D_{4d} | E | 2S_{8} | 2C_{4} | 2(S_{8})^{3} | C_{2} | 4C'_{2} | 4_{d} | rotations |
functions |
functions |
---|---|---|---|---|---|---|---|---|---|---|
A_{1} | +1 | +1 | +1 | +1 | +1 | +1 | +1 | - | x^{2}+y^{2}, z^{2} | - |
A_{2} | +1 | +1 | +1 | +1 | +1 | -1 | -1 | R_{z} | - | - |
B_{1} | +1 | -1 | +1 | -1 | +1 | +1 | -1 | - | - | - |
B_{2} | +1 | -1 | +1 | -1 | +1 | -1 | +1 | z | - | z^{3}, z(x^{2}+y^{2}) |
E_{1} | +2 | +(2)^{½} | 0 | -(2)^{½} | -2 | 0 | 0 | (x, y) | - | (xz^{2}, yz^{2}) [x(x^{2}+y^{2}), y(x^{2}+y^{2})] |
E_{2} | +2 | 0 | -2 | 0 | +2 | 0 | 0 | - | (x^{2}-y^{2}, xy) | [xyz, z(x^{2}-y^{2})] |
E_{3} | +2 | -(2)^{½} | 0 | +(2)^{½} | -2 | 0 | 0 | (R_{x}, R_{y}) | (xz, yz) | [y(3x^{2}-y^{2}), x(x^{2}-3y^{2})] |
Number of symmetry elements | h = 16 |
Number of irreducible representations | n = 7 |
Abelian group | no |
Number of subgroups | 9 |
Number of distinct subgroups | 8 |
Subgroups
(Number of different orientations) |
C_{s} , C_{2} (2) , C_{4} , D_{2} , D_{4} , C_{2v} , C_{4v} , S_{8} |
---|---|
Optical Isomerism (Chirality) | no |
Polar | no |
Sulfur S_{8} |
---|
dipole (p) | B_{2}+E_{1} |
---|---|
quadrupole (d) | A_{1}+E_{2}+E_{3} |
octopole (f) | B_{2}+E_{1}+E_{2}+E_{3} |
hexadecapole (g) | A_{1}+B_{1}+B_{2}+E_{1}+E_{2}+E_{3} |
32-pole (h) | A_{1}+A_{2}+B_{2}+E_{1}+E_{2}+2E_{3} |
64-pole (i) | A_{1}+B_{1}+B_{2}+2E_{1}+2E_{2}+E_{3} |
128-pole (j) | A_{1}+A_{2}+B_{2}+2E_{1}+2E_{2}+2E_{3} |
256-pole(k) | 2A_{1}+A_{2}+B_{1}+B_{2}+2E_{1}+2E_{2}+2E_{3} |
512-pole (l) | A_{1}+A_{2}+B_{1}+2B_{2}+3E_{1}+2E_{2}+2E_{3} |