# Permutation kernel

Suppose $G$ is a finite group, $R = \operatorname{Rad}(G)$ is its solvable radical, and $S = \operatorname{Soc}^*(G)$ is its socle over solvable radical, i.e., $S/R$ is the socle of $G/R$. $S/R$ can be expressed uniquely as a direct product of simple non-abelian groups.
The permutation kernel of $G$, denoted $\operatorname{PKer}(G)$, is the kernel of the action of $G$ on these factors induced by the action of $G$ on $S/R$ by conjugation. $\operatorname{PKer}(G)$ contains the socle over solvable radical $S$.
The permutation kernel is part of the Babai-Beals filtration of $G$.