Character table for point group D6h

Character table for point group D_6h

(x axis coincident with C'₂ axis)

D_6h	E	2C₆ (z)	2C₃	C₂	3C'₂	3C''₂	i	2S₃	2S₆	_h (xy)	3_d	3_v	linear functions, rotations	quadratic functions	cubic functions
A_1g	+1	+1	+1	+1	+1	+1	+1	+1	+1	+1	+1	+1	-	x²+y², z²	-
A_2g	+1	+1	+1	+1	-1	-1	+1	+1	+1	+1	-1	-1	R_z	-	-
B_1g	+1	-1	+1	-1	+1	-1	+1	-1	+1	-1	+1	-1	-	-	-
B_2g	+1	-1	+1	-1	-1	+1	+1	-1	+1	-1	-1	+1	-	-	-
E_1g	+2	+1	-1	-2	0	0	+2	+1	-1	-2	0	0	(R_x, R_y)	(xz, yz)	-
E_2g	+2	-1	-1	+2	0	0	+2	-1	-1	+2	0	0	-	(x²-y², xy)	-
A_1u	+1	+1	+1	+1	+1	+1	-1	-1	-1	-1	-1	-1	-	-	-
A_2u	+1	+1	+1	+1	-1	-1	-1	-1	-1	-1	+1	+1	z	-	z³, z(x²+y²)
B_1u	+1	-1	+1	-1	+1	-1	-1	+1	-1	+1	-1	+1	-	-	x(x²-3y²)
B_2u	+1	-1	+1	-1	-1	+1	-1	+1	-1	+1	+1	-1	-	-	y(3x²-y²)
E_1u	+2	+1	-1	-2	0	0	-2	-1	+1	+2	0	0	(x, y)	-	(xz², yz²) [x(x²+y²), y(x²+y²)]
E_2u	+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
Number of subgroups	30
Number of distinct subgroups	18
Subgroups (Number of different orientations)	C_s (3) , C_i , C₂ (3) , C₃ , C₆ , D₂ , D₃ (2) , D₆ , C_2v (3) , C_3v (2) , C_6v , C_2h (3) , C_3h , C_6h , D_2h , D_3h (2) , D_3d (2) , S₆
Optical Isomerism (Chirality)	no
Polar	no

Reduction formula for point group D_6h

Examples

Benzene	Bis(benzene)chromium (eclipsed)	[18]Annulene
Coronene	Superphane	Kekulene
Cucurbit[6]uril

Multipoles

dipole (p)	A_2u+E_1u
quadrupole (d)	A_1g+E_1g+E_2g
octopole (f)	A_2u+B_1u+B_2u+E_1u+E_2u
hexadecapole (g)	A_1g+B_1g+B_2g+E_1g+2E_2g
32-pole (h)	A_2u+B_1u+B_2u+2E_1u+2E_2u
64-pole (i)	2A_1g+A_2g+B_1g+B_2g+2E_1g+2E_2g
128-pole (j)	A_1u+2A_2u+B_1u+B_2u+3E_1u+2E_2u
256-pole(k)	2A_1g+A_2g+B_1g+B_2g+3E_1g+3E_2g
512-pole (l)	A_1u+2A_2u+2B_1u+2B_2u+3E_1u+3E_2u

First nonvanishing multipole: quadrupole

Literature

A. Gelessus, W. Thiel and W. Weber, J. Chem. Educ. 72, 505 (1995)
Multipoles and Symmetry

Character tables for chemically important point groups

Computational Laboratory for Analysis, Modeling and Visualization

Constructor University Bremen

Last update November, 13^th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement