S_{8} | E | S_{8} | C_{4} (z) | (S_{8})^{3} | C_{2} | (S_{8})^{5} | (C_{4})^{3} | (S_{8})^{7} | rotations |
functions |
functions |
---|---|---|---|---|---|---|---|---|---|---|---|
A | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | R_{z} | x^{2}+y^{2}, z^{2} | - |
B | +1 | -1 | +1 | -1 | +1 | -1 | +1 | -1 | z | - | z^{3}, z(x^{2}+y^{2}) |
E_{1} | +1 +1 |
+ +^{*} | +i -i | -^{*} - | -1 -1 | - -^{*} | -i +i | +^{*} + | x+iy x-iy |
- | (xz^{2}, yz^{2}) [x(x^{2}+y^{2}), y(x^{2}+y^{2})] |
E_{2} | +1 +1 |
+i -i | -1 -1 | -i +i | +1 +1 | +i -i | -1 -1 | -i +i | - | (x^{2}-y^{2}, xy) | [xyz, z(x^{2}-y^{2})] |
E_{3} | +1 +1 |
- -^{*} | +i -i | +^{*} + | -1 -1 | + +^{*} | -i +i | -^{*} - | R_{x}+iR_{y} R_{x}-iR_{y} |
(xz, yz) | [y(3x^{2}-y^{2}), x(x^{2}-3y^{2})] |
Number of symmetry elements | h = 8 |
Number of irreducible representations | n = 8 |
Number of real irreducible representations | n = 5 |
Abelian group | yes |
Number of subgroups | 2 |
Subgroups | C_{2} , C_{4} |
---|---|
Optical Isomerism (Chirality) | no |
dipole (p) | B+E_{1} |
---|---|
quadrupole (d) | A+E_{2}+E_{3} |
octopole (f) | B+E_{1}+E_{2}+E_{3} |
hexadecapole (g) | A+2B+E_{1}+E_{2}+E_{3} |
32-pole (h) | 2A+B+E_{1}+E_{2}+2E_{3} |
64-pole (i) | A+2B+2E_{1}+2E_{2}+E_{3} |
128-pole (j) | 2A+B+2E_{1}+2E_{2}+2E_{3} |
256-pole(k) | 3A+2B+2E_{1}+2E_{2}+2E_{3} |
512-pole (l) | 2A+3B+3E_{1}+2E_{2}+2E_{3} |