ih - Point Group Symmetry Character Tables - Chemistry Online Education

IH - POINT GROUP SYMMETRY CHARACTER TABLES

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Nonaxial
C₁
C_s
C_i

C_n
C₂
C₃
C₄
C₅
C₆

C_nv
C_2v
C_3v
C_4v
C_5v
C_6v

C_nh
C_2h
C_3h
C_4h
C_5h
C_6h

D_n
D₂
D₃
D₄
D₅
D₆

D_nh
D_2h
D_3h
D_4h
D_5h
D_6h

D_nd
D_2d
D_3d
D_4d
D_5d
D_6d

S_n
S₄
S₆
S₈
S₁₀

Higher
T_d
O_h
I_h

Linear
C_∞v
D_∞h

I_h Point Group

not Abelian, 10(32) irreducible representations

Character table for I_h point group

	E	12C₅	12(C₅)²	20C₃	15C₂	i	12S₁₀	12(S₁₀)³	20S₆	15σ	linear, rotations	quadratic
A_g	1	1	1	1	1	1	1	1	1	1		x²+y²+z²
T_1g	3	-2cos(4&pi/5)	-2cos(2&pi/5)	0	-1	3	-2cos(2&pi/5)	-2cos(4&pi/5)	0	-1	(R_x, R_y, R_z)
T_2g	3	-2cos(2&pi/5)	-2cos(4&pi/5)	0	-1	3	-2cos(4&pi/5)	-2cos(2&pi/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&pi/5)	-2cos(2&pi/5)	0	-1	-3	2cos(2&pi/5)	2cos(4&pi/5)	0	1	(x, y, z)
T_2u	3	-2cos(2&pi/5)	-2cos(4&pi/5)	0	-1	-3	2cos(4&pi/5)	2cos(2&pi/5)	0	1
G_u	4	-1	-1	1	0	-4	1	1	-1	0
H_u	5	0	0	-1	1	-5	0	0	1	-1

Product table for I_h point group

	A_g	T_1g	T_2g	G_g	H_g	A_u	T_1u	T_2u	G_u	H_u
A_g	A_g	T_1g	T_2g	G_g	H_g	A_u	T_1u	T_2u	G_u	H_u
T_1g	T_1g	A_g+T_1g+H_g	G_g+H_g	T_2g+G_g+H_g	T_1g+T_2g+G_g+H_g	T_1u	A_u+T_1u+H_u	G_u+H_u	T_2u+G_u+H_u	T_1u+T_2u+G_u+H_u
T_2g	T_2g	G_g+H_g	A_g+T_2g+H_g	T_1g+G_g+H_g	T_1g+T_2g+G_g+H_g	T_2u	G_u+H_u	A_u+T_2u+H_u	T_1u+G_u+H_u	T_1u+T_2u+G_u+H_u
G_g	G_g	T_2g+G_g+H_g	T_1g+G_g+H_g	A_g+T_1g+T_2g+G_g+H_g	T_1g+T_2g+G_g+2H_g	G_u	T_2u+G_u+H_u	T_1u+G_u+H_u	A_u+T_1u+T_2u+G_u+H_u	T_1u+T_2u+G_u+2H_u
H_g	H_g	T_1g+T_2g+G_g+H_g	T_1g+T_2g+G_g+H_g	T_1g+T_2g+G_g+2H_g	A_g+T_1g+T_2g+2G_g+2H_g	H_u	T_1u+T_2u+G_u+H_u	T_1u+T_2u+G_u+H_u	T_1u+T_2u+G_u+2H_u	A_u+T_1u+T_2u+2G_u+2H_u
A_u	A_u	T_1u	T_2u	G_u	H_u	A_g	T_1g	T_2g	G_g	H_g
T_1u	T_1u	A_u+T_1u+H_u	G_u+H_u	T_2u+G_u+H_u	T_1u+T_2u+G_u+H_u	T_1g	A_g+T_1g+H_g	G_g+H_g	T_2g+G_g+H_g	T_1g+T_2g+G_g+H_g
T_2u	T_2u	G_u+H_u	A_u+T_2u+H_u	T_1u+G_u+H_u	T_1u+T_2u+G_u+H_u	T_2g	G_g+H_g	A_g+T_2g+H_g	T_1g+G_g+H_g	T_1g+T_2g+G_g+H_g
G_u	G_u	T_2u+G_u+H_u	T_1u+G_u+H_u	A_u+T_1u+T_2u+G_u+H_u	T_1u+T_2u+G_u+2H_u	G_g	T_2g+G_g+H_g	T_1g+G_g+H_g	A_g+T_1g+T_2g+G_g+H_g	T_1g+T_2g+G_g+2H_g
H_u	H_u	T_1u+T_2u+G_u+H_u	T_1u+T_2u+G_u+H_u	T_1u+T_2u+G_u+2H_u	A_u+T_1u+T_2u+2G_u+2H_u	H_g	T_1g+T_2g+G_g+H_g	T_1g+T_2g+G_g+H_g	T_1g+T_2g+G_g+2H_g	A_g+T_1g+T_2g+2G_g+2H_g

Printable: character table for I_h product table for I_h I_h point group

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