Cubic Groups
The cubic groups are the groups of symmetries of the platonic solids, T, Td, O, Oh , I and Ih.
On this page:
Cases treated elsewhere:
- T ≅ A4 (T is the group of rotations of the tetrahedron)
- Td ≅ O ≅ S4 (Td is the group of all symmetries of the tetrahedron)
- I ≅ A5 (I is the group of rotations of the icosahedron)
The final column marked "Mull" is the Mulliken symbol used in Physics and Chemistry.
Oh is the group of all symmetries of the cube
Oh has order 48 and is isomorphic to \(\mathbf{S}_4\times\mathbb{Z}_2^c\), where \(\mathbb{Z}_2^c\) is the centre of O(3), generated by `inversion' \(i=-I:\mathbf{x}\mapsto-\mathbf{x}\), and often denoted Ci
Notation for elements:
- Ck is a rotation of order k (C4 is a rotation by π/2; C2 is a rotation by π around a line through mid points of opposite edges)
- iCk is Ck composed with i.
| Oh | e | C4 | C42 | C3 | C2 | i | iC4 | iC42 | iC3 | iC2 | notes | Mull. |
| # | 1 | 6 | 3 | 8 | 6 | 1 | 6 | 3 | 8 | 6 | |Oh| = 48 | |
| A0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial rep | A1g |
| A1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | alternating rep | A1u |
| A2 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | = A1 ⊗ A3 | A2g |
| A3 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | = A1 ⊗ A2 | A2u |
| E1 | 2 | 0 | 2 | -1 | 0 | 2 | 0 | 2 | -1 | 0 | Eg | |
| E2 | 2 | 0 | 2 | -1 | 0 | -2 | 0 | -2 | 1 | 0 | = E1 ⊗ A1 | Eu |
| T1 | 3 | 1 | -1 | 0 | -1 | -3 | -1 | 1 | 0 | 1 | symmetry rep of cube | T1u |
| T2 | 3 | 1 | -1 | 0 | -1 | 3 | 1 | -1 | 0 | -1 | = T1 ⊗ A1 | T1g |
| T3 | 3 | -1 | -1 | 0 | 1 | -3 | 1 | 1 | 0 | -1 | = T1 ⊗ A2 | T2u |
| T4 | 3 | -1 | -1 | 0 | 1 | 3 | -1 | -1 | 0 | 1 | = T1 ⊗ A3 | T2g |
- All reps are absolutely irreducible — even over Q
- The permutation representation on the set of 8 vertices of the cube is A0 + A3 + T1 + T4 = (A0 + A3)⊗(A0 + T1)
- The permutation representation on the set of 6 vertices of the octahedron is A0 + E1 + T1
- The permutation representation on the set of 12 edges of either is A0 + E2 + T1 + T3 + T4
- The permutation representation on the set of 3 diagonals of the octahedron is A0 + E1
- The permutation representation on the set of 4 diagonals of the cube is A0 + T4
- The "orientation permutation" representation on the set of 6 faces of the cube is A1 +??
- The "orientation permutation" representation on the set of 8 faces of the octahedron is T1 + ??
- The cube contains 2 inscribed tetrahedra; the permutation rep of this set is A0 + A3
Ih is the group of all symmetries of the icosahedron
| Ih | e | C5 | C52 | C3 | C2 | i | iC5 | iC52 | iC3 | iC2 | notes | Mull. |
| # | 1 | 12 | 12 | 20 | 15 | 1 | 12 | 12 | 20 | 15 | |Ih|=120 | - |
| A0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial rep | A1g |
| A1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | alternating rep | A1u |
| T1 | 3 | γ+ | γ- | 0 | -1 | -3 | -γ+ | -γ- | 0 | 1 | Symetry of icosahedron | T1u |
| T2 | 3 | γ+ | γ- | 0 | -1 | 3 | γ+ | γ- | 0 | -1 | = T1 ⊗ A1 | T1g |
| T3 | 3 | γ- | γ+ | 0 | -1 | -3 | -γ- | -γ+ | 0 | 1 | T2u | |
| T4 | 3 | γ- | γ+ | 0 | -1 | 3 | γ- | γ+ | 0 | -1 | = T3 ⊗ A1 | T2g |
| G1 | 4 | -1 | -1 | 1 | 0 | -4 | 1 | 1 | -1 | 0 | Gu | |
| G2 | 4 | -1 | -1 | 1 | 0 | 4 | -1 | -1 | 1 | 0 | = G1 ⊗ A1 | Gg |
| H1 | 5 | 0 | 0 | -1 | 1 | -5 | 0 | 0 | 1 | -1 | Hu | |
| H2 | 5 | 0 | 0 | -1 | 1 | 5 | 0 | 0 | -1 | 1 | = H1 ⊗ A1 | Hg |
- γ+ = 2cos(π/5) = ½(1+√5) (=golden ratio), and γ- = -2cos(2π/5) = ½(1-√5) ( = -(γ+)-1)
- All reps are absolutely irreducible.
- As a subgroup of O(3), Ih contains the central element i:x → -x (inversion in the origin).
So showing that Ih = I x Ci = A5 x Ci.