C_{6h} | E | C_{6}(z) | C_{3} | C_{2} | (C_{3})^{2} | (C_{6})^{5} | i | (S_{3})^{5} | (S_{6})^{5} | _{h} | S_{6} | S_{3} | rotations |
functions |
functions |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A_{g} | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | R_{z} | x^{2}+y^{2}, z^{2} | - |
B_{g} | +1 | -1 | +1 | -1 | +1 | -1 | +1 | -1 | +1 | -1 | +1 | -1 | - | - | - |
E_{1g} | +1 +1 |
+ +^{*} |
-^{*} - |
-1 -1 |
- -^{*} |
+^{*} + |
+1 +1 |
+ +^{*} |
-^{*} - |
-1 -1 |
- -^{*} |
+^{*} + |
R_{x}+iR_{y} R_{x}-iR_{y} |
(xz, yz) | - |
E_{2g} | +1 +1 |
-^{*} - |
- -^{*} |
+1 +1 |
-^{*} - |
- -^{*} |
+1 +1 |
-^{*} - |
- -^{*} |
+1 +1 |
-^{*} - |
- -^{*} |
- | (x^{2}-y^{2}, xy) | - |
A_{u} | +1 | +1 | +1 | +1 | +1 | +1 | -1 | -1 | -1 | -1 | -1 | -1 | z | - | z^{3}, z(x^{2}+y^{2}) |
B_{u} | +1 | -1 | +1 | -1 | +1 | -1 | -1 | +1 | -1 | +1 | -1 | +1 | - | - | y(3x^{2}-y^{2}), x(x^{2}-3y^{2}) |
E_{1u} | +1 +1 |
+ +^{*} |
-^{*} - |
-1 -1 |
- -^{*} |
+^{*} + |
-1 -1 |
- -^{*} |
+^{*} + |
+1 +1 |
+ +^{*} |
-^{*} - |
x+iy x-iy |
- | (xz^{2}, yz^{2}) [x(x^{2}+y^{2}), y(x^{2}+y^{2})] |
E_{2u} | +1 +1 |
-^{*} - |
- -^{*} |
+1 +1 |
-^{*} - |
- -^{*} |
-1 -1 |
+^{*} + |
+ +^{*} |
-1 -1 |
+^{*} + |
+ +^{*} |
- | - | [xyz, z(x^{2}-y^{2})] |
Number of symmetry elements | h = 12 |
Number of irreducible representations | n = 12 |
Number of real irreducible representations | n = 8 |
Abelian group | yes |
Number of subgroups | 8 |
Subgroups | C_{s} , C_{i} , C_{2} , C_{3} , C_{6} , C_{2h} , C_{3h} , S_{6} |
---|---|
Optical Isomerism (Chirality) | no |
Polar | no |
dipole (p) | A_{u}+E_{1u} |
---|---|
quadrupole (d) | A_{g}+E_{1g}+E_{2g} |
octopole (f) | A_{u}+2B_{u}+E_{1u}+E_{2u} |
hexadecapole (g) | A_{g}+2B_{g}+E_{1g}+2E_{2g} |
32-pole (h) | A_{u}+2B_{u}+2E_{1u}+2E_{2u} |
64-pole (i) | 3A_{g}+2B_{g}+2E_{1g}+2E_{2g} |
128-pole (j) | 3A_{u}+2B_{u}+3E_{1u}+2E_{2u} |
256-pole(k) | 3A_{g}+2B_{g}+3E_{1g}+3E_{2g} |
512-pole (l) | 3A_{u}+4B_{u}+3E_{1u}+3E_{2u} |