D6h Point Group
not Abelian, 12(16) irreducible representationsSubgroups of D6h point group: Cs, Ci, C2, C3, C6, D2, D3, D6, C2v, C3v, C6v, C2h, C3h, C6h, D2h, D3h, D3d, S6
Character table for D6h point group
| E | 2C6 | 2C3 | C2 | 3C'2 | 3C'2 | i | 2S3 | 2S6 | σh | 3σd | 3σv | Linear, rotations | Quadratic | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 |
Product table for D6h point group
| A1g | A2g | B1g | B2g | E1g | E2g | A1u | A2u | B1u | B2u | E1u | E2u | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A1g | A1g | A2g | B1g | B2g | E1g | E2g | A1u | A2u | B1u | B2u | E1u | E2u |
| A2g | A2g | A1g | B2g | B1g | E1g | E2g | A2u | A1u | B2u | B1u | E1u | E2u |
| B1g | B1g | B2g | A1g | A2g | E2g | E1g | B1u | B2u | A1u | A2u | E2u | E1u |
| B2g | B2g | B1g | A2g | A1g | E2g | E1g | B2u | B1u | A2u | A1u | E2u | E1u |
| E1g | E1g | E1g | E2g | E2g | A1g+A2g+E2g | B1g+B2g+E1g | E1u | E1u | E2u | E2u | A1u+A2u+E2u | B1u+B2u+E1u |
| E2g | E2g | E2g | E1g | E1g | B1g+B2g+E1g | A1g+A2g+E2g | E2u | E2u | E1u | E1u | B1u+B2u+E1u | A1u+A2u+E2u |
| A1u | A1u | A2u | B1u | B2u | E1u | E2u | A1g | A2g | B1g | B2g | E1g | E2g |
| A2u | A2u | A1u | B2u | B1u | E1u | E2u | A2g | A1g | B2g | B1g | E1g | E2g |
| B1u | B1u | B2u | A1u | A2u | E2u | E1u | B1g | B2g | A1g | A2g | E2g | E1g |
| B2u | B2u | B1u | A2u | A1u | E2u | E1u | B2g | B1g | A2g | A1g | E2g | E1g |
| E1u | E1u | E1u | E2u | E2u | A1u+A2u+E2u | B1u+B2u+E1u | E1g | E1g | E2g | E2g | A1g+A2g+E2g | B1g+B2g+E1g |
| E2u | E2u | E2u | E1u | E1u | B1u+B2u+E1u | A1u+A2u+E2u | E2g | E2g | E1g | E1g | B1g+B2g+E1g | A1g+A2g+E2g |
| C1 | Cs | Ci | ||
| C2 | C3 | C4 | C5 | C6 |
| C2v | C3v | C4v | C5v | C6v |
| C2h | C3h | C4h | C5h | C6h |
| D2 | D3 | D4 | D5 | D6 |
| D2h | D3h | D4h | D5h | D6h |
| D2d | D3d | D4d | D5d | D6d |
| S4 | S6 | S8 | S10 | |
| Td | Oh | Ih | ||
| C∞v | D∞h |
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