A group G is an internal semi-direct product of a normal subgroup N and subgroup H if G = NH and N ∩ H = {e}. This is written G = N ⋊ H. In this case every g ε‎ G can be uniquely written g = nh, where n ε‎ N and h ε‎ H. Multiplication in the group is given by

Such an example is the dihedral group D_2n, where N is the subgroup of rotations and H is a subgroup generated by a reflection.

For h ε‎ H, the map φ‎_h: n ↦hnh^-1 is an automorphism of N and φ‎: h ↦ φ‎_h is a homomorphism from H to Aut(N). More generally, given two groups N and H and a homomorphism φ‎: H→Aut(N), the external semi-direct product N⋊_φ‎H can be formed by multiplying elements of the Cartesian product according to rule

The semi-direct product is a means of creating a larger group from two groups other than the direct product. If the map φ‎ is the constant map to the identity map of N, then the semi-direct product agrees with the direct product.