External central product
Suppose and are groups. Suppose we identify a central subgroup of with a central subgroup of via an isomorphism of groups . The external central product of and with respect to is the quotient of the external direct product by the subgroup .
In particular, this is a group that has normal subgroups and isomorphic to and respectively, such that , and centralize each other, and is like when viewed as a subgroup of and like when viewed as a subgroup of . This is basically the definition of internal central product.