Character Tables for Common Symmetry Groups - The Dnh Groups
| D2h | E | C2(z) | C2(y) | C2(x) | i | σ(xy) | σ(xz) | σ(yz) | | |
| Ag | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | | x2, y2, z2 |
| B1g | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | Rz | xy |
| B2g | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | Ry | xz |
| B3g | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | Rx | yz |
| Au | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | | |
| B1u | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | z | |
| B2u | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | y | |
| B3u | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | x | |
| D3h | E | 2C3 | 3C2 | σh | 2S3 | 3σv | | |
| A1' | 1 | 1 | 1 | 1 | 1 | 1 | | x2 + y2, z2 |
| A2' | 1 | 1 | -1 | 1 | 1 | -1 | Rz | |
| E | 2 | -1 | 0 | 2 | -1 | 0 | (x, y) | (x2 - y2, xy) |
| A1" | 1 | 1 | 1 | -1 | -1 | -1 | | |
| A2" | 1 | 1 | -1 | -1 | -1 | 1 | z | |
| E' | 2 | -1 | 0 | -2 | 1 | 0 | (Rx, Ry) | (xz, yz) |
| D4h | E | 2C4 | C2 | 2C2' | 2C2" | i | 2S4 | σh | 2σv | 2σd | |
| 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 | Rz | |
| B1g | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | | x2 - y2 |
| B2g | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | | xy |
| Eg | 2 | 0 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | (Rx, Ry) | (xz, yz) |
| A1u | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | | |
| A2u | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | z | |
| B1u | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | | |
| B2u | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | | |
| Eu | 2 | 0 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | (x, y) | |
| D5h | E | 2C5 | 2C52 | 5C2 | σh | 2S5 | 2S53 | 5σv | | |
| A1' | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | | x2 + y2, z2 |
| A2' | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | Rz | |
| E1 | 2 | 2cos72° | 2cos144° | 0 | 2 | 2cos72° | 2cos144° | 0 | (x, y) | |
| E2 | 2 | 2cos144° | 2cos72° | 0 | 2 | 2cos144° | 2cos72° | 0 | | (x2 - y2, xy) |
| A1" | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | | |
| A2" | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | z | |
| E1" | 2 | 2cos72° | 2cos144° | 0 | -2 | -2cos72° | -2cos144° | 0 | (Rx, Ry) | (xz, yz) |
| E2" | 2 | 2cos144° | 2cos72° | 0 | -2 | -2cos144° | -2cos72° | 0 | | |
| D6h | E | 2C6 | 2C3 | C2 | 3C2' | 3C2' | i | 2S3 | 2S6 | σh | 3σd | 3σv | | |
| A1g | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | | x2 + y2, z2 |
| A2g | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | Rz | |
| B1g | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | | |
| B2g | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | | |
| E1g | 2 | 1 | -1 | -2 | 0 | 0 | 2 | 1 | -1 | -2 | 0 | 0 | (Rx, Ry) | (xz, yz) |
| E2g | 2 | -1 | -1 | 2 | 0 | 0 | 2 | -1 | -1 | 2 | 0 | 0 | | (x2 - y2, xy) |
| A1u | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | | |
| A2u | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | z | |
| B1u | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | | |
| B2u | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | | |
| E1u | 2 | 1 | -1 | -2 | 0 | 0 | -2 | -1 | 1 | 2 | 0 | 0 | (x, y) | |
| E2u | 2 | -1 | -1 | 2 | 0 | 0 | -2 | 1 | 1 | -2 | 0 | 0 | | |
Reference: F.A. Cotton, "Chemical Applications of Group Theory," 3rd ed., Wiley-Interscience, NY, 1990.
