| C6h | E | C6 | C3 | C2 | C3² | C6⁵ | i | S3⁵ | S6⁵ | σh | S6 | S3 | 线性 / 转动linear, rotations | 二次quadratic |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ag | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | Rz | x²+y², z² |
| Bg | 1 | −1 | 1 | −1 | 1 | −1 | 1 | −1 | 1 | −1 | 1 | −1 | ||
| E1g | 12 | ε1 | −ε*−1 | −1−2 | −ε−1 | ε*1 | 12 | ε1 | −ε*−1 | −1−2 | −ε−1 | ε*1 | (Rx, Ry) | (xz, yz) |
| ↳ 共轭conj. | 1 | ε* | −ε | −1 | −ε* | ε | 1 | ε* | −ε | −1 | −ε* | ε | ||
| E2g | 12 | −ε*−1 | −ε−1 | 12 | −ε*−1 | −ε−1 | 12 | −ε*−1 | −ε−1 | 12 | −ε*−1 | −ε−1 | (x²−y², xy) | |
| ↳ 共轭conj. | 1 | −ε | −ε* | 1 | −ε | −ε* | 1 | −ε | −ε* | 1 | −ε | −ε* | ||
| Au | 1 | 1 | 1 | 1 | 1 | 1 | −1 | −1 | −1 | −1 | −1 | −1 | z | |
| Bu | 1 | −1 | 1 | −1 | 1 | −1 | −1 | 1 | −1 | 1 | −1 | 1 | ||
| E1u | 12 | ε1 | −ε*−1 | −1−2 | −ε−1 | ε*1 | −1−2 | −ε−1 | ε*1 | 12 | ε1 | −ε*−1 | (x, y) | |
| ↳ 共轭conj. | 1 | ε* | −ε | −1 | −ε* | ε | −1 | −ε* | ε | 1 | ε* | −ε | ||
| E2u | 12 | −ε*−1 | −ε−1 | 12 | −ε*−1 | −ε−1 | −1−2 | ε*1 | ε1 | −1−2 | ε*1 | ε1 | ||
| ↳ 共轭conj. | 1 | −ε | −ε* | 1 | −ε | −ε* | −1 | ε | ε* | −1 | ε | ε* |
\(\varepsilon = \exp(2\pi i/6)\)，\(\varepsilon^{*}\) 为其复共轭。合并后每对复共轭不可约表示相加成一个二维实表示（物理上简并）。; \(\varepsilon^{*}\) is its complex conjugate. Merging adds each conjugate pair into one two-dimensional real representation (physically degenerate).
| 阶 hOrder h | 12 |
|---|---|
| 类数Classes | 12 |
| 所属族Family | Cnh |
| 代表分子Examples | 六羟基苯（平面构象）hexahydroxybenzene (planar conformer) |
约化可约表示Reduce a representation
输入可约表示在各类上的特征标，按约化公式 \(n_i = \frac{1}{h}\sum_R g_R\,\chi(R)\,\chi_i(R)^{*}\) 求出各不可约表示出现的次数。
Enter the characters of a reducible representation; the reduction formula gives how many times each irreducible representation occurs.