Tetrahedral symmetry

The set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A4 of S4.

Seen in stereographic projection the edges of the tetrakis hexahedron form 6 circles (or centrally radial lines) in the plane.

Td, *332, [3,3] or 43m, of order 24 – achiral or full tetrahedral symmetry, also known as the (2,3,3) triangle group.

Td is the union of T and the set obtained by combining each element of O \ T with inversion.

[1] This group has the same rotation axes as T, with mirror planes through two of the orthogonal directions.

The symmetries correspond to the even permutations of the body diagonals and the same combined with inversion.

It is also the symmetry of a pyritohedron, which is extremely similar to the cube described, with each rectangle replaced by a pentagon with one symmetry axis and 4 equal sides and 1 different side (the one corresponding to the line segment dividing the cube's face); i.e., the cube's faces bulge out at the dividing line and become narrower there.