C15h | E | C15 | (C15)2 | C5 | (C15)4 | C3 | (C5)2 | (C15)7 | (C15)8 | (C5)3 | (C3)2 | (C15)11 | (C5)4 | (C15)13 | (C15)14 | σh | S15 | (S15)17 | S5 | (S15)19 | S3 | (S5)7 | (S15)7 | (S15)23 | (S5)3 | (S3)5 | (S15)11 | (S5)9 | (S15)13 | (S15)29 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A' | 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 |
E'1* | 2 | 2cos(2π/15) | 2cos(4π/15) | 2cos(2π/5) | 2cos(8π/15) | -1 | 2cos(4π/5) | 2cos(14π/15) | 2cos(14π/15) | 2cos(4π/5) | -1 | 2cos(8π/15) | 2cos(2π/5) | 2cos(4π/15) | 2cos(2π/15) | 2 | 2cos(2π/15) | 2cos(4π/15) | 2cos(2π/5) | 2cos(8π/15) | -1 | 2cos(4π/5) | 2cos(14π/15) | 2cos(14π/15) | 2cos(4π/5) | -1 | 2cos(8π/15) | 2cos(2π/5) | 2cos(4π/15) | 2cos(2π/15) |
E'2* | 2 | 2cos(4π/15) | 2cos(8π/15) | 2cos(4π/5) | 2cos(14π/15) | -1 | 2cos(2π/5) | 2cos(2π/15) | 2cos(2π/15) | 2cos(2π/5) | -1 | 2cos(14π/15) | 2cos(4π/5) | 2cos(8π/15) | 2cos(4π/15) | 2 | 2cos(4π/15) | 2cos(8π/15) | 2cos(4π/5) | 2cos(14π/15) | -1 | 2cos(2π/5) | 2cos(2π/15) | 2cos(2π/15) | 2cos(2π/5) | -1 | 2cos(14π/15) | 2cos(4π/5) | 2cos(8π/15) | 2cos(4π/15) |
E'3* | 2 | 2cos(2π/5) | 2cos(4π/5) | 2cos(4π/5) | 2cos(2π/5) | 2 | 2cos(2π/5) | 2cos(4π/5) | 2cos(4π/5) | 2cos(2π/5) | 2 | 2cos(2π/5) | 2cos(4π/5) | 2cos(4π/5) | 2cos(2π/5) | 2 | 2cos(2π/5) | 2cos(4π/5) | 2cos(4π/5) | 2cos(2π/5) | 2 | 2cos(2π/5) | 2cos(4π/5) | 2cos(4π/5) | 2cos(2π/5) | 2 | 2cos(2π/5) | 2cos(4π/5) | 2cos(4π/5) | 2cos(2π/5) |
E'4* | 2 | 2cos(8π/15) | 2cos(14π/15) | 2cos(2π/5) | 2cos(2π/15) | -1 | 2cos(4π/5) | 2cos(4π/15) | 2cos(4π/15) | 2cos(4π/5) | -1 | 2cos(2π/15) | 2cos(2π/5) | 2cos(14π/15) | 2cos(8π/15) | 2 | 2cos(8π/15) | 2cos(14π/15) | 2cos(2π/5) | 2cos(2π/15) | -1 | 2cos(4π/5) | 2cos(4π/15) | 2cos(4π/15) | 2cos(4π/5) | -1 | 2cos(2π/15) | 2cos(2π/5) | 2cos(14π/15) | 2cos(8π/15) |
E'5* | 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 |
E'6* | 2 | 2cos(4π/5) | 2cos(2π/5) | 2cos(2π/5) | 2cos(4π/5) | 2 | 2cos(4π/5) | 2cos(2π/5) | 2cos(2π/5) | 2cos(4π/5) | 2 | 2cos(4π/5) | 2cos(2π/5) | 2cos(2π/5) | 2cos(4π/5) | 2 | 2cos(4π/5) | 2cos(2π/5) | 2cos(2π/5) | 2cos(4π/5) | 2 | 2cos(4π/5) | 2cos(2π/5) | 2cos(2π/5) | 2cos(4π/5) | 2 | 2cos(4π/5) | 2cos(2π/5) | 2cos(2π/5) | 2cos(4π/5) |
E'7* | 2 | 2cos(14π/15) | 2cos(2π/15) | 2cos(4π/5) | 2cos(4π/15) | -1 | 2cos(2π/5) | 2cos(8π/15) | 2cos(8π/15) | 2cos(2π/5) | -1 | 2cos(4π/15) | 2cos(4π/5) | 2cos(2π/15) | 2cos(14π/15) | 2 | 2cos(14π/15) | 2cos(2π/15) | 2cos(4π/5) | 2cos(4π/15) | -1 | 2cos(2π/5) | 2cos(8π/15) | 2cos(8π/15) | 2cos(2π/5) | -1 | 2cos(4π/15) | 2cos(4π/5) | 2cos(2π/15) | 2cos(14π/15) |
A'' | 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 |
E''1* | 2 | 2cos(2π/15) | 2cos(4π/15) | 2cos(2π/5) | 2cos(8π/15) | -1 | 2cos(4π/5) | 2cos(14π/15) | 2cos(14π/15) | 2cos(4π/5) | -1 | 2cos(8π/15) | 2cos(2π/5) | 2cos(4π/15) | 2cos(2π/15) | -2 | -2cos(2π/15) | -2cos(4π/15) | -2cos(2π/5) | -2cos(8π/15) | 1 | -2cos(4π/5) | -2cos(14π/15) | -2cos(14π/15) | -2cos(4π/5) | 1 | -2cos(8π/15) | -2cos(2π/5) | -2cos(4π/15) | -2cos(2π/15) |
E''2* | 2 | 2cos(4π/15) | 2cos(8π/15) | 2cos(4π/5) | 2cos(14π/15) | -1 | 2cos(2π/5) | 2cos(2π/15) | 2cos(2π/15) | 2cos(2π/5) | -1 | 2cos(14π/15) | 2cos(4π/5) | 2cos(8π/15) | 2cos(4π/15) | -2 | -2cos(4π/15) | -2cos(8π/15) | -2cos(4π/5) | -2cos(14π/15) | 1 | -2cos(2π/5) | -2cos(2π/15) | -2cos(2π/15) | -2cos(2π/5) | 1 | -2cos(14π/15) | -2cos(4π/5) | -2cos(8π/15) | -2cos(4π/15) |
E''3* | 2 | 2cos(2π/5) | 2cos(4π/5) | 2cos(4π/5) | 2cos(2π/5) | 2 | 2cos(2π/5) | 2cos(4π/5) | 2cos(4π/5) | 2cos(2π/5) | 2 | 2cos(2π/5) | 2cos(4π/5) | 2cos(4π/5) | 2cos(2π/5) | -2 | -2cos(2π/5) | -2cos(4π/5) | -2cos(4π/5) | -2cos(2π/5) | -2 | -2cos(2π/5) | -2cos(4π/5) | -2cos(4π/5) | -2cos(2π/5) | -2 | -2cos(2π/5) | -2cos(4π/5) | -2cos(4π/5) | -2cos(2π/5) |
E''4* | 2 | 2cos(8π/15) | 2cos(14π/15) | 2cos(2π/5) | 2cos(2π/15) | -1 | 2cos(4π/5) | 2cos(4π/15) | 2cos(4π/15) | 2cos(4π/5) | -1 | 2cos(2π/15) | 2cos(2π/5) | 2cos(14π/15) | 2cos(8π/15) | -2 | -2cos(8π/15) | -2cos(14π/15) | -2cos(2π/5) | -2cos(2π/15) | 1 | -2cos(4π/5) | -2cos(4π/15) | -2cos(4π/15) | -2cos(4π/5) | 1 | -2cos(2π/15) | -2cos(2π/5) | -2cos(14π/15) | -2cos(8π/15) |
E''5* | 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 |
E''6* | 2 | 2cos(4π/5) | 2cos(2π/5) | 2cos(2π/5) | 2cos(4π/5) | 2 | 2cos(4π/5) | 2cos(2π/5) | 2cos(2π/5) | 2cos(4π/5) | 2 | 2cos(4π/5) | 2cos(2π/5) | 2cos(2π/5) | 2cos(4π/5) | -2 | -2cos(4π/5) | -2cos(2π/5) | -2cos(2π/5) | -2cos(4π/5) | -2 | -2cos(4π/5) | -2cos(2π/5) | -2cos(2π/5) | -2cos(4π/5) | -2 | -2cos(4π/5) | -2cos(2π/5) | -2cos(2π/5) | -2cos(4π/5) |
E''7* | 2 | 2cos(14π/15) | 2cos(2π/15) | 2cos(4π/5) | 2cos(4π/15) | -1 | 2cos(2π/5) | 2cos(8π/15) | 2cos(8π/15) | 2cos(2π/5) | -1 | 2cos(4π/15) | 2cos(4π/5) | 2cos(2π/15) | 2cos(14π/15) | -2 | -2cos(14π/15) | -2cos(2π/15) | -2cos(4π/5) | -2cos(4π/15) | 1 | -2cos(2π/5) | -2cos(8π/15) | -2cos(8π/15) | -2cos(2π/5) | 1 | -2cos(4π/15) | -2cos(4π/5) | -2cos(2π/15) | -2cos(14π/15) |
Number of symmetry elements | h = 30 |
Number of classes, irreps | n = 30 |
Number of real-valued irreducible representations | n = 16 |
Abelian group | yes |
Optical Isomerism (Chirality) | no |
Polar | no |
Parity | no |