Character table for point group D6h

(x axis coincident with C'2 axis)
D6h E 2C6 (z) 2C3 C2 3C'2 3C''2 i 2S3 2S6 h (xy) 3d 3v
linear functions,
rotations
quadratic
functions
cubic
functions
A1g +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 - x2+y2, z2 -
A2g +1 +1 +1 +1 -1 -1 +1 +1 +1 +1 -1 -1 Rz - -
B1g +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 - - -
B2g +1 -1 +1 -1 -1 +1 +1 -1 +1 -1 -1 +1 - - -
E1g +2 +1 -1 -2 0 0 +2 +1 -1 -2 0 0 (Rx, Ry) (xz, yz) -
E2g +2 -1 -1 +2 0 0 +2 -1 -1 +2 0 0 - (x2-y2, xy) -
A1u +1 +1 +1 +1 +1 +1 -1 -1 -1 -1 -1 -1 - - -
A2u +1 +1 +1 +1 -1 -1 -1 -1 -1 -1 +1 +1 z - z3, z(x2+y2)
B1u +1 -1 +1 -1 +1 -1 -1 +1 -1 +1 -1 +1 - - x(x2-3y2)
B2u +1 -1 +1 -1 -1 +1 -1 +1 -1 +1 +1 -1 - - y(3x2-y2)
E1u +2 +1 -1 -2 0 0 -2 -1 +1 +2 0 0 (x, y) - (xz2, yz2) [x(x2+y2), y(x2+y2)]
E2u +2 -1 -1 +2 0 0 -2 +1 +1 -2 0 0 - - [xyz, z(x2-y2)]


Additional information

Number of symmetry elements h = 24
Number of irreducible representations n = 12
Abelian group no
Subgroups Cs , Ci , C2 , C3 , C6 , D2 , D3 , D6 , C2v , C3v , C6v , C2h , C3h , C6h , D2h , D3h , D3d , S6
Optical Isomerism (Chirality) no


Reduction formula for point group D6h

Type of representation

general 3N vib

E 2C6 (z) 2C3 C2 3C'2 3C''2 i 2S3 2S6 h (xy) 3d 3v




Multipoles

dipole (p) A2u+E1u
quadrupole (d) A1g+E1g+E2g
octopole (f) A2u+B1u+B2u+E1u+E2u
hexadecapole (g) A1g+B1g+B2g+E1g+2E2g
32-pole (h) A2u+B1u+B2u+2E1u+2E2u
64-pole (i) 2A1g+A2g+B1g+B2g+2E1g+2E2g
128-pole (j) A1u+2A2u+B1u+B2u+3E1u+2E2u
256-pole(k) 2A1g+A2g+B1g+B2g+3E1g+3E2g
512-pole (l) A1u+2A2u+2B1u+2B2u+3E1u+3E2u

First nonvanishing multipole: quadrupole

Literature



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