External central product

Definition

Suppose $A$ and $B$ are groups. Suppose we identify a central subgroup $C$ of $A$ with a central subgroup $D$ of $B$ via an isomorphism of groups $\varphi :C\to D$. The external central product of $A$ and $B$ with respect to $\varphi$ is the quotient of the external direct product $A\times B$ by the subgroup $\{g,\varphi (g)^{-1}\mid g\in C\}$.

In particular, this is a group $G$ that has normal subgroups $A_{1}$ and $B_{1}$isomorphic to $A$ and $B$ respectively, such that $A_{1}B_{1}=G$, $A_{1}$ and $B_{1}$ centralize each other, and $A_{1}\cap B_{1}$ is like $C\leq A$ when viewed as a subgroup of $A_{1}$ and like $D\leq B$ when viewed as a subgroup of $B_{1}$. This is basically the definition of internal central product.