A regular tetrahedron has a proper rotational symmetry group of order 12, isomorphic to the alternating group acting on its four vertices. Its full spatial symmetry group has order 24 and is isomorphic to the symmetric group ; the additional operations reverse orientation. Three-fold vertex/face-axis rotations and half-turns about opposite-edge midpoints generate the proper group. The distinction matters when a field symmetry uses only proper rotations, but a scalar density also admits reflections.
New to topics? Read the docs here!