[GAP Forum] Construction of a group

[Stefanos Aivazidis]

Dear Forum,

I wish to construct a family of groups parametrised by a variable n, to be
specified by the user. Let me describe the steps:

0) Specify a value for n,

1) construct the finite field F=F_q with q=2^(2n+1) elements,

2) construct the automorphism f of F mapping each element x to x^(2^a),
where a=n+1,

3) define an operation * on FxF (as a set) via the rule:
(x,y)*(x',y')=(x+x',y+y'+xf(x')), where the operations on the
right-hand-side are those of the underlying field F,

Now P=(FxF,*) is a group, but I don't know if I can assure GAP this is the
case. Perhaps I can ask her to test this somehow (GAP is a woman in my
world).

4) assuming that GAP now knows P is a group, define the semi-direct product
S=P]C, where C is the multiplicative group of F and acts on P (via
automorphisms) by the rule:c.(x,y)=(cx,cf(c)y), where f is as in 2) and
concatenation on the right-hand-side is just multiplication in F.

To give some motivation for the above construction, P thus constructed is
the Sylow 2-subgroup of Sz(q), while S is its normaliser in Sz(q). Of
course I can access both P and S via P:=SylowSubgroup(Sz(q),2) and
S:=Normaliser(Sz(q),P) respectively, but I would prefer to work with P and
S constructed as per the above "algorithm". The reason is that I mainly
want to count subgroups and conjugacy classes of subgroups of P and S for
(at least) n=4, since for smaller values of n, the order of C is a prime
(thus C uninteresting), but GAP is unable to handle this in terms of memory.

I have serious doubts that constructing P and S as above will be more
memory-efficient than the permutation representation already in
Sz(IsPermGroup,q), but I can't decide this issue in advance.

If it is indeed non-trivial to decide beforehand, I would appreciate any
help in how to write the code for the construction as explained above.

Many thanks,
Stefanos