Semidirect products

The semidirect product N ⋊ H is built from a normal subgroup N and a complement H (so G = NH and N ∩ H = 1), with H acting on N by conjugation. When the action is trivial this is just the direct product N × H; a non-trivial action gives a genuinely twisted group.

The name N ⋊ H is used here only when it is unambiguous: among all finite groups, exactly one non-direct N ⋊ H exists. Then N ⋊ H names that single group. (In the search bar the ASCII form N:H works too.)

Examples:

• C₇ ⋊ C₉ — the non-abelian group of order 63 (the only non-trivial C₇ ⋊ C₉; the trivial action gives the cyclic C₆₃).
• (C₂ × C₂) ⋊ C₃ — the alternating group A₄.

Wikipedia: Semidirect product