Character table for point group C6

Character table for point group C₆

=exp(2

i/6)

C₆	E	C₆	C₃	C₂	(C₃)²	(C₆)⁵	linear functions, rotations	quadratic functions	cubic functions
A	+1	+1	+1	+1	+1	+1	z, R_z	x²+y², z²	z³, z(x²+y²)
B	+1	-1	+1	-1	+1	-1	-	-	y(3x²-y²), x(x²-3y²)
E₁	+1 +1	+ +^*	-^* -	-1 -1	- -^*	+^* +	x+iy; R_x+iR_y x-iy; R_x-iR_y	(xz, yz)	(xz², yz²) [x(x²+y²), y(x²+y²)]
E₂	+1 +1	-^* -	- -^*	+1 +1	-^* -	- -^*	-	(x²-y², xy)	[xyz, z(x²-y²)]

Additional information

Number of symmetry elements	h = 6
Number of irreducible representations	n = 6
Number of real irreducible representations	n = 4
Abelian group	yes
Number of subgroups	2
Subgroups	C₂ , C₃
Optical Isomerism (Chirality)	yes
Polar	yes

Reduction formula for point group C₆

Multipoles

dipole (p)	A+E₁
quadrupole (d)	A+E₁+E₂
octopole (f)	A+2B+E₁+E₂
hexadecapole (g)	A+2B+E₁+2E₂
32-pole (h)	A+2B+2E₁+2E₂
64-pole (i)	3A+2B+2E₁+2E₂
128-pole (j)	3A+2B+3E₁+2E₂
256-pole(k)	3A+2B+3E₁+3E₂
512-pole (l)	3A+4B+3E₁+3E₂

First nonvanishing multipole: dipole

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