If $G$ is a $2$-group with cyclic center having order of $Z(G)=2^n$ and index of $|G/Z(G)|=4$, then how many groups of a particular order are possible?
I know that there are two possibilities for such a group:
- $G = \langle x,y,s \mid x^2 = y^2 = 1\rangle$ with $x,y \in G$ and $s$ is generator of center,
- $G = \langle x,y,s \mid x^2 = y^2 = s\rangle$, where $Z(G) = \langle s \rangle$ and $|G/Z(G)|=4$.
Further I know some properties of these groups:
- $x^2 \in Z(G)$ for every $x \in G$,
- $xy = yx$ or $xy = eyx$ for $e \in Z(G)$.
How to prove that these are the only two possibilities?