A group $G$ is called self-replicating, if there exists a finite index subgroup $H$, such that $H\cong G\times\dots\times G$. Maybe the most famous example of a self-replicating group is a subgroup $K$ of index 16 in the Grigorchuk group. For this group a very explicit description is given in (Grigorchuk, Just infinite branch groups, in: New horizons in pro-$p$ groups): We have that $K$ contains a normal subgroup $K_1$ with $K/K_1\cong C_4$ and $K_1\cong K\times K$. The Frattini subgroup of $K$ is of level 2, that is, it contains the normal subgroup $K_2\cong K\times K\times K\times K$. Moreover, a set of generators and their action on the infinite binary tree is given.
Unfortunately, we have not been able to find a similarly detailed description for the other famous branch groups, in particular the Gupta-Sidki and the Fabrikowsky-Gupta groups (see: Bartholdi-Grigorchuk-Sunik, Branch groups, arXiv:math/0510294, section 1.6). More precisely, we are interested in the congruence level $\ell$ of the Frattini subgroup, and the finite group $K/K_\ell$, where $K_\ell$ is the $\ell$-th principal congruence subgroup. Computing these data right from the definitions would be a huge amount of work, so we would be most thankful for a pointer to the literature.
Thank you in advance
Jan-Christoph Schlage-Puchta
