## Prove that the central product of $Z_4$ and $D_8$ is isomorphic to $Z_4$ and $Q_8$.

Let $$Y_1=\{(1,1),(x^2,r^2)\},\;Y_2=\{(1,1),(x^2,-1)\}$$, then prove $$Z_4\times D_8/Y_1\cong Z_4\times Q_8/Y_2$$.

Since both of them have order $$16$$, it’s not hard to list all of their elements, but is there any direct proof ?