Definition:Semidirect Product

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

Let $\Aut(N)$ denote the automorphism group of $N$.

Let $\phi : H\to \Aut(N)$ 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:
 * $(n_1, h_1) \circ (n_2, h_2) = (n_1\cdot \phi(h_1)(n_2), h_1\cdot h_2)$

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

Also see

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