Point Groups in Three Dimensions - The Seven Remaining Point Groups

The Seven Remaining Point Groups

The remaining point groups are said to be of very high or polyhedral symmetry because they have more than one rotation axis of order greater than 2. Here, C_n denotes an axis of rotation through 360°/n and S_n denotes an axis of improper rotation through the same. In parentheses are the orbifold notation, Coxeter notation, the full Hermann–Mauguin notation, and the abbreviated one if different. The groups are:

• T (332, +, 23) of order 12 – chiral tetrahedral symmetry. There are four C₃ axes, each through two vertices of a cube (body diagonals) or one of a regular tetrahedron, and three C₂ axes, through the centers of the cube's faces, or the midpoints of the tetrahedron's edges. This group is isomorphic to A₄, the alternating group on 4 elements, and is the rotation group for a regular tetrahedron.

• T_d (*332, 43m) of order 24 – full tetrahedral symmetry. This group has the same rotation axes as T, but with six mirror planes, each containing two edges of the cube or one edge of the tetrahedron, a single C₂ axis and two C₃ axes. The C₂ axes are now actually S₄ axes. This group is the symmetry group for a regular tetrahedron. T_d is isomorphic to S₄, the symmetric group on 4 letters. See also the isometries of the regular tetrahedron.

• T_h (3*2, 2/m3, m3) of order 24 – pyritohedral symmetry. This group has the same rotation axes as T, with mirror planes parallel to the cube faces. The C₃ axes become S₆ axes, and there is inversion symmetry. T_h is isomorphic to A₄ × C₂. 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 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.

• O (432, +, 432) of order 24 – chiral octahedral symmetry. This group is like T, but the C₂ axes are now C₄ axes, and additionally there are 6 C₂ axes, through the midpoints of the edges of the cube. This group is also isomorphic to S₄, and is the rotation group of the cube and octahedron.

• O_h (*432, 4/m32/m, m3m) of order 48 – full octahedral symmetry. This group has the same rotation axes as O, but with mirror planes, comprising both the mirror planes of T_d and T_h. This group is isomorphic to S₄ × C₂, and is the symmetry group of the cube and octahedron. See also the isometries of the cube.

• I (532, +, 532) of order 60 – chiral icosahedral symmetry; the rotation group of the icosahedron and the dodecahedron. It is a normal subgroup of index 2 in the full group of symmetries I_h. The group I is isomorphic to A₅, the alternating group on 5 letters. The group contains 10 versions of D₃ and 6 versions of D₅ (rotational symmetries like prisms and antiprisms).

• I_h (*532, 532/m, 53m) of order 120 – full icosahedral symmetry; the symmetry group of the icosahedron and the dodecahedron. The group I_h is isomorphic to A₅ × C₂. The group contains 10 versions of D_3d and 6 versions of D_5d (symmetries like antiprisms).

The continuous groups related to these groups are:

• K or SO(3), all possible rotations.
• K_h or O(3), all possible rotations and reflections.