Supergroups of dihedral group:D8

This article discusses some of the supergroups of the dihedral group of order eight.

Note that unlike the discussion of the subgroup structure of dihedral group:D8, this discussion is necessarily not comprehensive, because there are infinitely many groups containing the dihedral group of order eight. However, we provide a comprehensive discussion of all the groups of order sixteen containing this as a subgroup.

Analysis of groups containing this as a subgroup of index two

The general procedure

We can use cohomology theory to begin this analysis. Specifically, we are interested in groups of order sixteen where the dihedral group of order eight is the normal subgroup and the quotient group is the cyclic group of order two. We denote by $N$ the dihedral group of order eight and $Q$ the quotient group, which is cyclic of order two.

The classification proceeds in three steps:

Determine the set of possible homomorphisms $Q \to O u t (N)$ . In this case, both $Q$ and $A u t (N)$ (the outer automorphism group of $N$ ) are cyclic of order two.
For each such homomorphism, determine whether an extension exists.
If an extension exists, classify the extensions using the second homomology group $H^{2} (Q, Z (N))$ corresponding to the induced action on $Z (N)$ from the homomorphism to $O u t (N)$ .

The classification

Since both $Q$ and $O u t (N)$ are cyclic of order two, there are only two possible homomorphisms from $Q$ to $O u t (N)$ : the trivial homomorphism and the unique isomorphism.

For the trivial homomorphism, there are two extensions. The number two comes because $H^{2} (Z / 2 Z, Z / 2 Z)$ has order two.

The two extensions are:

Direct product of D8 and Z2
Central product of D8 and Z4: This is a central product where we assume the subgroup of order two in cyclic group:Z4 to equal the center of $D_{8}$ . Note that although this is not a direct product involving the dihedral group, it is still a semidirect product involving the dihedral group. Specifically, there is an element of order two outside $D_{8}$ .

For the nontrivial homomorphism, the induced action on $Z (N)$ is still trivial, so we again get two possibilities, because $H^{2} (Z / 2 Z, Z / 2 Z)$ . These two possibilities are: