Character table for point group C6h

=exp(2i/6)
C6h E C6(z) C3 C2 (C3)2 (C6)5 i (S3)5 (S6)5 h S6 S3
linear functions,
rotations
quadratic
functions
cubic
functions
Ag +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 Rz x2+y2, z2 -
Bg +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 - - -
E1g +1
+1
+
+*
-*
-
-1
-1
-
-*
+*
+
+1
+1
+
+*
-*
-
-1
-1
-
-*
+*
+
Rx+iRy
Rx-iRy
(xz, yz) -
E2g +1
+1
-*
-
-
-*
+1
+1
-*
-
-
-*
+1
+1
-*
-
-
-*
+1
+1
-*
-
-
-*
- (x2-y2, xy) -
Au +1 +1 +1 +1 +1 +1 -1 -1 -1 -1 -1 -1 z - z3, z(x2+y2)
Bu +1 -1 +1 -1 +1 -1 -1 +1 -1 +1 -1 +1 - - y(3x2-y2), x(x2-3y2)
E1u +1
+1
+
+*
-*
-
-1
-1
-
-*
+*
+
-1
-1
-
-*
+*
+
+1
+1
+
+*
-*
-
x+iy
x-iy
- (xz2, yz2) [x(x2+y2), y(x2+y2)]
E2u +1
+1
-*
-
-
-*
+1
+1
-*
-
-
-*
-1
-1
+*
+
+
+*
-1
-1
+*
+
+
+*
- - [xyz, z(x2-y2)]


Additional information

Number of symmetry elements h = 12
Number of irreducible representations n = 12
Number of real irreducible representations n = 8
Abelian group yes
Number of subgroups8
Subgroups Cs , Ci , C2 , C3 , C6 , C2h , C3h , S6
Optical Isomerism (Chirality) no


Reduction formula for point group C6h

Type of representation

Information for point groups with complex irreducible representations

general 3N vib

E C6(z) C3 C2 (C3)2 (C6)5 i (S3)5 (S6)5 h S6 S3




Multipoles

dipole (p) Au+E1u
quadrupole (d) Ag+E1g+E2g
octopole (f) Au+2Bu+E1u+E2u
hexadecapole (g) Ag+2Bg+E1g+2E2g
32-pole (h) Au+2Bu+2E1u+2E2u
64-pole (i) 3Ag+2Bg+2E1g+2E2g
128-pole (j) 3Au+2Bu+3E1u+2E2u
256-pole(k) 3Ag+2Bg+3E1g+3E2g
512-pole (l) 3Au+4Bu+3E1u+3E2u

First nonvanishing multipole: quadrupole

Literature



Character tables for chemically important point groups Computational Laboratory for Analysis, Modeling and Visualization Jacobs University Bremen