Alternative characterisation of group defined by nested semidirect products

Question

I am studying the group generated by the permutations $\{(1,2),(3,4),(5,6),(7,8),(1,3)(2,4), (5,7)(6,8),(1,5)(2,6)(3,7)(4,8)\}$ using the GAP system. GAP's StructureDescription of this order 128 group is "(((C2 x C2 x C2 x C2) : C2) : C2) : C2", i.e. a nested semidirect product. Is there any other way to characterise this group in terms of larger known groups, and if so is there a (systematic?) procedure to do so? The reason I am hopeful is that "(C2 x C2) : C2" is simply $D_8$.

Answer 1

In general, StructureDescription is a useless command when groups get bigger (and for $p$-groups "bigger" means more than $p^3$). One reason is that there are many different possible semidirect product structures.

In your case, however you also have a transitive permutation representation, and thus can identify it in the library of transitive groups (which describes not only the abstract structure, but also its permutation action):

gap> IsTransitive(g,[1..8]);
true
gap> TransitiveIdentification(g);
35
gap> TransitiveGroup(8,35);
[2^4]D(4)

This name means (reference: https://doi.org/10.1112/S1461157000000115 ) that there are 4 copies of a group $C_2$, acting on two points, and these copies are permuted by a dihedral group on 4 points (you call this dihedral group $D_8$ in your question).

Another way of calling the group thus would be $C_2\wr D_8$, the natural wreath product. It has a normal subgroup $C_2^4$ and a complement $D_8$ that acts on it via the permutation action on 4 points.