Character Tables for Common Symmetry Groups - The Cubic Groups
| T | E | 4C3 | 4C32 | 3C2 | ε = exp(2π/3) | |
|---|---|---|---|---|---|---|
| A | 1 | 1 | 1 | 1 | x2 + y2 + z2 | |
| E | { 1 ε ε+ 1 }
1 ε+ ε 1 | (2z2 - x2 - y2, x2 - y2) | ||||
| T | 3 | 0 | 0 | -1 | (Rx, Ry, Rz); (x, y, z) | (xy, xz, yz) |
| Td | E | 8C3 | 3C2 | 6S4 | 6σd | ||
|---|---|---|---|---|---|---|---|
| A1 | 1 | 1 | 1 | 1 | 1 | x2 + y2 + z2 | |
| A2 | 1 | 1 | 1 | -1 | -1 | ||
| E | 2 | -1 | 2 | 0 | 0 | (2z2 - x2 - y2, x2 - y2) | |
| T1 | 3 | 0 | -1 | 1 | -1 | (Rx, Ry, Rz) | |
| T2 | 3 | 0 | -1 | -1 | 1 | (x, y, z) | (xy, xz, yz) |
| Oh | E | 8C3 | 6C2 | 6C4 | 3C2(= C42) | i | 6S4 | 8S6 | 3σh | 6σd | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A1g | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | x2 + y2 + z2 | |
| A2g | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | ||
| Eg | 2 | -1 | 0 | 0 | 2 | 2 | 0 | -1 | 2 | 0 | (2z2 - x2y2, x2 - y2) | |
| T1g | 3 | 0 | -1 | 1 | -1 | 3 | 1 | 0 | -1 | -1 | (Rx, Ry, Rz) | |
| T2g | 3 | 0 | 1 | -1 | -1 | 3 | -1 | 0 | -1 | 1 | (xz, yz, xy) | |
| A1u | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | ||
| A2u | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | ||
| Eu | 2 | -1 | 0 | 0 | 2 | -2 | 0 | 1 | -2 | 0 | ||
| T1u | 3 | 0 | -1 | 1 | -1 | -3 | -1 | 0 | 1 | 1 | (x, y, z) | |
| T2u | 3 | 0 | 1 | -1 | -1 | -3 | 1 | 0 | 1 | -1 |
Reference: F.A. Cotton, "Chemical Applications of Group Theory," 3rd ed., Wiley-Interscience, NY, 1990.