Supergroups of Klein four-group

This article gives specific information, namely, supergroups, about a particular group, namely: Klein four-group.
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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	$1^{2}2$
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:

• elementary abelian group of order eight: This corresponds to the identity element of $H^{2}(Q,N)$.
• direct product of Z4 and Z2: This corresponds to the non-identity element of $H^{2}(Q,N)$.

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.