Character table for point group C6h

Character table for point group C_6h

=exp(2

i/6)

C_6h	E	C₆(z)	C₃	C₂	(C₃)²	(C₆)⁵	i	(S₃)⁵	(S₆)⁵	_h	S₆	S₃	linear functions, rotations	quadratic functions	cubic functions
A_g	+1	+1	+1	+1	+1	+1	+1	+1	+1	+1	+1	+1	R_z	x²+y², z²	-
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	-^* -	- -^*	-	(x²-y², xy)	-
A_u	+1	+1	+1	+1	+1	+1	-1	-1	-1	-1	-1	-1	z	-	z³, z(x²+y²)
B_u	+1	-1	+1	-1	+1	-1	-1	+1	-1	+1	-1	+1	-	-	y(3x²-y²), x(x²-3y²)
E_1u	+1 +1	+ +^*	-^* -	-1 -1	- -^*	+^* +	-1 -1	- -^*	+^* +	+1 +1	+ +^*	-^* -	x+iy x-iy	-	(xz², yz²) [x(x²+y²), y(x²+y²)]
E_2u	+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 subgroups	8
Subgroups	C_s , C_i , C₂ , C₃ , C₆ , C_2h , C_3h , S₆
Optical Isomerism (Chirality)	no
Polar	no

Reduction formula for point group C_6h

Multipoles

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

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