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

We are interested in the subgroup: $H := \langle y \rangle = \{ e, y, y^2, y^3 \} = \{ e, y, a^2, a^2y \}$

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

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]

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] fully invariant subgroup(G,H); using fully invariant subgroup