Extensions for trivial outer action of Z2 on Q8

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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 trivial.

Description in terms of cohomology groups

We have the induced outer action which is trivial:

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 extensions for the trivial outer action of Q on N correspond to the elements of the second cohomology group for trivial group action:

\! H^2(Q;Z(N))

The correspondence is as follows: an element of H^2(Q;Z(N)) gives an extension with base Z(N) and quotient Q. We take the central product of this extension group with N, identifying the common Z(N).

See second cohomology group for trivial group action of Z2 on Z2, which is isomorphic to cyclic group:Z2.

Extensions

Cohomology class type Number of cohomology classes Corresponding group extension for Q on Z(N) Second part of GAP ID (order is 4) Corresponding group extension for Q on N (obtained by taking the central product with N of the extension of Q on Z(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 semidirect product? Nilpotency class of whole group Derived length of whole group
trivial 1 Klein four-group 2 direct product of Q8 and Z2 12 Yes No 2 2
nontrivial 1 cyclic group:Z4 1 central product of D8 and Z4 13 Yes No 2 2