Babai-Beals filtration

Definition
Let $$G$$ be a finite group. The Babai-Beals filtration of $$G$$ is the following ascending chain of subgroups of $$G$$:

$$1 \le \operatorname{Sol}(G) \le \operatorname{Soc}^*(G) \le \operatorname{PKer}(G) \le G$$

where:


 * $$1$$ denotes the trivial subgroup of $$G$$.
 * $$\operatorname{Sol}(G)$$ denotes the defining ingredient::solvable radical of $$G$$.
 * $$\operatorname{Soc}^*(G)$$ denotes the defining ingredient::socle over solvable radical of $$G$$, i.e., $$\operatorname{Soc}^*(G)/\operatorname{Sol}(G) = \operatorname{Soc}(G/\operatorname{Sol}(G))$$.
 * $$\operatorname{PKer}(G)$$ denotes the defining ingredient::permutation kernel of $$G$$.

Here is what we can say about the successive quotients:


 * $$\operatorname{Sol}(G)$$ is a solvable group.
 * $$\operatorname{Soc}^*(G)/\operatorname{Sol}(G)$$ is a direct product of simple non-abelian groups (with some of them possibly isomorphic).
 * $$\operatorname{PKer}(G)/\operatorname{Soc}^*(G)$$ is isomorphic to a subgroup of the external direct product of the outer automorphism groups of some simple non-abelian groups (the same ones that appear in the quotient $$\operatorname{Soc}^*(G)/\operatorname{Sol}(G)$$, with the same repetitions). Hence, by Schreier's conjecture (which has been proved conditional to the classification of finite simple groups) it is a solvable group.
 * $$G/\operatorname{PKer}(G)$$ has order at most $$\log_{60}|G|$$, hence is relatively small compared to the size of $$G$$.