Class two product

Definition
The class two product of groups is defined as their defining ingredient::varietal product corresponding to the variety of groups of nilpotency class two. It is the class two case of the defining ingredient::nilpotent product.

For two groups
Suppose $$G_1$$ and $$G_2$$ are groups. Let $$G$$ be the external free product $$G_1 * G_2$$, and identify the groups $$G_1$$ and $$G_2$$ with their respective images in $$G$$. The class two product of $$G_1$$ and $$G_2$$ can be defined as:

$$G/[[G_1,G_2],G]$$

For two groups
The subgroup $$[G_1,G_2]$$ in the class two product is isomorphic to the exterior product of $$G_1$$ and $$G_2$$ viewed as subgroups inside $$G_1 \times G_2$$, which in turn is isomorphic to the exterior product of their abelianizations. The quotient group by $$[G_1,G_2]$$ is isomorphic to the external direct product $$G_1 \times G_2$$.