Let $G_1,G_2, H$ three groups and two epimorphisms $\mu_i\colon G_i\to H$. Define \begin{equation} G_1\triangle G_2=\{(g_1,g_2)\in G_1\times G_2\mid \mu_1(g_1)=\mu_2(g_2)\}.\end{equation} Call $N_i=Ker(\mu_i)$ for $i=1,2$ and $R_1=\{(1_{N_1},n_2)\mid n_2\in N_2\}$ and $R_2=\{(n_1,1_{N_2})\mid n_1\in N_1\}$. The group $G_1\triangle G_2$ has this properties
- $N_j\simeq R_i\unlhd G_1\triangle G_2$ for $i\neq j$
- $G_i\simeq G_1\triangle G_2/R_i$ for $i=1,2$
- $G_1\triangle G_2/R_1\times R_2\simeq H$
These groups are described in Satz I9.11 of Endliche Gruppen I, B. Huppert. The author calls them
das direkte produkte von $G_1,G_2$ mit vereinigter Faktorgruppe $H$
This resemble me a sort of dual concept of the amalgam product, even though the traslation would be "the direct product of $G_1,G_2$ with identified quotient". In my research, I ended up with these groups, that I've described only for completeness. I would be thankful to find some other source in literature, I don't know their standard name in english. Moreover, I can't find the latex symbol used to identify them. The latex symbol for the central products would be helpful too, because I could rotate it.