Character table for point group Ih

Character table for point group I_h

I_h	E	12C₅	12(C₅)²	20C₃	15C₂	i	12S₁₀	12(S₁₀)³	20S₆	15	linear functions, rotations	quadratic functions	cubic functions
A_g	+1	+1	+1	+1	+1	+1	+1	+1	+1	+1	-	x²+y²+z²	-
T_1g	+3	-2cos(4/5)	-2cos(2/5)	0	-1	+3	-2cos(2/5)	-2cos(4/5)	0	-1	(R_x, R_y, R_z)	-	-
T_2g	+3	-2cos(2/5)	-2cos(4/5)	0	-1	+3	-2cos(4/5)	-2cos(2/5)	0	-1	-	-	-
G_g	+4	-1	-1	+1	0	+4	-1	-1	+1	0	-	-	-
H_g	+5	0	0	-1	+1	+5	0	0	-1	+1	-	[2z²-x²-y², x²-y², xy, xz, yz]	-
A_u	+1	+1	+1	+1	+1	-1	-1	-1	-1	-1	-	-	-
T_1u	+3	-2cos(4/5)	-2cos(2/5)	0	-1	-3	+2cos(2/5)	+2cos(4/5)	0	+1	(x, y, z)	-	[x(z²+y²), y(z²+x²), z(x²+y²)]
T_2u	+3	-2cos(2/5)	-2cos(4/5)	0	-1	-3	+2cos(4/5)	+2cos(2/5)	0	+1	-	-	[x³, y³, z³]
G_u	+4	-1	-1	+1	0	-4	+1	+1	-1	0	-	-	[x(z²-y²), y(z²-x²), z(x²-y²), xyz]
H_u	+5	0	0	-1	+1	-5	0	0	+1	-1	-	-	-

Information for point groups with fivefold rotational axis

Additional information

Number of symmetry elements	h = 120
Number of irreducible representations	n = 10
Abelian group	no
Number of subgroups	20
Subgroups	C_s , C_i , C₂ , C₃ , C₅ , D₂ , D₃ , D₅ , C_2v , C_3v , C_5v , C_2h , D_2h , D_3d , D_5d , S₆ , S₁₀ , T , T_h , I
Optical Isomerism (Chirality)	no
Polar	no

Reduction formula for point group I_h

Examples

[Pb₁₂]^2--Ion	Fullerene C₂₀	[B₁₂H₁₂]^2--Ion
Dodecahedrane	Fullerene C₆₀	Fullerene C₈₀ : 7

Multipoles

dipole (p)	T_1u
quadrupole (d)	H_g
octopole (f)	T_2u+G_u
hexadecapole (g)	G_g+H_g
32-pole (h)	T_1u+T_2u+H_u
64-pole (i)	A_g+T_1g+G_g+H_g
128-pole (j)	T_1u+T_2u+G_u+H_u
256-pole(k)	T_2g+G_g+2H_g
512-pole (l)	T_1u+T_2u+2G_u+H_u

First nonvanishing multipole: 64-pole

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