D_{6d} | E | 2S_{12} | 2C_{6} | 2S_{4} | 2C_{3} | 2(S_{12})^{5} | C_{2} | 6C'_{2} | 6_{d} | rotations |
functions |
functions |
---|---|---|---|---|---|---|---|---|---|---|---|---|
A_{1} | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | - | x^{2}+y^{2}, z^{2} | - |
A_{2} | +1 | +1 | +1 | +1 | +1 | +1 | +1 | -1 | -1 | R_{z} | - | - |
B_{1} | +1 | -1 | +1 | -1 | +1 | -1 | +1 | +1 | -1 | - | - | - |
B_{2} | +1 | -1 | +1 | -1 | +1 | -1 | +1 | -1 | +1 | z | - | z^{3}, z(x^{2}+y^{2}) |
E_{1} | +2 | +(3)^{½} | +1 | 0 | -1 | -(3)^{½} | -2 | 0 | 0 | (x, y) | - | (xz^{2}, yz^{2}) [x(x^{2}+y^{2}), y(x^{2}+y^{2})] |
E_{2} | +2 | +1 | -1 | -2 | -1 | +1 | +2 | 0 | 0 | - | (x^{2}-y^{2}, xy) | - |
E_{3} | +2 | 0 | -2 | 0 | +2 | 0 | -2 | 0 | 0 | - | - | [y(3x^{2}-y^{2}), x(x^{2}-3y^{2})] |
E_{4} | +2 | -1 | -1 | +2 | -1 | -1 | +2 | 0 | 0 | - | - | [xyz, z(x^{2}-y^{2})] |
E_{5} | +2 | -(3)^{½} | +1 | 0 | -1 | +(3)^{½} | -2 | 0 | 0 | (R_{x}, R_{y}) | (xz, yz) | - |
Number of symmetry elements | h = 24 |
Number of irreducible representations | n = 9 |
Abelian group | no |
Number of subgroups | 14 |
Number of distinct subgroups | 13 |
Subgroups
(Number of different orientations) |
C_{s} , C_{2} (2) , C_{3} , C_{6} , D_{2} , D_{3} , D_{6} , C_{2v} , C_{3v} , C_{6v} , D_{2d} , S_{4} , S_{12} |
---|---|
Optical Isomerism (Chirality) | no |
dipole (p) | B_{2}+E_{1} |
---|---|
quadrupole (d) | A_{1}+E_{2}+E_{5} |
octopole (f) | B_{2}+E_{1}+E_{3}+E_{4} |
hexadecapole (g) | A_{1}+E_{2}+E_{3}+E_{4}+E_{5} |
32-pole (h) | B_{2}+E_{1}+E_{2}+E_{3}+E_{4}+E_{5} |
64-pole (i) | A_{1}+B_{1}+B_{2}+E_{1}+E_{2}+E_{3}+E_{4}+E_{5} |
128-pole (j) | A_{1}+A_{2}+B_{2}+E_{1}+E_{2}+E_{3}+E_{4}+2E_{5} |
256-pole(k) | A_{1}+B_{1}+B_{2}+2E_{1}+E_{2}+E_{3}+2E_{4}+E_{5} |
512-pole (l) | A_{1}+A_{2}+B_{2}+E_{1}+2E_{2}+2E_{3}+E_{4}+2E_{5} |