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, Cn denotes an axis of rotation through 360°/n and Sn 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.
- Td (*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. Td is isomorphic to S4, the symmetric group on 4 letters. See also the isometries of the regular tetrahedron.
- Th (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. Th 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.
- Oh (*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 Td and Th. 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 Ih. 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).
- Ih (*532, 532/m, 53m) of order 120 – full icosahedral symmetry; the symmetry group of the icosahedron and the dodecahedron. The group Ih is isomorphic to A5 × C2. The group contains 10 versions of D3d and 6 versions of D5d (symmetries like antiprisms).
The continuous groups related to these groups are:
- K or SO(3), all possible rotations.
- Kh or O(3), all possible rotations and reflections.
Read more about this topic: Point Groups In Three Dimensions
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