C16h | E | C16 | C8 | (C16)3 | C4 | (C16)5 | (C8)3 | (C16)7 | C2 | (C16)9 | (C8)5 | (C16)11 | (C4)3 | (C16)13 | (C8)7 | (C16)15 | i | (S16)9 | (S8)5 | (S16)11 | (S4)3 | (S16)13 | (S8)7 | (S16)15 | σh | S16 | S8 | (S16)3 | S4 | (S16)5 | (S8)3 | (S16)7 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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 |
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 |
E1g* | 2 | 2cos(π/8) | 2cos(π/4) | 2cos(3π/8) | 0 | -2cos(3π/8) | -2cos(π/4) | -2cos(π/8) | -2 | -2cos(π/8) | -2cos(π/4) | -2cos(3π/8) | 0 | 2cos(3π/8) | 2cos(π/4) | 2cos(π/8) | 2 | 2cos(π/8) | 2cos(π/4) | 2cos(3π/8) | 0 | -2cos(3π/8) | -2cos(π/4) | -2cos(π/8) | -2 | -2cos(π/8) | -2cos(π/4) | -2cos(3π/8) | 0 | 2cos(3π/8) | 2cos(π/4) | 2cos(π/8) |
E2g* | 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 | -2cos(π/4) | 0 | 2cos(π/4) | 2 | 2cos(π/4) | 0 | -2cos(π/4) | -2 | -2cos(π/4) | 0 | 2cos(π/4) |
E3g* | 2 | 2cos(3π/8) | -2cos(π/4) | -2cos(π/8) | 0 | 2cos(π/8) | 2cos(π/4) | -2cos(3π/8) | -2 | -2cos(3π/8) | 2cos(π/4) | 2cos(π/8) | 0 | -2cos(π/8) | -2cos(π/4) | 2cos(3π/8) | 2 | 2cos(3π/8) | -2cos(π/4) | -2cos(π/8) | 0 | 2cos(π/8) | 2cos(π/4) | -2cos(3π/8) | -2 | -2cos(3π/8) | 2cos(π/4) | 2cos(π/8) | 0 | -2cos(π/8) | -2cos(π/4) | 2cos(3π/8) |
E4g* | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 |
E5g* | 2 | -2cos(3π/8) | -2cos(π/4) | 2cos(π/8) | 0 | -2cos(π/8) | 2cos(π/4) | 2cos(3π/8) | -2 | 2cos(3π/8) | 2cos(π/4) | -2cos(π/8) | 0 | 2cos(π/8) | -2cos(π/4) | -2cos(3π/8) | 2 | -2cos(3π/8) | -2cos(π/4) | 2cos(π/8) | 0 | -2cos(π/8) | 2cos(π/4) | 2cos(3π/8) | -2 | 2cos(3π/8) | 2cos(π/4) | -2cos(π/8) | 0 | 2cos(π/8) | -2cos(π/4) | -2cos(3π/8) |
E6g* | 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 | 2cos(π/4) | 0 | -2cos(π/4) | 2 | -2cos(π/4) | 0 | 2cos(π/4) | -2 | 2cos(π/4) | 0 | -2cos(π/4) |
E7g* | 2 | -2cos(π/8) | 2cos(π/4) | -2cos(3π/8) | 0 | 2cos(3π/8) | -2cos(π/4) | 2cos(π/8) | -2 | 2cos(π/8) | -2cos(π/4) | 2cos(3π/8) | 0 | -2cos(3π/8) | 2cos(π/4) | -2cos(π/8) | 2 | -2cos(π/8) | 2cos(π/4) | -2cos(3π/8) | 0 | 2cos(3π/8) | -2cos(π/4) | 2cos(π/8) | -2 | 2cos(π/8) | -2cos(π/4) | 2cos(3π/8) | 0 | -2cos(3π/8) | 2cos(π/4) | -2cos(π/8) |
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 |
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 |
E1u* | 2 | 2cos(π/8) | 2cos(π/4) | 2cos(3π/8) | 0 | -2cos(3π/8) | -2cos(π/4) | -2cos(π/8) | -2 | -2cos(π/8) | -2cos(π/4) | -2cos(3π/8) | 0 | 2cos(3π/8) | 2cos(π/4) | 2cos(π/8) | -2 | -2cos(π/8) | -2cos(π/4) | -2cos(3π/8) | 0 | 2cos(3π/8) | 2cos(π/4) | 2cos(π/8) | 2 | 2cos(π/8) | 2cos(π/4) | 2cos(3π/8) | 0 | -2cos(3π/8) | -2cos(π/4) | -2cos(π/8) |
E2u* | 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 | 2cos(π/4) | 0 | -2cos(π/4) | -2 | -2cos(π/4) | 0 | 2cos(π/4) | 2 | 2cos(π/4) | 0 | -2cos(π/4) |
E3u* | 2 | 2cos(3π/8) | -2cos(π/4) | -2cos(π/8) | 0 | 2cos(π/8) | 2cos(π/4) | -2cos(3π/8) | -2 | -2cos(3π/8) | 2cos(π/4) | 2cos(π/8) | 0 | -2cos(π/8) | -2cos(π/4) | 2cos(3π/8) | -2 | -2cos(3π/8) | 2cos(π/4) | 2cos(π/8) | 0 | -2cos(π/8) | -2cos(π/4) | 2cos(3π/8) | 2 | 2cos(3π/8) | -2cos(π/4) | -2cos(π/8) | 0 | 2cos(π/8) | 2cos(π/4) | -2cos(3π/8) |
E4u* | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | 0 |
E5u* | 2 | -2cos(3π/8) | -2cos(π/4) | 2cos(π/8) | 0 | -2cos(π/8) | 2cos(π/4) | 2cos(3π/8) | -2 | 2cos(3π/8) | 2cos(π/4) | -2cos(π/8) | 0 | 2cos(π/8) | -2cos(π/4) | -2cos(3π/8) | -2 | 2cos(3π/8) | 2cos(π/4) | -2cos(π/8) | 0 | 2cos(π/8) | -2cos(π/4) | -2cos(3π/8) | 2 | -2cos(3π/8) | -2cos(π/4) | 2cos(π/8) | 0 | -2cos(π/8) | 2cos(π/4) | 2cos(3π/8) |
E6u* | 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 | -2cos(π/4) | 0 | 2cos(π/4) | -2 | 2cos(π/4) | 0 | -2cos(π/4) | 2 | -2cos(π/4) | 0 | 2cos(π/4) |
E7u* | 2 | -2cos(π/8) | 2cos(π/4) | -2cos(3π/8) | 0 | 2cos(3π/8) | -2cos(π/4) | 2cos(π/8) | -2 | 2cos(π/8) | -2cos(π/4) | 2cos(3π/8) | 0 | -2cos(3π/8) | 2cos(π/4) | -2cos(π/8) | -2 | 2cos(π/8) | -2cos(π/4) | 2cos(3π/8) | 0 | -2cos(3π/8) | 2cos(π/4) | -2cos(π/8) | 2 | -2cos(π/8) | 2cos(π/4) | -2cos(3π/8) | 0 | 2cos(3π/8) | -2cos(π/4) | 2cos(π/8) |
Number of symmetry elements | h = 32 |
Number of classes, irreps | n = 32 |
Number of real-valued irreducible representations | n = 18 |
Abelian group | yes |
Optical Isomerism (Chirality) | no |
Polar | no |
Parity | yes |