>> 2. An alternative is to use the library of perfect groups in GAP.
>That's a good method, but how do you know that the
>group in question is perfect?
Roughly speaking, there is no abelian group "on top"...
One can, for instance, rule out, on case-by-case basis,
all the possibilities for the commutator subgroup to be
a proper normal subgroup. There are not that many of them.
>> 3. A third possibility is the construction of the semidirect
>> product as a group of 4 by 4 matrices over the field with
>> 5 elements.
>I can't see why this construction gives the group in question.
> Can you say a bit
>more or give some references about that?
The group in question is an index two subgroup in the point
stabilizer of Sp_4(5) acting naturally on its 4-dim. module.
See e.g. Atlas of Finite Groups by Conway et al.
I posted 4x4-matrices that generate it to the Forum, but the message
apparently got lost somewhere...