C18h | E | C18 | C9 | C6 | (C9)2 | (C18)5 | C3 | (C18)7 | (C9)4 | C2 | (C9)5 | (C18)11 | (C3)2 | (C18)13 | (C9)7 | (C6)5 | (C9)8 | (C18)17 | i | (S9)5 | (S18)11 | (S3)5 | (S18)13 | (S9)7 | (S6)5 | (S9)17 | (S18)17 | σh | S18 | S9 | S6 | (S9)11 | (S18)5 | S3 | (S18)7 | (S9)13 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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 | 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 | 1 | -1 | 1 | -1 | 1 | -1 |
E1g* | 2 | 2cos(π/9) | 2cos(2π/9) | 1 | 2cos(4π/9) | -2cos(4π/9) | -1 | -2cos(2π/9) | -2cos(π/9) | -2 | -2cos(π/9) | -2cos(2π/9) | -1 | -2cos(4π/9) | 2cos(4π/9) | 1 | 2cos(2π/9) | 2cos(π/9) | 2 | 2cos(π/9) | 2cos(2π/9) | 1 | 2cos(4π/9) | -2cos(4π/9) | -1 | -2cos(2π/9) | -2cos(π/9) | -2 | -2cos(π/9) | -2cos(2π/9) | -1 | -2cos(4π/9) | 2cos(4π/9) | 1 | 2cos(2π/9) | 2cos(π/9) |
E2g* | 2 | 2cos(2π/9) | 2cos(4π/9) | -1 | -2cos(π/9) | -2cos(π/9) | -1 | 2cos(4π/9) | 2cos(2π/9) | 2 | 2cos(2π/9) | 2cos(4π/9) | -1 | -2cos(π/9) | -2cos(π/9) | -1 | 2cos(4π/9) | 2cos(2π/9) | 2 | 2cos(2π/9) | 2cos(4π/9) | -1 | -2cos(π/9) | -2cos(π/9) | -1 | 2cos(4π/9) | 2cos(2π/9) | 2 | 2cos(2π/9) | 2cos(4π/9) | -1 | -2cos(π/9) | -2cos(π/9) | -1 | 2cos(4π/9) | 2cos(2π/9) |
E3g* | 2 | 1 | -1 | -2 | -1 | 1 | 2 | 1 | -1 | -2 | -1 | 1 | 2 | 1 | -1 | -2 | -1 | 1 | 2 | 1 | -1 | -2 | -1 | 1 | 2 | 1 | -1 | -2 | -1 | 1 | 2 | 1 | -1 | -2 | -1 | 1 |
E4g* | 2 | 2cos(4π/9) | -2cos(π/9) | -1 | 2cos(2π/9) | 2cos(2π/9) | -1 | -2cos(π/9) | 2cos(4π/9) | 2 | 2cos(4π/9) | -2cos(π/9) | -1 | 2cos(2π/9) | 2cos(2π/9) | -1 | -2cos(π/9) | 2cos(4π/9) | 2 | 2cos(4π/9) | -2cos(π/9) | -1 | 2cos(2π/9) | 2cos(2π/9) | -1 | -2cos(π/9) | 2cos(4π/9) | 2 | 2cos(4π/9) | -2cos(π/9) | -1 | 2cos(2π/9) | 2cos(2π/9) | -1 | -2cos(π/9) | 2cos(4π/9) |
E5g* | 2 | -2cos(4π/9) | -2cos(π/9) | 1 | 2cos(2π/9) | -2cos(2π/9) | -1 | 2cos(π/9) | 2cos(4π/9) | -2 | 2cos(4π/9) | 2cos(π/9) | -1 | -2cos(2π/9) | 2cos(2π/9) | 1 | -2cos(π/9) | -2cos(4π/9) | 2 | -2cos(4π/9) | -2cos(π/9) | 1 | 2cos(2π/9) | -2cos(2π/9) | -1 | 2cos(π/9) | 2cos(4π/9) | -2 | 2cos(4π/9) | 2cos(π/9) | -1 | -2cos(2π/9) | 2cos(2π/9) | 1 | -2cos(π/9) | -2cos(4π/9) |
E6g* | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 |
E7g* | 2 | -2cos(2π/9) | 2cos(4π/9) | 1 | -2cos(π/9) | 2cos(π/9) | -1 | -2cos(4π/9) | 2cos(2π/9) | -2 | 2cos(2π/9) | -2cos(4π/9) | -1 | 2cos(π/9) | -2cos(π/9) | 1 | 2cos(4π/9) | -2cos(2π/9) | 2 | -2cos(2π/9) | 2cos(4π/9) | 1 | -2cos(π/9) | 2cos(π/9) | -1 | -2cos(4π/9) | 2cos(2π/9) | -2 | 2cos(2π/9) | -2cos(4π/9) | -1 | 2cos(π/9) | -2cos(π/9) | 1 | 2cos(4π/9) | -2cos(2π/9) |
E8g* | 2 | -2cos(π/9) | 2cos(2π/9) | -1 | 2cos(4π/9) | 2cos(4π/9) | -1 | 2cos(2π/9) | -2cos(π/9) | 2 | -2cos(π/9) | 2cos(2π/9) | -1 | 2cos(4π/9) | 2cos(4π/9) | -1 | 2cos(2π/9) | -2cos(π/9) | 2 | -2cos(π/9) | 2cos(2π/9) | -1 | 2cos(4π/9) | 2cos(4π/9) | -1 | 2cos(2π/9) | -2cos(π/9) | 2 | -2cos(π/9) | 2cos(2π/9) | -1 | 2cos(4π/9) | 2cos(4π/9) | -1 | 2cos(2π/9) | -2cos(π/9) |
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 | -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 | -1 | 1 | -1 | 1 | -1 | 1 |
E1u* | 2 | 2cos(π/9) | 2cos(2π/9) | 1 | 2cos(4π/9) | -2cos(4π/9) | -1 | -2cos(2π/9) | -2cos(π/9) | -2 | -2cos(π/9) | -2cos(2π/9) | -1 | -2cos(4π/9) | 2cos(4π/9) | 1 | 2cos(2π/9) | 2cos(π/9) | -2 | -2cos(π/9) | -2cos(2π/9) | -1 | -2cos(4π/9) | 2cos(4π/9) | 1 | 2cos(2π/9) | 2cos(π/9) | 2 | 2cos(π/9) | 2cos(2π/9) | 1 | 2cos(4π/9) | -2cos(4π/9) | -1 | -2cos(2π/9) | -2cos(π/9) |
E2u* | 2 | 2cos(2π/9) | 2cos(4π/9) | -1 | -2cos(π/9) | -2cos(π/9) | -1 | 2cos(4π/9) | 2cos(2π/9) | 2 | 2cos(2π/9) | 2cos(4π/9) | -1 | -2cos(π/9) | -2cos(π/9) | -1 | 2cos(4π/9) | 2cos(2π/9) | -2 | -2cos(2π/9) | -2cos(4π/9) | 1 | 2cos(π/9) | 2cos(π/9) | 1 | -2cos(4π/9) | -2cos(2π/9) | -2 | -2cos(2π/9) | -2cos(4π/9) | 1 | 2cos(π/9) | 2cos(π/9) | 1 | -2cos(4π/9) | -2cos(2π/9) |
E3u* | 2 | 1 | -1 | -2 | -1 | 1 | 2 | 1 | -1 | -2 | -1 | 1 | 2 | 1 | -1 | -2 | -1 | 1 | -2 | -1 | 1 | 2 | 1 | -1 | -2 | -1 | 1 | 2 | 1 | -1 | -2 | -1 | 1 | 2 | 1 | -1 |
E4u* | 2 | 2cos(4π/9) | -2cos(π/9) | -1 | 2cos(2π/9) | 2cos(2π/9) | -1 | -2cos(π/9) | 2cos(4π/9) | 2 | 2cos(4π/9) | -2cos(π/9) | -1 | 2cos(2π/9) | 2cos(2π/9) | -1 | -2cos(π/9) | 2cos(4π/9) | -2 | -2cos(4π/9) | 2cos(π/9) | 1 | -2cos(2π/9) | -2cos(2π/9) | 1 | 2cos(π/9) | -2cos(4π/9) | -2 | -2cos(4π/9) | 2cos(π/9) | 1 | -2cos(2π/9) | -2cos(2π/9) | 1 | 2cos(π/9) | -2cos(4π/9) |
E5u* | 2 | -2cos(4π/9) | -2cos(π/9) | 1 | 2cos(2π/9) | -2cos(2π/9) | -1 | 2cos(π/9) | 2cos(4π/9) | -2 | 2cos(4π/9) | 2cos(π/9) | -1 | -2cos(2π/9) | 2cos(2π/9) | 1 | -2cos(π/9) | -2cos(4π/9) | -2 | 2cos(4π/9) | 2cos(π/9) | -1 | -2cos(2π/9) | 2cos(2π/9) | 1 | -2cos(π/9) | -2cos(4π/9) | 2 | -2cos(4π/9) | -2cos(π/9) | 1 | 2cos(2π/9) | -2cos(2π/9) | -1 | 2cos(π/9) | 2cos(4π/9) |
E6u* | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | -2 | 1 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | -2 | 1 | 1 |
E7u* | 2 | -2cos(2π/9) | 2cos(4π/9) | 1 | -2cos(π/9) | 2cos(π/9) | -1 | -2cos(4π/9) | 2cos(2π/9) | -2 | 2cos(2π/9) | -2cos(4π/9) | -1 | 2cos(π/9) | -2cos(π/9) | 1 | 2cos(4π/9) | -2cos(2π/9) | -2 | 2cos(2π/9) | -2cos(4π/9) | -1 | 2cos(π/9) | -2cos(π/9) | 1 | 2cos(4π/9) | -2cos(2π/9) | 2 | -2cos(2π/9) | 2cos(4π/9) | 1 | -2cos(π/9) | 2cos(π/9) | -1 | -2cos(4π/9) | 2cos(2π/9) |
E8u* | 2 | -2cos(π/9) | 2cos(2π/9) | -1 | 2cos(4π/9) | 2cos(4π/9) | -1 | 2cos(2π/9) | -2cos(π/9) | 2 | -2cos(π/9) | 2cos(2π/9) | -1 | 2cos(4π/9) | 2cos(4π/9) | -1 | 2cos(2π/9) | -2cos(π/9) | -2 | 2cos(π/9) | -2cos(2π/9) | 1 | -2cos(4π/9) | -2cos(4π/9) | 1 | -2cos(2π/9) | 2cos(π/9) | -2 | 2cos(π/9) | -2cos(2π/9) | 1 | -2cos(4π/9) | -2cos(4π/9) | 1 | -2cos(2π/9) | 2cos(π/9) |
Number of symmetry elements | h = 36 |
Number of classes, irreps | n = 36 |
Number of real-valued irreducible representations | n = 20 |
Abelian group | yes |
Optical Isomerism (Chirality) | no |
Polar | no |
Parity | yes |