✅
D20d | E | 2S40 | 2C20 | 2(S40)3 | 2C10 | 2S8 | 2(C20)3 | 2(S40)7 | 2C5 | 2(S40)9 | 2C4 | 2(S40)11 | 2(C10)3 | 2(S40)13 | 2(C20)7 | 2(S8)3 | 2(C5)2 | 2(S40)17 | 2(C20)9 | 2(S40)19 | C2 | 20C'2 | 20σd |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
A2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 |
B1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 |
B2 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 |
E1 | 2 | 2cos(π/20) | 2cos(π/10) | 2cos(3π/20) | 2cos(π/5) | 2cos(π/4) | 2cos(3π/10) | 2cos(7π/20) | 2cos(2π/5) | 2cos(9π/20) | 0 | -2cos(9π/20) | -2cos(2π/5) | -2cos(7π/20) | -2cos(3π/10) | -2cos(π/4) | -2cos(π/5) | -2cos(3π/20) | -2cos(π/10) | -2cos(π/20) | -2 | 0 | 0 |
E2 | 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 | -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 | 0 | 0 |
E3 | 2 | 2cos(3π/20) | 2cos(3π/10) | 2cos(9π/20) | -2cos(2π/5) | -2cos(π/4) | -2cos(π/10) | -2cos(π/20) | -2cos(π/5) | -2cos(7π/20) | 0 | 2cos(7π/20) | 2cos(π/5) | 2cos(π/20) | 2cos(π/10) | 2cos(π/4) | 2cos(2π/5) | -2cos(9π/20) | -2cos(3π/10) | -2cos(3π/20) | -2 | 0 | 0 |
E4 | 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 | 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 | 0 | 0 |
E5 | 2 | 2cos(π/4) | 0 | -2cos(π/4) | -2 | -2cos(π/4) | 0 | 2cos(π/4) | 2 | 2cos(π/4) | 0 | -2cos(π/4) | -2 | -2cos(π/4) | 0 | 2cos(π/4) | 2 | 2cos(π/4) | 0 | -2cos(π/4) | -2 | 0 | 0 |
E6 | 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 | -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 | 0 | 0 |
E7 | 2 | 2cos(7π/20) | -2cos(3π/10) | -2cos(π/20) | -2cos(2π/5) | 2cos(π/4) | 2cos(π/10) | 2cos(9π/20) | -2cos(π/5) | -2cos(3π/20) | 0 | 2cos(3π/20) | 2cos(π/5) | -2cos(9π/20) | -2cos(π/10) | -2cos(π/4) | 2cos(2π/5) | 2cos(π/20) | 2cos(3π/10) | -2cos(7π/20) | -2 | 0 | 0 |
E8 | 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 | 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 | 0 | 0 |
E9 | 2 | 2cos(9π/20) | -2cos(π/10) | -2cos(7π/20) | 2cos(π/5) | 2cos(π/4) | -2cos(3π/10) | -2cos(3π/20) | 2cos(2π/5) | 2cos(π/20) | 0 | -2cos(π/20) | -2cos(2π/5) | 2cos(3π/20) | 2cos(3π/10) | -2cos(π/4) | -2cos(π/5) | 2cos(7π/20) | 2cos(π/10) | -2cos(9π/20) | -2 | 0 | 0 |
E10 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | 0 |
E11 | 2 | -2cos(9π/20) | -2cos(π/10) | 2cos(7π/20) | 2cos(π/5) | -2cos(π/4) | -2cos(3π/10) | 2cos(3π/20) | 2cos(2π/5) | -2cos(π/20) | 0 | 2cos(π/20) | -2cos(2π/5) | -2cos(3π/20) | 2cos(3π/10) | 2cos(π/4) | -2cos(π/5) | -2cos(7π/20) | 2cos(π/10) | 2cos(9π/20) | -2 | 0 | 0 |
E12 | 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 | -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 | 0 | 0 |
E13 | 2 | -2cos(7π/20) | -2cos(3π/10) | 2cos(π/20) | -2cos(2π/5) | -2cos(π/4) | 2cos(π/10) | -2cos(9π/20) | -2cos(π/5) | 2cos(3π/20) | 0 | -2cos(3π/20) | 2cos(π/5) | 2cos(9π/20) | -2cos(π/10) | 2cos(π/4) | 2cos(2π/5) | -2cos(π/20) | 2cos(3π/10) | 2cos(7π/20) | -2 | 0 | 0 |
E14 | 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 | 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 | 0 | 0 |
E15 | 2 | -2cos(π/4) | 0 | 2cos(π/4) | -2 | 2cos(π/4) | 0 | -2cos(π/4) | 2 | -2cos(π/4) | 0 | 2cos(π/4) | -2 | 2cos(π/4) | 0 | -2cos(π/4) | 2 | -2cos(π/4) | 0 | 2cos(π/4) | -2 | 0 | 0 |
E16 | 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 | -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 | 0 | 0 |
E17 | 2 | -2cos(3π/20) | 2cos(3π/10) | -2cos(9π/20) | -2cos(2π/5) | 2cos(π/4) | -2cos(π/10) | 2cos(π/20) | -2cos(π/5) | 2cos(7π/20) | 0 | -2cos(7π/20) | 2cos(π/5) | -2cos(π/20) | 2cos(π/10) | -2cos(π/4) | 2cos(2π/5) | 2cos(9π/20) | -2cos(3π/10) | 2cos(3π/20) | -2 | 0 | 0 |
E18 | 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 | 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 | 0 | 0 |
E19 | 2 | -2cos(π/20) | 2cos(π/10) | -2cos(3π/20) | 2cos(π/5) | -2cos(π/4) | 2cos(3π/10) | -2cos(7π/20) | 2cos(2π/5) | -2cos(9π/20) | 0 | 2cos(9π/20) | -2cos(2π/5) | 2cos(7π/20) | -2cos(3π/10) | 2cos(π/4) | -2cos(π/5) | 2cos(3π/20) | -2cos(π/10) | 2cos(π/20) | -2 | 0 | 0 |
Number of symmetry elements | h = 80 |
Number of classes, irreps | n = 23 |
Abelian group | no |
Optical Isomerism (Chirality) | no |
Polar | no |
Parity | no |