Central Product/Examples/Q with Q

Example of Central Product
Let $G$ be the quaternion group $Q$ whose group presentation is:
 * $Q = \gen {x, y: x^4 = e_H, y^2 = x^2, x y = y x^{-1} }$

From Center of Quaternion Group, the center of $H$ is:
 * $\map Z H = \set {e_G, x^2}$

Let:
 * $Z = W = \set {e_G, x^2}$

Let $\theta: Z \to W$ be the mapping defined as:
 * $\map \theta g = \begin{cases} e_G & : g = e_G \\ x^2 & : g = x^2 \end{cases}$

Let $X$ be the set defined as:
 * $X = \set {\tuple {z, \map \theta z^{-1} }: z \in Z}$

The central product of $G$ and $H$ via $\theta$ has $19$ elements.