C20h | E | 2C20 | 2C10 | 2(C20)3 | 2C5 | 2C4 | 2(C10)3 | 2(C20)7 | 2(C5)2 | 2(C20)9 | C2 | i | 2(S20)9 | 2(S5)7 | 2(S20)7 | 2(S10)3 | 2S4 | 2S5 | 2(S20)3 | 2S10 | 2S20 | σh |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Ag | 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 |
E1g* | 2 | 2cos(π/10) | 2cos(π/5) | 2cos(3π/10) | 2cos(2π/5) | 0 | -2cos(2π/5) | -2cos(3π/10) | -2cos(π/5) | -2cos(π/10) | -2 | 2 | 2cos(π/10) | 2cos(π/5) | 2cos(3π/10) | 2cos(2π/5) | 0 | -2cos(2π/5) | -2cos(3π/10) | -2cos(π/5) | -2cos(π/10) | -2 |
E2g* | 2 | 2cos(π/5) | 2cos(2π/5) | -2cos(2π/5) | -2cos(π/5) | -2 | -2cos(π/5) | -2cos(2π/5) | 2cos(2π/5) | 2cos(π/5) | 2 | 2 | 2cos(π/5) | 2cos(2π/5) | -2cos(2π/5) | -2cos(π/5) | -2 | -2cos(π/5) | -2cos(2π/5) | 2cos(2π/5) | 2cos(π/5) | 2 |
E3g* | 2 | 2cos(3π/10) | -2cos(2π/5) | -2cos(π/10) | -2cos(π/5) | 0 | 2cos(π/5) | 2cos(π/10) | 2cos(2π/5) | -2cos(3π/10) | -2 | 2 | 2cos(3π/10) | -2cos(2π/5) | -2cos(π/10) | -2cos(π/5) | 0 | 2cos(π/5) | 2cos(π/10) | 2cos(2π/5) | -2cos(3π/10) | -2 |
E4g* | 2 | 2cos(2π/5) | -2cos(π/5) | -2cos(π/5) | 2cos(2π/5) | 2 | 2cos(2π/5) | -2cos(π/5) | -2cos(π/5) | 2cos(2π/5) | 2 | 2 | 2cos(2π/5) | -2cos(π/5) | -2cos(π/5) | 2cos(2π/5) | 2 | 2cos(2π/5) | -2cos(π/5) | -2cos(π/5) | 2cos(2π/5) | 2 |
E5g* | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 |
E6g* | 2 | -2cos(2π/5) | -2cos(π/5) | 2cos(π/5) | 2cos(2π/5) | -2 | 2cos(2π/5) | 2cos(π/5) | -2cos(π/5) | -2cos(2π/5) | 2 | 2 | -2cos(2π/5) | -2cos(π/5) | 2cos(π/5) | 2cos(2π/5) | -2 | 2cos(2π/5) | 2cos(π/5) | -2cos(π/5) | -2cos(2π/5) | 2 |
E7g* | 2 | -2cos(3π/10) | -2cos(2π/5) | 2cos(π/10) | -2cos(π/5) | 0 | 2cos(π/5) | -2cos(π/10) | 2cos(2π/5) | 2cos(3π/10) | -2 | 2 | -2cos(3π/10) | -2cos(2π/5) | 2cos(π/10) | -2cos(π/5) | 0 | 2cos(π/5) | -2cos(π/10) | 2cos(2π/5) | 2cos(3π/10) | -2 |
E8g* | 2 | -2cos(π/5) | 2cos(2π/5) | 2cos(2π/5) | -2cos(π/5) | 2 | -2cos(π/5) | 2cos(2π/5) | 2cos(2π/5) | -2cos(π/5) | 2 | 2 | -2cos(π/5) | 2cos(2π/5) | 2cos(2π/5) | -2cos(π/5) | 2 | -2cos(π/5) | 2cos(2π/5) | 2cos(2π/5) | -2cos(π/5) | 2 |
E9g* | 2 | -2cos(π/10) | 2cos(π/5) | -2cos(3π/10) | 2cos(2π/5) | 0 | -2cos(2π/5) | 2cos(3π/10) | -2cos(π/5) | 2cos(π/10) | -2 | 2 | -2cos(π/10) | 2cos(π/5) | -2cos(3π/10) | 2cos(2π/5) | 0 | -2cos(2π/5) | 2cos(3π/10) | -2cos(π/5) | 2cos(π/10) | -2 |
Au | 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 |
E1u* | 2 | 2cos(π/10) | 2cos(π/5) | 2cos(3π/10) | 2cos(2π/5) | 0 | -2cos(2π/5) | -2cos(3π/10) | -2cos(π/5) | -2cos(π/10) | -2 | -2 | -2cos(π/10) | -2cos(π/5) | -2cos(3π/10) | -2cos(2π/5) | 0 | 2cos(2π/5) | 2cos(3π/10) | 2cos(π/5) | 2cos(π/10) | 2 |
E2u* | 2 | 2cos(π/5) | 2cos(2π/5) | -2cos(2π/5) | -2cos(π/5) | -2 | -2cos(π/5) | -2cos(2π/5) | 2cos(2π/5) | 2cos(π/5) | 2 | -2 | -2cos(π/5) | -2cos(2π/5) | 2cos(2π/5) | 2cos(π/5) | 2 | 2cos(π/5) | 2cos(2π/5) | -2cos(2π/5) | -2cos(π/5) | -2 |
E3u* | 2 | 2cos(3π/10) | -2cos(2π/5) | -2cos(π/10) | -2cos(π/5) | 0 | 2cos(π/5) | 2cos(π/10) | 2cos(2π/5) | -2cos(3π/10) | -2 | -2 | -2cos(3π/10) | 2cos(2π/5) | 2cos(π/10) | 2cos(π/5) | 0 | -2cos(π/5) | -2cos(π/10) | -2cos(2π/5) | 2cos(3π/10) | 2 |
E4u* | 2 | 2cos(2π/5) | -2cos(π/5) | -2cos(π/5) | 2cos(2π/5) | 2 | 2cos(2π/5) | -2cos(π/5) | -2cos(π/5) | 2cos(2π/5) | 2 | -2 | -2cos(2π/5) | 2cos(π/5) | 2cos(π/5) | -2cos(2π/5) | -2 | -2cos(2π/5) | 2cos(π/5) | 2cos(π/5) | -2cos(2π/5) | -2 |
E5u* | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 |
E6u* | 2 | -2cos(2π/5) | -2cos(π/5) | 2cos(π/5) | 2cos(2π/5) | -2 | 2cos(2π/5) | 2cos(π/5) | -2cos(π/5) | -2cos(2π/5) | 2 | -2 | 2cos(2π/5) | 2cos(π/5) | -2cos(π/5) | -2cos(2π/5) | 2 | -2cos(2π/5) | -2cos(π/5) | 2cos(π/5) | 2cos(2π/5) | -2 |
E7u* | 2 | -2cos(3π/10) | -2cos(2π/5) | 2cos(π/10) | -2cos(π/5) | 0 | 2cos(π/5) | -2cos(π/10) | 2cos(2π/5) | 2cos(3π/10) | -2 | -2 | 2cos(3π/10) | 2cos(2π/5) | -2cos(π/10) | 2cos(π/5) | 0 | -2cos(π/5) | 2cos(π/10) | -2cos(2π/5) | -2cos(3π/10) | 2 |
E8u* | 2 | -2cos(π/5) | 2cos(2π/5) | 2cos(2π/5) | -2cos(π/5) | 2 | -2cos(π/5) | 2cos(2π/5) | 2cos(2π/5) | -2cos(π/5) | 2 | -2 | 2cos(π/5) | -2cos(2π/5) | -2cos(2π/5) | 2cos(π/5) | -2 | 2cos(π/5) | -2cos(2π/5) | -2cos(2π/5) | 2cos(π/5) | -2 |
E9u* | 2 | -2cos(π/10) | 2cos(π/5) | -2cos(3π/10) | 2cos(2π/5) | 0 | -2cos(2π/5) | 2cos(3π/10) | -2cos(π/5) | 2cos(π/10) | -2 | -2 | 2cos(π/10) | -2cos(π/5) | 2cos(3π/10) | -2cos(2π/5) | 0 | 2cos(2π/5) | -2cos(3π/10) | 2cos(π/5) | -2cos(π/10) | 2 |
Number of symmetry elements | h = 40 |
Number of classes, irreps | n = 40 |
Number of real-valued irreducible representations | n = 22 |
Abelian group | yes |
Optical Isomerism (Chirality) | no |
Polar | no |
Parity | yes |