Center of nontrivial semidirect product of Z4 and Z4

Definition
Consider the group:

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

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:

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

This is the center. In particular, it is a normal subgroup isomorphic to the Klein four-group and the quotient group is also isomorphic to the Klein four-group.

The multiplication table for $$H$$ is given as follows:

Cosets
$$H$$ is a normal subgroup, so its left cosets coincide with its right cosets. There are four cosets, because the index of $$H$$ in $$G$$ is $$16/4 = 4$$. These are:

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

The quotient group is isomorphic to a Klein four-group, and the multiplication table is given as follows:

GAP implementation
The group and subgroup pair can be constructed using GAP's SmallGroup and Center functions:

G := SmallGroup(16,4); H := Center(G);

Here is the GAP display:

gap> G := SmallGroup(16,4); H := Center(G);  Group([ f3, f4 ])

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

gap> Order(G); 16 gap> Order(H); 4 gap> Index(G,H); 4 gap> StructureDescription(H); "C2 x C2" gap> StructureDescription(G/H); "C2 x C2" gap> H = Socle(G); true gap> H = Agemo(G,2,1); true gap> IsNormal(G,H); true gap> IsCharacteristicSubgroup(G,H); true gap> IsFullinvariant(G,H); true