Supergroups of Klein four-group

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This article gives specific information, namely, supergroups, about a particular group, namely: Klein four-group.
View supergroups of particular groups | View other specific information about Klein four-group

This article discusses possible supergroups of the Klein four-group.

Note that unlike the discussion of the subgroup structure of Klein four-group, thisdiscussion is necessarily not comprehensive, because there are infinitely many groups containing the Klein four-group as a subgroup.

General lists

Direct products

Each of the groups listed below arises as the external direct product of Klein four-group and some nontrivial group. In particular, each of these groups contains the Klein four-group as a direct factor -- and hence as both a normal subgroup and a quotient group.

Note that since order of direct product is product of orders, if the other group has order a, the direct product has order 4a.

Other factor of direct product Order Second part of GAP ID Value of direct product Order Second part of GAP ID Hall-Senior symbol (if applicable)
cyclic group:Z2 2 1 elementary abelian group:E8 8 5 (1)^3
cyclic group:Z3 3 1 direct product of Z6 and Z2 12 5
cyclic group:Z4 4 1 direct product of Z4 and V4 16 10 1^22
Klein four-group 4 2 elementary abelian group:E16 16 14 1^4
cyclic group:Z5 5 1 direct product of Z10 and Z2 20 5
symmetric group:S3 6 1 direct product of D12 and Z2 24 14
cyclic group:Z6 6 2 direct product of E8 and Z3 24 15

Classification of supergroups of order eight

The general procedure

Suppose G is a group of order eight containing a subgroup N of order four isomorphic to the Klein four-group. Note that since N is of index two, it is a normal subgroup. Further, the quotient group is isomorphic to Q, the cyclic group of order two.

The classification has two steps:

  • Find all homomorphisms from Q to \operatorname{Aut}(N).
  • For each such homomorphism, find all possible extensions. These are classified by the elements of H^2(Q,N) for the action.

The classification

\operatorname{Aut}(N) is isomorphic to the symmetric group of degree three. This has three subgroups of order two, all of which are conjugate. Thus, up to equivalence under automorphisms of N, there is only one subgroup of order two.

Thus, there are (up to equivalence under automorphisms) two homomorphisms from Q to \operatorname{Aut}(N): the trivial homomorphism and an isomorphism to one of the subgroups of order two.

For the trivial map, the cohomology group H^2(Q,N) has order two, and the two extensions are:

For the nontrivial map, the cohomology group H^2(Q,N) is trivial, and the unique extension corresponding to it is the dihedral group of order eight.