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