Central product of D8 and Z4

Definition

The central product of the dihedral group of order eight and cyclic group of order four is a central product of these groups, over a common central subgrou)p of order two.

It is given by the presentation:

$G:=\langle a,x,y\mid a^{4}=x^{2}=e,xax=a^{-1},ay=ya,xy=yx\rangle$.

Here, $\langle a,x\rangle$ is the dihedral group of order eight and $\langle y\rangle$ is the cyclic group of order four.

Subgroups

Further information: Subgroup structure of central product of D8 and Z4

1. The trivial subgroup. Isomorphic to trivial group. (1)
2. The subgroup $\langle a^{2}\rangle$. This is the unique normal subgroup of order two, and is contained in the center. Isomorphic to cyclic group:Z2. (1)
3. The subgroups $\langle x\rangle$, $\langle ax\rangle$, $\langle a^{2}x\rangle$, $\langle a^{3}x\rangle$. These come in two conjugacy classes of 2-subnormal subgroups, one comprising $\langle x\rangle$ and $\langle a^{2}x\rangle$ and the other comprising $\langle ax\rangle$ and $\langle a^{3}x\rangle$. However, they are all automorphic subgroups. Isomorphic to cyclic group:Z2. (4)
4. The subgroups $\langle ay\rangle$ and $\langle a^{3}y\rangle$. These form a single conjugacy class of 2-subnormal subgroups. Isomorphic to cyclic group:Z2. (2)
5. The subgroup $\langle y\rangle$ of order four. This is the center. Isomorphic to cyclic group:Z4. (1)
6. The subgroups $\langle a\rangle$, $\langle xy\rangle$ and $\langle axy\rangle$. These are normal subgroups but are automorphic subgroups: they are related by outer automorphisms. Isomorphic to cyclic group:Z4. (3)
7. The subgroup $\langle a^{2},x\rangle$, $\langle a^{2},ax\rangle$ and $\langle a^{2},ay\rangle$. These are all normal subgroups but are related by outer automorphisms. Isomorphic to Klein four-group. (3)
8. The subgroup $\langle a,xy\rangle$. This is an isomorph-free subgroup of order eight, containing the three non-characteristic cyclic subgroups of order four. Isomorphic to quaternion group. (3)
9. The subgroups $\langle a,y\rangle$, $\langle x,y\rangle$ and $\langle ax,y\rangle$. These are all normal and related by outer automorphisms. Isomorphic to direct product of Z4 and Z2. (3)
10. The subgroups $\langle a,x\rangle$, $\langle xy,ay\rangle$ and $\langle axy,ay\rangle$. These are all normal and are related by outer automorphisms. Isomorphic to dihedral group:D8. (3)
11. The whole group. (1)

GAP implementation

Group ID

The ID of this group in GAP's list of groups of order sixteen is $13$. The group can thus be described using GAP's SmallGroup function as:

SmallGroup(16,13)