Three-step group

Definition
A finite group $$G$$ is said to be a three-step group if it satisfies all the following conditions:


 * The commutator subgroup $$G'$$ has a permutable complement $$Q$$ which is a cyclic Hall subgroup
 * $$G'' \le F(G) \le G'$$ where $$F(G)$$ is the Fitting subgroup of $$G$$
 * If $$H$$ is a largest Hall subgroup of $$G$$ contained in $$F(G)$$, then $$H$$ is non-cyclic and $$F(G) = HC_G(H)$$
 * $$H$$ contains a cyclic subgroup $$R$$ such that $$C_{G'}(y) = R$$ for all non-identity elements $$y$$ of $$H$$.