dickson, I believe) that classifies the subgroups of PSL_2(q). The theorem is for example in Huppert's first volume but the book is in my office, so I can't find the exact page.
The only simple nonabelian subgroups are the obvious PSL's and A_5. Thus M10 cannot be a subgroup of GL2(19).

Ignoring this, in GAP you probably would use
h:=GL(2,19);
u:=List(ConjugacyClassesSubgroups(h),Representative);
to get a list of representatives of all subgroups up to conjugacy,
u:=Filtered(u,x->IsomorphismGroups(x,g)<>fail);
to find those which are isomorphic to your group g.

To classify the extensions abstractly (assuming that by `*' you mean direct product) you would need cohomology for nonsolvable factor groups, which the `cohomolo' package provides. However, given the small size, you also would find such a group (apart from a few obvioud degenerate cases) in the libary of perfect groups.

Regards,

   Alexander Hulpke