Center of central product of D8 and Z4

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.