Definition:Semidirect Product

Definition
Let $H$ and $N$ be groups.

Let $\operatorname{Aut} \left({N}\right)$ denote the automorphism group of $N$.

Let $\phi: H \to \operatorname{Aut} \left({N}\right)$ be a group homomorphism, that is, let $H$ act on $N$.

The (outer) semidirect product of $N$ and $H$ with respect to $\phi$ is the cartesian product $N \times H$ with the group law defined as:
 * $\left({n_1, h_1}\right) \circ \left({n_2, h_2}\right) = \left({n_1 \cdot \phi \left({h_1}\right) \left({n_2}\right), h_1 \cdot h_2}\right)$

It is denoted $N \rtimes_\phi H$.

Also see

 * Semidirect Product of Groups is Group
 * Inverse of Element in Semidirect Product