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.

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
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

Subgroups

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

The trivial subgroup. Isomorphic to trivial group. (1)
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)
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)
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)
The subgroup $\langle y\rangle$ of order four. This is the center. Isomorphic to cyclic group:Z4. (1)
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)
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)
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)
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)
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)
The whole group. (1)