d3d - Point Group Symmetry Character Tables - Chemistry Online Education

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Linear
C_∞v
D_∞h

D_3d Point Group

not Abelian, 6(8) irreducible representations

Subgroups of D_3d point group: C_s, C_i, C₂, C₃, D₃, C_3v, S₆

Character table for D_3d point group

	E	2C₃	3C'₂	i	2S₆	3σ_d	linear, rotations	quadratic
A_1g	1	1	1	1	1	1		x²+y², z²
A_2g	1	1	-1	1	1	-1	R_z
E_g	2	-1	0	2	-1	0	(R_x, R_y)	(x²-y², xy) (xz, yz)
A_1u	1	1	1	-1	-1	-1
A_2u	1	1	-1	-1	-1	1	z
E_u	2	-1	0	-2	1	0	(x, y)

Product table for D_3d point group

	A_1g	A_2g	E_g	A_1u	A_2u	E_u
A_1g	A_1g	A_2g	E_g	A_1u	A_2u	E_u
A_2g	A_2g	A_1g	E_g	A_2u	A_1u	E_u
E_g	E_g	E_g	A_1g+A_2g+E_g	E_u	E_u	A_1u+A_2u+E_u
A_1u	A_1u	A_2u	E_u	A_1g	A_2g	E_g
A_2u	A_2u	A_1u	E_u	A_2g	A_1g	E_g
E_u	E_u	E_u	A_1u+A_2u+E_u	E_g	E_g	A_1g+A_2g+E_g

Printable: character table for D_3d product table for D_3d D_3d point group