Given two groups$G_1$ and $G_2$ and a joint subgroup$C \xhookrightarrow{\iota_i} Z(G_i)$ in each of their centers, then the corresponding (“external”) central product is the quotient group

Beware that there is no widely accepted conventnotation for the central product, and that most notational conventions supporess the choices of central subgroups involved. The “$\circ$”-notation is popular in finite group-theory, while in Riemannian geometry people tend to use “$\cdot$” (see Sp(n).Sp(1)) or no symbol at all.

Also beware that most texts insists on stating the choices as that of two separate subgroups $C_i \xhookrightarrow{\iota_i} G_i$ together with an isomorphism$C_1 \xrightarrow[\simeq]{\phi} C_2$ between them.