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Extensions for nontrivial outer action of Z2 on Q8

This article describes all the group extensions corresponding to a particular outer action with normal subgroup quaternion group and quotient group cyclic group:Z2.

We consider here the group extensions where the base normal subgroup N is dihedral group:D8, the quotient group Q is cyclic group:Z2, and the induced outer action of the quotient group on the normal subgroup is a nontrivial map. There are three such possible maps, but they are all conjugate to each other, and hence there is essentially only one type of map.

More explicitly, note that \operatorname{Out}(N) is isomorphic to symmetric group:S3, with three conjugate copies of cyclic group:Z2 in it (the three S2 in S3s), and thus there is a unique (up to conjugacy) nontrivial map from Q to the group.

Description in terms of cohomology groups

We have the induced outer action which is nontrivial:

Q \to \operatorname{Out}(N)

Composing with the natural mapping \operatorname{Out}(N) \to \operatorname{Aut}(Z(N)), we get a trivial map:

Q \to \operatorname{Aut}(Z(N))

Thus, the number of extensions for the trivial outer action of Q on N equals the number of elements in the second cohomology group for trivial group action H^2(Q;Z(N)) for the trivial group action. More explicitly, H^2(Q;Z(N)) acts on the set of extensions (possibly with repetitions) in a manner that is equivalent to the regular group action. However, the extension set does not have a natural choice of extension corresponding to the identity element.

H^2(Q;Z(N)) is the second cohomology group for trivial group action of Z2 on Z2, and is isomorphic to cyclic group:Z2. The extension set is thus a set of size two with this group acting on it.


Number of cohomology classes giving the extension Corresponding group extension for Q on N Second part of GAP ID (order is 16) Is the extension a semidirect product of N by Q? Is the base characteristic in the whole group? Nilpotency class of extension group Derived length of extension group Minimum size of generating set of whole group
1 semidihedral group:SD16 8 Yes No 3 2 2
1 generalized quaternion group:Q16 9 No Yes 3 2 2