Center of direct product of D8 and Z2

Definition
The group $$G$$ is direct product of D8 and Z2, given as follows:

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

The subgroup $$\langle a,x \rangle$$ is the first direct factor (dihedral group:D8) and the subgroup $$\langle y \rangle$$ is the second direct factor (cyclic group:Z2).

$$G$$ has 16 elements:

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

The subgroup we are interested in is:

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

$$H$$ is the center of $$G$$ and is isomorphic to the Klein four-group.

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

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

The quotient group is also isomorphic to the Klein four-group.