Tetrahedral symmetry

Selected point groups in three dimensions

Involutional symmetry Cs, (*) [ ] =	Cyclic symmetry Cnv, (*nn) [n] =	Dihedral symmetry Dnh, (*n22) [n,2] =
Polyhedral group, [n,3], (*n32)
Tetrahedral symmetry Td, (*332) [3,3] =	Octahedral symmetry Oh, (*432) [4,3] =	Icosahedral symmetry Ih, (*532) [5,3] =

A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.

The group of all (not necessarily orientation preserving) symmetries is isomorphic to the group S₄, the symmetric group of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A₄ of S₄.