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 $A_1B_1 = G$, $A_1$ and $B_1$ centralize each other, and $A_1 \cap B_1$ is like $C \le A$ when viewed as a subgroup of $A_1$ and like $D \le B$ when viewed as a subgroup of $B_1$. This is basically the definition of internal central product.