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.
Th is the group of symmetries of a `decorated cube'
The 'decorated cube' is as follows. Take a cube and draw a single new line across each face that divides the square face into two equal rectangles - and this should be done in such a way that each edge of the cube only meets one of these lines (so each edge meets exactly 3 of the rectangles).
Th has order 24 and is isomorphic to \(\mathbf{A}_4\times\mathbb{Z}_2^c\), 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. The representations are therefore obtained from those of A4 tensored with those of \(\mathbb{Z}_2\).
| Th | e | C2C2 | C3 | C32 | i | iC2C2 | iC3 | iC32 | notes |
| # | 1 | 3 | 4 | 4 | 1 | 3 | 4 | 4 | |Th|=24 |
| A0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial rep |
| A1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | alternating rep |
| E1 | 2 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | |
| E2 | 2 | 2 | -1 | -1 | -2 | -2 | 1 | 1 | A1 ⊗ E1 |
| T1 | 3 | -1 | 0 | 0 | 3 | -1 | 0 | 0 | A1 ⊗ T2 |
| T2 | 3 | -1 | 0 | 0 | -3 | 1 | 0 | 0 | natural rep on cube |
- Th is the symmetry group of a volleyball with its markings, and T2 is the representation in question.
- T1 is the representation where i acts trivially, so factors through the rotation group T.
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 4 oriented diagonals of the cube is A3 + T1
- 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 | \(\varphi^+\) | \(\varphi^-\) | 0 | -1 | -3 | -\(\varphi^+\) | -\(\varphi^-\) | 0 | 1 | Symetry of icosahedron | T1u |
| T2 | 3 | \(\varphi^+\) | \(\varphi^-\) | 0 | -1 | 3 | \(\varphi^+\) | \(\varphi^-\) | 0 | -1 | = T1 ⊗ A1 | T1g |
| T3 | 3 | \(\varphi^-\) | \(\varphi^+\) | 0 | -1 | -3 | -\(\varphi^-\) | -\(\varphi^+\) | 0 | 1 | T2u | |
| T4 | 3 | \(\varphi^-\) | \(\varphi^+\) | 0 | -1 | 3 | \(\varphi^-\) | \(\varphi^+\) | 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 |
- \(\varphi^+\) = 2cos(π/5) = ½(1+√5) (=golden ratio), and \(\varphi^-\) = -2cos(2π/5) = ½(1-√5) ( = -(\(\varphi^+\))-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.