Tetrahedral Symmetry - Pyritohedral Symmetry

T_h, 3*2, or m3, of order 24 - pyritohedral symmetry. This group has the same rotation axes as T, with mirror planes through two of the orthogonal directions. The 3-fold axes are now S₆ axes, and there is inversion symmetry. T_h is isomorphic to T × Z₂: every element of T_h is either an element of T, or one combined with inversion. Apart from these two normal subgroups, there is also a normal subgroup D_2h (that of a cuboid), of type Dih₂ × Z₂ = Z₂ × Z₂ × Z₂ . It is the direct product of the normal subgroup of T (see above) with C_i. The quotient group is the same as above: of type Z₃. The three elements of the latter are the identity, "clockwise rotation", and "anti-clockwise rotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation.

It is the symmetry of a cube with on each face a line segment dividing the face into two equal rectangles, such that the line segments of adjacent faces do not meet at the edge. 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. It is a subgroup of the full icosahedral symmetry group (as isometry group, not just as abstract group), with 4 of the 10 3-fold axes.

The conjugacy classes of T_h include those of T, with the two classes of 4 combined, and each with inversion:

• identity
• 8 × rotation by 120°
• 3 × rotation by 180°
• inversion
• 8 × rotoreflection by 60°
• 3 × reflection in a plane