Reference for central product I am reading central product of groups from text Group Theory I by M Suzuki. But I am neither able to understand nor does getting a feel on what is happening. I will be thankful to you if you can tell me some easy reference regarding the central product. I will be happy to see some computations that is how to construct central product of two groups?
 A: Let $G$ and $H$ be groups, and let $\phi$ be an isomorphism from a subgroup $Z$ of $Z(G)$ to a subgroup of $Z(H)$. Then we can define the central product of $G$ and $H$ with respect to $\phi$ as the quotient group
$G \circ_\phi H = (G \times H)/\{(z,\phi(z^{-1})) : z \in Z \}.$
The way to think of it is being the direct product, but with the elements $z$ and $\phi(z)$ identified for $z \in Z$.
With this general definition, the direct product itself is a special case with $Z=1$. But usually, the isomorphism is from $Z(G)$ to $Z(H)$, and the map $\phi$ is often left unspecified, particularly when all isomorphisms $Z(G) \to Z(H)$ define isomorphic central products.
As an example, let $G$ and $H$ be nonabelian groups of order 8, i.e. $Q_8$ or $D_8$. Then $|Z(G)|=|Z(H)|=2$, and their central product has order $8^2/2 = 32$ with centre of order 2. You can repeat this process, by taking central products of $n$ copies of $D_8$ or $Q_8$, giving groups $E$ of order $2^{2n+1}$ with centre of order 2 and $E/Z(E)$ elementary abelian. These are the so-called extraspecial groups, and there are two isomorphism types.
