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 subgroup of order two.

It is given by the presentation:

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

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

Arithmetic functions

Function	Value	Explanation
order	16
exponent	4	Cyclic subgroup of order four.
nilpotency class	2
derived length	2
Frattini length	2
Fitting length	1
minimum size of generating set	3
subgroup rank	2	All proper subgroups are cyclic, dihedral, or Klein four-groups.
max-length	4
rank as p-group	2	There exist Klein four-subgroups.
normal rank	2
characteristic rank of a p-group	1

Group properties

Property	Satisfied	Explanation	Comment
Abelian group	No	$a$ and $x$ don't commute
Nilpotent group	Yes	Prime power order implies nilpotent
Metacyclic group	No
Supersolvable group	Yes
Solvable group	Yes	Nilpotent implies solvable
T-group	No	$\langle a^{4},x\rangle \triangleleft \langle a^{2},x\rangle$, which is normal, but $\langle a^{4},x\rangle$ is not normal
Monolithic group	Yes	Unique minimal normal subgroup of order two
One-headed group	No	Seven distinct maximal normal subgroups of order eight
Directly indecomposable group	Yes
Centrally indecomposable group	No
Splitting-simple group	No

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

This finite group has order 16 and has ID 13 among the groups of order 16 in GAP's SmallGroup library. For context, there are groups of order 16. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(16,13)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(16,13);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [16,13]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.