Character table for point group D6d

(x axis coincident with C'2 axis)
D6d E 2S12 2C6 2S4 2C3 2(S12)5 C2 6C'2 6d
linear functions,
rotations
quadratic
functions
cubic
functions
A1 +1 +1 +1 +1 +1 +1 +1 +1 +1 - x2+y2, z2 -
A2 +1 +1 +1 +1 +1 +1 +1 -1 -1 Rz - -
B1 +1 -1 +1 -1 +1 -1 +1 +1 -1 - - -
B2 +1 -1 +1 -1 +1 -1 +1 -1 +1 z - z3, z(x2+y2)
E1 +2 +(3)½ +1 0 -1 -(3)½ -2 0 0 (x, y) - (xz2, yz2) [x(x2+y2), y(x2+y2)]
E2 +2 +1 -1 -2 -1 +1 +2 0 0 - (x2-y2, xy) -
E3 +2 0 -2 0 +2 0 -2 0 0 - - [y(3x2-y2), x(x2-3y2)]
E4 +2 -1 -1 +2 -1 -1 +2 0 0 - - [xyz, z(x2-y2)]
E5 +2 -(3)½ +1 0 -1 +(3)½ -2 0 0 (Rx, Ry) (xz, yz) -


Additional information

Number of symmetry elements h = 24
Number of irreducible representations n = 9
Abelian group no
Number of subgroups14
Number of distinct subgroups13
Subgroups
(Number of different orientations)
Cs , C2 (2) , C3 , C6 , D2 , D3 , D6 , C2v , C3v , C6v , D2d , S4 , S12
Optical Isomerism (Chirality) no


Reduction formula for point group D6d

Type of representation

general 3N vib

E 2S12 2C6 2S4 2C3 2(S12)5 C2 6C'2 6d




Multipoles

dipole (p) B2+E1
quadrupole (d) A1+E2+E5
octopole (f) B2+E1+E3+E4
hexadecapole (g) A1+E2+E3+E4+E5
32-pole (h) B2+E1+E2+E3+E4+E5
64-pole (i) A1+B1+B2+E1+E2+E3+E4+E5
128-pole (j) A1+A2+B2+E1+E2+E3+E4+2E5
256-pole(k) A1+B1+B2+2E1+E2+E3+2E4+E5
512-pole (l) A1+A2+B2+E1+2E2+2E3+E4+2E5

First nonvanishing multipole: quadrupole

Literature



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