I'm interested in a concrete example of an infinite metabelian quotient of the free product $C\_2 \* C\_2 \* C\_2$, where $C\_2$ is the cyclic group of order $2$. In particular, I would be interested in a homomorphism onto a wreath product of abelian groups. How would this look like on the standard generating set $\left\{a,b,c\right\}$ where $a,b,c$ are the generators of one copy of $C\_2$ each.


What are the infinite nilpotent quotients of $C\_2 \* C\_2 \* C\_2$?



Infinite metabelian/nilpotent quotients of $C_2 * C_2 * C_2$
Perhaps the easiest metabalian quotient of $C\_2\ast C\_2\ast C\_2$ is $C\_2\ast C\_2 \cong \mathbb{Z}\rtimes C\_2$ the infinite dihedral group.


