Kaustuv M. Das asked:
I am running GAP 3.4 on a Pentium 133 machine with 32 megs of memory (in
a DOS window from within Windows 95). I am interested in getting hold of
all the Sylow_3 subgroups of Sp_6(5), but whenever I try the GAP command:
Three people have responded to this query, but none of them seem to have
attempted to answer what was asked. He wants to find ALL Sylow 3-subgroups,
not just one of them!
The way to do that is to find the normaliser of one Sylow subgroup, find
coset representatives for it in the whole group, then conjugate your
Sylow subgroup by these coset representatives.
Akos Seress would probably want to compute the normaliser within the matrix
group, using standard constructions for classical groups, but currently it
is probably easiest to work in the permutation representation - there is
in fact nothing lost by working in Psp(6,5), since that only loses the
centre of order 2.
Well I computed the normaliser, and it has order 324 in PSp(6,5) (and hence 648 in Sp(6,5)), which means index 705250000000 in the whole group.
Oh dear! There appear to be 705250000000 Sylow 3-subgroups. No wonder
your computer was struggling. Are you sure you want them all?