| C28h | E | 2C28 | 2C14 | 2(C28)3 | 2C7 | 2(C28)5 | 2(C14)3 | 2C4 | 2(C7)2 | 2(C28)9 | 2(C14)5 | 2(C28)11 | 2(C7)3 | 2(C28)13 | C2 | i | 2(S28)13 | 2(S7)3 | 2(S28)11 | 2(S14)5 | 2(S28)9 | 2(S7)9 | 2S4 | 2(S14)3 | 2(S28)5 | 2S7 | 2(S28)3 | 2S14 | 2S28 | σh |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ag | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| Bg | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 |
| E1g* | 2 | 2cos(π/14) | 2cos(π/7) | 2cos(3π/14) | 2cos(2π/7) | 2cos(5π/14) | 2cos(3π/7) | 0 | -2cos(3π/7) | -2cos(5π/14) | -2cos(2π/7) | -2cos(3π/14) | -2cos(π/7) | -2cos(π/14) | -2 | 2 | 2cos(π/14) | 2cos(π/7) | 2cos(3π/14) | 2cos(2π/7) | 2cos(5π/14) | 2cos(3π/7) | 0 | -2cos(3π/7) | -2cos(5π/14) | -2cos(2π/7) | -2cos(3π/14) | -2cos(π/7) | -2cos(π/14) | -2 |
| E2g* | 2 | 2cos(π/7) | 2cos(2π/7) | 2cos(3π/7) | -2cos(3π/7) | -2cos(2π/7) | -2cos(π/7) | -2 | -2cos(π/7) | -2cos(2π/7) | -2cos(3π/7) | 2cos(3π/7) | 2cos(2π/7) | 2cos(π/7) | 2 | 2 | 2cos(π/7) | 2cos(2π/7) | 2cos(3π/7) | -2cos(3π/7) | -2cos(2π/7) | -2cos(π/7) | -2 | -2cos(π/7) | -2cos(2π/7) | -2cos(3π/7) | 2cos(3π/7) | 2cos(2π/7) | 2cos(π/7) | 2 |
| E3g* | 2 | 2cos(3π/14) | 2cos(3π/7) | -2cos(5π/14) | -2cos(π/7) | -2cos(π/14) | -2cos(2π/7) | 0 | 2cos(2π/7) | 2cos(π/14) | 2cos(π/7) | 2cos(5π/14) | -2cos(3π/7) | -2cos(3π/14) | -2 | 2 | 2cos(3π/14) | 2cos(3π/7) | -2cos(5π/14) | -2cos(π/7) | -2cos(π/14) | -2cos(2π/7) | 0 | 2cos(2π/7) | 2cos(π/14) | 2cos(π/7) | 2cos(5π/14) | -2cos(3π/7) | -2cos(3π/14) | -2 |
| E4g* | 2 | 2cos(2π/7) | -2cos(3π/7) | -2cos(π/7) | -2cos(π/7) | -2cos(3π/7) | 2cos(2π/7) | 2 | 2cos(2π/7) | -2cos(3π/7) | -2cos(π/7) | -2cos(π/7) | -2cos(3π/7) | 2cos(2π/7) | 2 | 2 | 2cos(2π/7) | -2cos(3π/7) | -2cos(π/7) | -2cos(π/7) | -2cos(3π/7) | 2cos(2π/7) | 2 | 2cos(2π/7) | -2cos(3π/7) | -2cos(π/7) | -2cos(π/7) | -2cos(3π/7) | 2cos(2π/7) | 2 |
| E5g* | 2 | 2cos(5π/14) | -2cos(2π/7) | -2cos(π/14) | -2cos(3π/7) | 2cos(3π/14) | 2cos(π/7) | 0 | -2cos(π/7) | -2cos(3π/14) | 2cos(3π/7) | 2cos(π/14) | 2cos(2π/7) | -2cos(5π/14) | -2 | 2 | 2cos(5π/14) | -2cos(2π/7) | -2cos(π/14) | -2cos(3π/7) | 2cos(3π/14) | 2cos(π/7) | 0 | -2cos(π/7) | -2cos(3π/14) | 2cos(3π/7) | 2cos(π/14) | 2cos(2π/7) | -2cos(5π/14) | -2 |
| E6g* | 2 | 2cos(3π/7) | -2cos(π/7) | -2cos(2π/7) | 2cos(2π/7) | 2cos(π/7) | -2cos(3π/7) | -2 | -2cos(3π/7) | 2cos(π/7) | 2cos(2π/7) | -2cos(2π/7) | -2cos(π/7) | 2cos(3π/7) | 2 | 2 | 2cos(3π/7) | -2cos(π/7) | -2cos(2π/7) | 2cos(2π/7) | 2cos(π/7) | -2cos(3π/7) | -2 | -2cos(3π/7) | 2cos(π/7) | 2cos(2π/7) | -2cos(2π/7) | -2cos(π/7) | 2cos(3π/7) | 2 |
| E7g* | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 |
| E8g* | 2 | -2cos(3π/7) | -2cos(π/7) | 2cos(2π/7) | 2cos(2π/7) | -2cos(π/7) | -2cos(3π/7) | 2 | -2cos(3π/7) | -2cos(π/7) | 2cos(2π/7) | 2cos(2π/7) | -2cos(π/7) | -2cos(3π/7) | 2 | 2 | -2cos(3π/7) | -2cos(π/7) | 2cos(2π/7) | 2cos(2π/7) | -2cos(π/7) | -2cos(3π/7) | 2 | -2cos(3π/7) | -2cos(π/7) | 2cos(2π/7) | 2cos(2π/7) | -2cos(π/7) | -2cos(3π/7) | 2 |
| E9g* | 2 | -2cos(5π/14) | -2cos(2π/7) | 2cos(π/14) | -2cos(3π/7) | -2cos(3π/14) | 2cos(π/7) | 0 | -2cos(π/7) | 2cos(3π/14) | 2cos(3π/7) | -2cos(π/14) | 2cos(2π/7) | 2cos(5π/14) | -2 | 2 | -2cos(5π/14) | -2cos(2π/7) | 2cos(π/14) | -2cos(3π/7) | -2cos(3π/14) | 2cos(π/7) | 0 | -2cos(π/7) | 2cos(3π/14) | 2cos(3π/7) | -2cos(π/14) | 2cos(2π/7) | 2cos(5π/14) | -2 |
| E10g* | 2 | -2cos(2π/7) | -2cos(3π/7) | 2cos(π/7) | -2cos(π/7) | 2cos(3π/7) | 2cos(2π/7) | -2 | 2cos(2π/7) | 2cos(3π/7) | -2cos(π/7) | 2cos(π/7) | -2cos(3π/7) | -2cos(2π/7) | 2 | 2 | -2cos(2π/7) | -2cos(3π/7) | 2cos(π/7) | -2cos(π/7) | 2cos(3π/7) | 2cos(2π/7) | -2 | 2cos(2π/7) | 2cos(3π/7) | -2cos(π/7) | 2cos(π/7) | -2cos(3π/7) | -2cos(2π/7) | 2 |
| E11g* | 2 | -2cos(3π/14) | 2cos(3π/7) | 2cos(5π/14) | -2cos(π/7) | 2cos(π/14) | -2cos(2π/7) | 0 | 2cos(2π/7) | -2cos(π/14) | 2cos(π/7) | -2cos(5π/14) | -2cos(3π/7) | 2cos(3π/14) | -2 | 2 | -2cos(3π/14) | 2cos(3π/7) | 2cos(5π/14) | -2cos(π/7) | 2cos(π/14) | -2cos(2π/7) | 0 | 2cos(2π/7) | -2cos(π/14) | 2cos(π/7) | -2cos(5π/14) | -2cos(3π/7) | 2cos(3π/14) | -2 |
| E12g* | 2 | -2cos(π/7) | 2cos(2π/7) | -2cos(3π/7) | -2cos(3π/7) | 2cos(2π/7) | -2cos(π/7) | 2 | -2cos(π/7) | 2cos(2π/7) | -2cos(3π/7) | -2cos(3π/7) | 2cos(2π/7) | -2cos(π/7) | 2 | 2 | -2cos(π/7) | 2cos(2π/7) | -2cos(3π/7) | -2cos(3π/7) | 2cos(2π/7) | -2cos(π/7) | 2 | -2cos(π/7) | 2cos(2π/7) | -2cos(3π/7) | -2cos(3π/7) | 2cos(2π/7) | -2cos(π/7) | 2 |
| E13g* | 2 | -2cos(π/14) | 2cos(π/7) | -2cos(3π/14) | 2cos(2π/7) | -2cos(5π/14) | 2cos(3π/7) | 0 | -2cos(3π/7) | 2cos(5π/14) | -2cos(2π/7) | 2cos(3π/14) | -2cos(π/7) | 2cos(π/14) | -2 | 2 | -2cos(π/14) | 2cos(π/7) | -2cos(3π/14) | 2cos(2π/7) | -2cos(5π/14) | 2cos(3π/7) | 0 | -2cos(3π/7) | 2cos(5π/14) | -2cos(2π/7) | 2cos(3π/14) | -2cos(π/7) | 2cos(π/14) | -2 |
| Au | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
| Bu | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 |
| E1u* | 2 | 2cos(π/14) | 2cos(π/7) | 2cos(3π/14) | 2cos(2π/7) | 2cos(5π/14) | 2cos(3π/7) | 0 | -2cos(3π/7) | -2cos(5π/14) | -2cos(2π/7) | -2cos(3π/14) | -2cos(π/7) | -2cos(π/14) | -2 | -2 | -2cos(π/14) | -2cos(π/7) | -2cos(3π/14) | -2cos(2π/7) | -2cos(5π/14) | -2cos(3π/7) | 0 | 2cos(3π/7) | 2cos(5π/14) | 2cos(2π/7) | 2cos(3π/14) | 2cos(π/7) | 2cos(π/14) | 2 |
| E2u* | 2 | 2cos(π/7) | 2cos(2π/7) | 2cos(3π/7) | -2cos(3π/7) | -2cos(2π/7) | -2cos(π/7) | -2 | -2cos(π/7) | -2cos(2π/7) | -2cos(3π/7) | 2cos(3π/7) | 2cos(2π/7) | 2cos(π/7) | 2 | -2 | -2cos(π/7) | -2cos(2π/7) | -2cos(3π/7) | 2cos(3π/7) | 2cos(2π/7) | 2cos(π/7) | 2 | 2cos(π/7) | 2cos(2π/7) | 2cos(3π/7) | -2cos(3π/7) | -2cos(2π/7) | -2cos(π/7) | -2 |
| E3u* | 2 | 2cos(3π/14) | 2cos(3π/7) | -2cos(5π/14) | -2cos(π/7) | -2cos(π/14) | -2cos(2π/7) | 0 | 2cos(2π/7) | 2cos(π/14) | 2cos(π/7) | 2cos(5π/14) | -2cos(3π/7) | -2cos(3π/14) | -2 | -2 | -2cos(3π/14) | -2cos(3π/7) | 2cos(5π/14) | 2cos(π/7) | 2cos(π/14) | 2cos(2π/7) | 0 | -2cos(2π/7) | -2cos(π/14) | -2cos(π/7) | -2cos(5π/14) | 2cos(3π/7) | 2cos(3π/14) | 2 |
| E4u* | 2 | 2cos(2π/7) | -2cos(3π/7) | -2cos(π/7) | -2cos(π/7) | -2cos(3π/7) | 2cos(2π/7) | 2 | 2cos(2π/7) | -2cos(3π/7) | -2cos(π/7) | -2cos(π/7) | -2cos(3π/7) | 2cos(2π/7) | 2 | -2 | -2cos(2π/7) | 2cos(3π/7) | 2cos(π/7) | 2cos(π/7) | 2cos(3π/7) | -2cos(2π/7) | -2 | -2cos(2π/7) | 2cos(3π/7) | 2cos(π/7) | 2cos(π/7) | 2cos(3π/7) | -2cos(2π/7) | -2 |
| E5u* | 2 | 2cos(5π/14) | -2cos(2π/7) | -2cos(π/14) | -2cos(3π/7) | 2cos(3π/14) | 2cos(π/7) | 0 | -2cos(π/7) | -2cos(3π/14) | 2cos(3π/7) | 2cos(π/14) | 2cos(2π/7) | -2cos(5π/14) | -2 | -2 | -2cos(5π/14) | 2cos(2π/7) | 2cos(π/14) | 2cos(3π/7) | -2cos(3π/14) | -2cos(π/7) | 0 | 2cos(π/7) | 2cos(3π/14) | -2cos(3π/7) | -2cos(π/14) | -2cos(2π/7) | 2cos(5π/14) | 2 |
| E6u* | 2 | 2cos(3π/7) | -2cos(π/7) | -2cos(2π/7) | 2cos(2π/7) | 2cos(π/7) | -2cos(3π/7) | -2 | -2cos(3π/7) | 2cos(π/7) | 2cos(2π/7) | -2cos(2π/7) | -2cos(π/7) | 2cos(3π/7) | 2 | -2 | -2cos(3π/7) | 2cos(π/7) | 2cos(2π/7) | -2cos(2π/7) | -2cos(π/7) | 2cos(3π/7) | 2 | 2cos(3π/7) | -2cos(π/7) | -2cos(2π/7) | 2cos(2π/7) | 2cos(π/7) | -2cos(3π/7) | -2 |
| E7u* | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 |
| E8u* | 2 | -2cos(3π/7) | -2cos(π/7) | 2cos(2π/7) | 2cos(2π/7) | -2cos(π/7) | -2cos(3π/7) | 2 | -2cos(3π/7) | -2cos(π/7) | 2cos(2π/7) | 2cos(2π/7) | -2cos(π/7) | -2cos(3π/7) | 2 | -2 | 2cos(3π/7) | 2cos(π/7) | -2cos(2π/7) | -2cos(2π/7) | 2cos(π/7) | 2cos(3π/7) | -2 | 2cos(3π/7) | 2cos(π/7) | -2cos(2π/7) | -2cos(2π/7) | 2cos(π/7) | 2cos(3π/7) | -2 |
| E9u* | 2 | -2cos(5π/14) | -2cos(2π/7) | 2cos(π/14) | -2cos(3π/7) | -2cos(3π/14) | 2cos(π/7) | 0 | -2cos(π/7) | 2cos(3π/14) | 2cos(3π/7) | -2cos(π/14) | 2cos(2π/7) | 2cos(5π/14) | -2 | -2 | 2cos(5π/14) | 2cos(2π/7) | -2cos(π/14) | 2cos(3π/7) | 2cos(3π/14) | -2cos(π/7) | 0 | 2cos(π/7) | -2cos(3π/14) | -2cos(3π/7) | 2cos(π/14) | -2cos(2π/7) | -2cos(5π/14) | 2 |
| E10u* | 2 | -2cos(2π/7) | -2cos(3π/7) | 2cos(π/7) | -2cos(π/7) | 2cos(3π/7) | 2cos(2π/7) | -2 | 2cos(2π/7) | 2cos(3π/7) | -2cos(π/7) | 2cos(π/7) | -2cos(3π/7) | -2cos(2π/7) | 2 | -2 | 2cos(2π/7) | 2cos(3π/7) | -2cos(π/7) | 2cos(π/7) | -2cos(3π/7) | -2cos(2π/7) | 2 | -2cos(2π/7) | -2cos(3π/7) | 2cos(π/7) | -2cos(π/7) | 2cos(3π/7) | 2cos(2π/7) | -2 |
| E11u* | 2 | -2cos(3π/14) | 2cos(3π/7) | 2cos(5π/14) | -2cos(π/7) | 2cos(π/14) | -2cos(2π/7) | 0 | 2cos(2π/7) | -2cos(π/14) | 2cos(π/7) | -2cos(5π/14) | -2cos(3π/7) | 2cos(3π/14) | -2 | -2 | 2cos(3π/14) | -2cos(3π/7) | -2cos(5π/14) | 2cos(π/7) | -2cos(π/14) | 2cos(2π/7) | 0 | -2cos(2π/7) | 2cos(π/14) | -2cos(π/7) | 2cos(5π/14) | 2cos(3π/7) | -2cos(3π/14) | 2 |
| E12u* | 2 | -2cos(π/7) | 2cos(2π/7) | -2cos(3π/7) | -2cos(3π/7) | 2cos(2π/7) | -2cos(π/7) | 2 | -2cos(π/7) | 2cos(2π/7) | -2cos(3π/7) | -2cos(3π/7) | 2cos(2π/7) | -2cos(π/7) | 2 | -2 | 2cos(π/7) | -2cos(2π/7) | 2cos(3π/7) | 2cos(3π/7) | -2cos(2π/7) | 2cos(π/7) | -2 | 2cos(π/7) | -2cos(2π/7) | 2cos(3π/7) | 2cos(3π/7) | -2cos(2π/7) | 2cos(π/7) | -2 |
| E13u* | 2 | -2cos(π/14) | 2cos(π/7) | -2cos(3π/14) | 2cos(2π/7) | -2cos(5π/14) | 2cos(3π/7) | 0 | -2cos(3π/7) | 2cos(5π/14) | -2cos(2π/7) | 2cos(3π/14) | -2cos(π/7) | 2cos(π/14) | -2 | -2 | 2cos(π/14) | -2cos(π/7) | 2cos(3π/14) | -2cos(2π/7) | 2cos(5π/14) | -2cos(3π/7) | 0 | 2cos(3π/7) | -2cos(5π/14) | 2cos(2π/7) | -2cos(3π/14) | 2cos(π/7) | -2cos(π/14) | 2 |
| Number of symmetry elements | h = 56 |
| Number of classes, irreps | n = 56 |
| Number of real-valued irreducible representations | n = 30 |
| Abelian group | yes |
| Optical Isomerism (Chirality) | no |
| Polar | no |
| Parity | yes |