Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 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:Z2 and the group is (up to isomorphism) nontrivial semidirect product of Z4 and Z4 (see subgroup structure of nontrivial semidirect product of Z4 and Z4).
The subgroup is a normal subgroup and the quotient group is isomorphic to dihedral group:D8.
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Definition

Consider the group (here $e$ denotes the identity element):

$G := \langle x,y \mid x^4 = y^4 = e, yxy^{-1} = x^3 \rangle$.

This is a group of order 16 with elements:

$\! \{ e,x,x^2,x^3,y,xy,x^2y,x^3y,y^2,xy^2,x^2y^2,x^3y^2,y^3,xy^3,x^2y^3,x^3y^3 \}$

We are interested in the subgroup:

$\! H = \{ e, y^2 \}$

This is a subgroup generated by the square $y^2$ which is not a commutator (the only commutators are $e$ and $x^2$).

The quotient group $G/H$ is isomorphic to dihedral group:D8. We can see this by noting that the presentation of $G/H$ is the same as that of $G$ but now with the additional constraint that $y^2$ is the identity element. This is precisely the presentation of the dihedral group of order 8.

Cosets

The subgroup is a normal subgroup, so the left and right cosets coincide. The 8 cosets are:

$\! \{ e, y^2 \}, \{ x, xy^2 \}, \{ x^2, x^2y^2\}, \{ x^3, x^3y^2 \}, \{ y, y^3 \}, \{ xy, xy^3 \}, \{ x^2y, x^2y^3 \}, \{ x^3y, x^3y^3 \}$

Arithmetic functions

Function	Value	Explanation
order of whole group	16
order of subgroup	2
index	8
size of conjugacy class	1
number of conjugacy classes in automorphism class	1

Subgroup properties

Invariance under automorphisms and endomorphisms

Property	Meaning	Satisfied?	Explanation
normal subgroup	invariant under inner automorphisms	Yes	central implies normal
characteristic subgroup	invariant under all automorphisms	Yes	it is the only subgroup generated by a square that is not a commutator
fully invariant subgroup	invariant under all endomorphisms	No	The endomorphism $x \mapsto e, y \mapsto x$ sends this subgroup to $\{ e, x^2 \}$.
isomorph-characteristic subgroup	Every isomorphic subgroup is characteristic	Yes	The two other isomorphic subgroups are both characteristic, see central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4 and derived subgroup of nontrivial semidirect product of Z4 and Z4.
characteristic-isomorph-free subgroup	no other isomorphic characteristic subgroup	No	$\{ y^2, e \}$ and $\{ x^2y^2, e\}$ are isomorphic characteristic subgroups. Hence also not a isomorph-free subgroup, isomorph-containing subgroup, or normal-isomorph-free subgroup.
isomorph-normal subgroup	Every isomorphic subgroup is normal	Yes	Follows from being isomorph-characteristic.

Centrality and related properties

Property	Meaning	Satisfied?	Explanation
central subgroup	invariant under inner automorphisms	Yes	The center is $\langle x^2, y^2 \rangle$.
central factor	product with centralizer is whole group	Yes	central implies central factor

GAP implementation

The group and subgroup pair can be constructed as follows:

G := SmallGroup(16,4); H := Group(Difference(Set(List(G,x -> x^2)),Set(DerivedSubgroup(G))));

Here is the GAP display for this:

gap> G := SmallGroup(16,4); H := Group(Difference(Set(List(G,x -> x^2)),Set(DerivedSubgroup(G))));
<pc group of size 16 with 4 generators>
<pc group with 1 generators>

Here is GAP code to verify some of the assertions in this page:

gap> Order(G);
16
gap> Order(H);
2
gap> Index(G,H);
8
gap> StructureDescription(H);
"C2"
gap> StructureDescription(G/H);
"D8"
gap> IsNormal(G,H);
true
gap> IsCharacteristicSubgroup(G,H);
true
gap> IsFullinvariant(G,H);
false