Extensions for trivial outer action of Z2 on D16

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This article describes all the group extensions corresponding to a particular outer action with normal subgroup dihedral group:D8 and quotient group cyclic group:Z2.

We consider here the group extensions where the base normal subgroup N is dihedral group:D16, 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))

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 Second part of GAP ID (order is 32) Is the extension a semidirect product of N by Q? Is the base characteristic in the semidirect product?
trivial 1 Klein four-group 2 direct product of D16 and Z2 39 Yes No
nontrivial 1 cyclic group:Z4 1 central product of D16 and Z4 42 Yes No