External central product

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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_1isomorphic 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.