Past exam of the options II course of the University of Oxford / 2010 / B / B2 / a

With $N◃G$, the internal semidirect product conditions are $G=NH$ and $N\cap H=\{1\}$. Every element then has a unique expression $nh$. The extra data needed to reconstruct the group are the conjugation actionOn the set $N\times H$, the external semidirect product $N{⋊}_{\varphi }H$ has multiplicationThe map $(n,h)\mapsto nh$ identifies it with $G$. A direct product results when this action is trivial.

For example, $S_3=A_3⋊⟨(12)⟩\cong C_3⋊C_2$. The two subgroups intersect trivially and their orders multiply to six, so they generate $S_3$. Conjugation by the transposition inverts the three-cycle. This action is nontrivial, and $S_3$ is nonabelian whereas $C_3\times C_2$ is abelian, so this is not their direct product.