Wreath product (reviewed)

If $S\leq S_n$ and $H$ are groups, the wreath product of $H$ and $S$ is the semidirect product $$H \wr S = H^n \rtimes S$$ where $S$ acts on $H^n$ via $S\subseteq S_n$, permuting the $n$ copies of $H$.

If $S$ is a transitive subgroup of $S_n$, we say that the wreath product is transitive.