Center of central product of D8 and Z4

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) cyclic group:Z4 and the group is (up to isomorphism) central product of D8 and Z4 (see subgroup structure of central product of D8 and Z4).
The subgroup is a normal subgroup and the quotient group is isomorphic to Klein four-group.
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Definition

The group $G$ is central product of D8 and Z4:

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

The group has 16 elements:

$\!e,a,a^{2},a^{3},x,ax,a^{2}x,a^{3}x,y,ay,a^{2}y,a^{3}y,xy,axy,a^{2}xy,a^{3}xy$

We are interested in the subgroup:

$H:=\langle y\rangle =\{e,y,y^{2},y^{3}\}=\{e,y,a^{2},a^{2}y\}$

This subgroup is isomorphic to cyclic group:Z4 and is the center of $G$.

Cosets

$H$ is a normal subgroup of $G$ and has four cosets:

$\{e,y,a^{2},a^{2}y\},\{a,ay,a^{3},a^{3}y\},\{x,xy,a^{2}x,a^{2}xy\},\{ax,axy,a^{3}x,a^{3}xy\}$

The quotient group is isomorphic to a Klein four-group.

Subgroup-defining functions

Subgroup-defining function	Meaning in general	Why it takes this value
center	elements that commute with every element	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.

Subgroup properties

Invariance under automorphisms and endomorphisms: basic properties

Property	Meaning	Satisfied?	Explanation	GAP verification (set G := SmallGroup(16,13);H := Center(G))
normal subgroup	invariant under inner automorphisms	Yes	center is normal	IsNormal(G,H); using IsNormal
characteristic subgroup	invariant under all automorphisms	Yes	center is characteristic	IsCharacteristicSubgroup(G,H); using IsCharacteristicSubgroup
fully invariant subgroup	invariant under all endomorphisms	No	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	fully invariant subgroup(G,H); using fully invariant subgroup