D4h Point Group
not Abelian, 10(12) irreducible representationsSubgroups of D4h point group: Cs, Ci, C2, C4, D2, D4, C2v, C4v, C2h, C4h, D2h, D2d, S4
Character table for D4h point group
| E | 2C4 (z) | C2 | 2C'2 | 2C''2 | i | 2S4 | σh | 2σv | 2σd | linears, rotations | quadratic | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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) |
Product table for D4h point group
| A1g | A2g | B1g | B2g | Eg | A1u | A2u | B1u | B2u | Eu | |
|---|---|---|---|---|---|---|---|---|---|---|
| A1g | A1g | A2g | B1g | B2g | Eg | A1u | A2u | B1u | B2u | Eu |
| A2g | A2g | A1g | B2g | B1g | Eg | A2u | A1u | B2u | B1u | Eu |
| B1g | B1g | B2g | A1g | A2g | Eg | B1u | B2u | A1u | A2u | Eu |
| B2g | B2g | B1g | A2g | A1g | Eg | B2u | B1u | A2u | A1u | Eu |
| Eg | Eg | Eg | Eg | Eg | A1g+A2g+B1g+B2g | Eu | Eu | Eu | Eu | A1u+A2u+B1u+B2u |
| A1u | A1u | A2u | B1u | B2u | Eu | A1g | A2g | B1g | B2g | Eg |
| A2u | A2u | A1u | B2u | B1u | Eu | A2g | A1g | B2g | B1g | Eg |
| B1u | B1u | B2u | A1u | A2u | Eu | B1g | B2g | A1g | A2g | Eg |
| B2u | B2u | B1u | A2u | A1u | Eu | B2g | B1g | A2g | A1g | Eg |
| Eu | Eu | Eu | Eu | Eu | A1u+A2u+B1u+B2u | Eg | Eg | Eg | Eg | A1g+A2g+B1g+B2g |
| 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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