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
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