Let $H$ and $N$ be groups and let $\phi: H\to \text{Aut}(N)$ be a group homomorphism from $H$ to the automorphism group of $N$. The semidirect product $H\ltimes N$ is a group on the Cartesian product $H\times N$ with operation $$ given by \((h_1,n_1)*(h_2,n_2) = (h_1h_2, n_1\phi(h_1)n_2).\) The identity element of $H\ltimes N$ is the ordered pair consisting of the identity elements of $H$ and $N$, respectively. The inverse with respect to $$ is given by \((h,n)^{-1} = (h^{-1},\phi(h^{-1})n^{-1}).\)
Wikidata ID: Q291126