Character Tables for Common Symmetry Groups - The Cubic Groups

Chemistry

Character Tables for Common Symmetry Groups - The Cubic Groups

T	E	4C₃	4C₃²	3C₂		ε = exp(2π/3)
A	1	1	1	1		x² + y² + z²
E	{ 1 ε ε⁺ 1 } 1 ε⁺ ε 1					(2z² - x² - y², x² - y²)
T	3	0	0	-1	(R_x, R_y, R_z); (x, y, z)	(xy, xz, yz)

T_d	E	8C₃	3C₂	6S₄	6σ_d
A₁	1	1	1	1	1		x² + y² + z²
A₂	1	1	1	-1	-1
E	2	-1	2	0	0		(2z² - x² - y², x² - y²)
T₁	3	0	-1	1	-1	(R_x, R_y, R_z)
T₂	3	0	-1	-1	1	(x, y, z)	(xy, xz, yz)

O_h	E	8C₃	6C₂	6C₄	3C₂(= C₄²)	i	6S₄	8S₆	3σ_h	6σ_d
A_1g	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
E_g	2	-1	0	0	2	2	0	-1	2	0		(2z² - x²y², x² - y²)
T_1g	3	0	-1	1	-1	3	1	0	-1	-1	(R_x, R_y, R_z)
T_2g	3	0	1	-1	-1	3	-1	0	-1	1		(xz, yz, xy)
A_1u	1	1	1	1	1	-1	-1	-1	-1	-1
A_2u	1	1	-1	-1	1	-1	1	-1	-1	1
E_u	2	-1	0	0	2	-2	0	1	-2	0
T_1u	3	0	-1	1	-1	-3	-1	0	1	1	(x, y, z)
T_2u	3	0	1	-1	-1	-3	1	0	1	-1

Reference: F.A. Cotton, "Chemical Applications of Group Theory," 3rd ed., Wiley-Interscience, NY, 1990.