Perfect groups (awaiting review)

A group $G$ is perfect if it is equal to its own commutator subgroup $G'$. Equivalently, $G$ has no non-trivial cyclic (equivalently abelian, equivalently soluble) quotients. All non-abelian simple and quasisimple groups are perfect. A perfect group may have cyclic composition factors, for example, $\mathrm{SL}(2,5)=C_2.A_5$ is perfect.