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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 $D_8$.

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  • $\begingroup$ In general there’s no reason to expect such characterizations for “generic” $p$-groups; there are just too many of them (asymptotically $p^{\frac{2}{27} n^3 + O(n^{8/3})}$ of order $p^n$). In this particular case there are $2328$ groups of order $128$ so most of them can’t have very simple descriptions in terms of “known” $2$-groups. $\endgroup$ – Qiaochu Yuan Oct 26 '20 at 9:00
  • $\begingroup$ @QiaochuYuan thanks for your response. I realise this and was also put off by the 2328 candidate groups, however, I was hoping that the rather specific "algebraic" expression for the structure would allow some form of "expansion". $\endgroup$ – BeMuSeD Oct 26 '20 at 13:21
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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.

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