[GAP Forum] Galois Group question

[Igor Schein]

Dear Gap Forum,

Given a small group G of order 2^n, I would like to know whether or not 
there exists a polynomial P in x^4 whose Galois group is G.  Let me
illustrate:

gap> IdGroup(TransitiveGroup(8,GaloisType(x^8+3*x^4+1)));
[ 8, 3 ]

So the answer is yes for small group [8,3],  P: x -> x^2+3*x+1

However, if I consider small group [8,4], such P clearly doesn't
exist, so the answer is no.

So my question is how I can answer this question using GAP commands
and intrinsic properties of groups.  Specifically, I need to know the
answer for [64,64] and [64,122].

Thanks in advance and please forgive any (unintentional) ambiguity
and/or terminology misuse.

Igor