Character Tables

Nonaxial Groups

These groups are characterized by a lack of a proper rotation axis.

\(C_1\)	E
A	1

\(C_s\)	E	σ_h
A'	1	1	x, y, R_z	x², y², z², xy
A"	1	-1	z, R_x, R_y	yz, xz

\(C_i\)	E	i
Ag	1	1	R_x, R_y, R_z	x², y², z², xy, yz, zx
A_u	1	-1	x,y,z

Cyclic \(C_n\) Groups

These groups are characterized by an n-fold proper rotation axis \(C_n\).

C₂	E	C₂
A	1	1	z, R_z	x², y², z², xy
B	1	-1	x, y, R_x, Ry	yz,xz

C₃	E	C₃	C₃²		ε=exp(2π/3)
A	1	1	1	z, R_z	x²+y², z²
E	\(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\left. \begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix} \right\}\)	(x,y), (R_x,R_y)	(x²-y², xy), (xz, yz)

C₄	E	C₄	C₂	C₄³
A	1	1	1	1	z, R_z	x²+y², z²
B	1	-1	1	-1		x²-y², xy
E	\(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\)	\(\begin{matrix} \;i \\ -i\end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\left. \begin{matrix} -i \\ \;i \end{matrix} \right\}\)	(x,y), (R_x,R_y)	(xz, yz)

C₅	E	C₅	C₅²	C₅³	C₅⁴		ε=exp(i2π/5)
A	1	1	1	1	1	Z, R_z	x²+y², z²
E₁	\(\left\{ \begin{matrix}\sf 1 \\ 1 \end{matrix} \right.\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} \epsilon^2\; \\ \epsilon^{*2} \end{matrix}\)	\(\begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix}\)	\(\left. \begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix} \right\}\)	(x, y), (R_x,R_y)	(xz, yz)
E₂	\(\left\{ \begin{matrix}\sf 1 \\ 1 \end{matrix} \right.\)	\(\begin{matrix} \epsilon^2\; \\ \epsilon^{*2} \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\left. \begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix} \right\}\)		(x²-y², xy)

C₆	E	C₆	C₃	C₂	C₃²	C₆⁵		ε=exp(i2π/6)
A	1	1	1	1	1	1	z, R_z	x²+y², z²
B	1	-1	1	-1	1	-1
E₁	\(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} -\epsilon^* \\ -\epsilon\;\end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\)	\(\left. \begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix} \right\}\)	(R_x,R_y), (x,y)	(xz, yz)
E₂	\(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\)	\(\begin{matrix} -\epsilon^*\; \\ -\epsilon\; \end{matrix}\)	\(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\)	\(\begin{matrix} 1 \\ 1 \end{matrix}\)	\(\begin{matrix} -\epsilon^*\; \\ -\epsilon\; \end{matrix}\)	\(\left. \begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix} \right\}\)		(x²-y², xy)

C₇	E	C₇	C₇²	C₇³	C₇⁴	C₇⁵	C₇⁶		ε=exp(i2π/7)
A	1	1	1	1	1	1	1	z, R_z	x²+y², z²
E₁	\(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} \epsilon^2\; \\ \epsilon^{*2} \end{matrix}\)	\(\begin{matrix} \epsilon^3\; \\ \epsilon^{*3} \end{matrix}\)	\(\begin{matrix} \epsilon^{*3}\; \\ \epsilon^3\; \end{matrix}\)	\(\begin{matrix} \epsilon^{*2}\; \\ \epsilon^2\; \end{matrix}\)	\(\left. \begin{matrix} \epsilon^{*} \\ \epsilon\; \end{matrix} \right\}\)	(R_x,R_y), (x,y)	(xz, yz)
E₂	\(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\)	\(\begin{matrix} \epsilon^2\; \\ \epsilon^{*2} \end{matrix}\)	\(\begin{matrix} \epsilon^{*3}\; \\ \epsilon^3\; \end{matrix}\)	\(\begin{matrix} \epsilon^{*} \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} \epsilon^3\; \\ \epsilon^{*3} \end{matrix}\)	\(\left. \begin{matrix} \epsilon^{*2}\; \\ \epsilon^2\; \end{matrix} \right\}\)		(x²-y², xy)
E₃	\(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\)	\(\begin{matrix} \epsilon^3\; \\ \epsilon^{*3} \end{matrix}\)	\(\begin{matrix} \epsilon^{*} \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} \epsilon^2\; \\ \epsilon^{*2} \end{matrix}\)	\(\begin{matrix} \epsilon^{*2}\; \\ \epsilon^2\; \end{matrix}\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\left. \begin{matrix} \epsilon^{*3}\; \\ \epsilon^3\; \end{matrix} \right\}\)

C₈	E	C₈	C₄	C₈³	C₂	C₈⁵	C₄³	C₈⁷		ε=exp(i2π/8)
A	1	1	1	1	1	1	1	1	z, R_z	x²+y², z²
B	1	-1	1	-1	1	-1	1	-1
E₁	\(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\)	\(\begin{matrix} \epsilon\; \\ \epsilon^*\end{matrix}\)	\(\begin{matrix} \;i \\ -i \end{matrix}\)	\(\begin{matrix} -\epsilon^* \\ -\epsilon\; \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\)	\(\begin{matrix} -i \\ \;i \end{matrix}\)	\(\left. \begin{matrix} \epsilon^{*} \\ \epsilon\; \end{matrix} \right\}\)	(R_x,R_y), (x,y)	(xz, yz)
E₂	\(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\)	\(\begin{matrix} \;i \\ -i \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} -i \\ \;i \end{matrix}\)	\(\begin{matrix} 1 \\ 1 \end{matrix}\)	\(\begin{matrix} \;i \\ -i \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\left. \begin{matrix} -i \\ \;i \end{matrix} \right\}\)		(x²-y², xy)
E₃	\(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\)	\(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\)	\(\begin{matrix} \;i \\ -i\; \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} -i \\ \;i \end{matrix}\)	\(\left. \begin{matrix} -\epsilon^* \\ -\epsilon\; \end{matrix} \right\}\)

Reflection \(C_{nh}\) Groups

These groups are characterized by an n-fold proper rotation axis \(C_n\) and a mirror plane \(\sigma_h\) normal to \(C_n\).

\(C_{2h}\)	E	C₂	i	σ_h
A_g	1	1	1	1	R_z	x², y², z²
B_g	1	-1	1	-1	R_x, R_y	xz, yz
A_u	1	1	-1	-1	z
B_u	1	-1	-1	1	x,y

\(C_{3h}\)	E	C₃	C₃²	σ_h	S₃	S₃⁵		ε=exp(i2π/3)
A'	1	1	1	1	1	1	R_z	x²+y², z²
E'	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} 1 \\ 1 \end{matrix}\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\left.\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\right\}\)	(x,y)	(x²-y², xy)
A"	1	1	1	-1	-1	-1	z
E"	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\)	\(\left.\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\right\}\)	(R_x, R_y)	(xz, yz)

\(C_{4h}\)	E	C₄	C₂	C₄³	i	S₄³	σ_h	S₄
A_g	1	1	1	1	1	1	1	1	R_z	x²+y², z²
B_g	1	-1	1	-1	1	-1	1	-1		x²-y², xy
E_g	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} \; i \\ -i \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} -i \\ \; i \end{matrix}\)	\(\begin{matrix} 1 \\ 1 \end{matrix}\)	\(\begin{matrix} \; i \\ -i \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\left.\begin{matrix} -i \\ \; i \end{matrix}\right\}\)	(R_x, R_y)	(xz, yz)
A_u	1	1	1	1	-1	-1	-1	-1	z
B_u	1	-1	1	-1	-1	1	-1	1
E_u	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} \; i \\ -i \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} -i \\ \; i \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} -i \\ \; i \end{matrix}\)	\(\begin{matrix} 1 \\ 1 \end{matrix}\)	\(\left.\begin{matrix} \; i \\ -i \end{matrix}\right\}\)	(x,y)

\(C_{5h}\)	E	C₅	C₅²	C₅³	C₅⁴	σ_h	S₅	S₅⁷	S₅³	S₅⁹		ε=exp(i2π/5)
A'	1	1	1	1	1	1	1	1	1	1	R_z	x²+y², z²
E₁'	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} \epsilon^{2}\; \\ \epsilon^{*2} \end{matrix}\)	\(\begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} 1 \\ 1 \end{matrix}\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} \epsilon^{2}\; \\ \epsilon^{*2} \end{matrix}\)	\(\begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix}\)	\(\left.\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\right\}\)	(x, y)
E₂'	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} \epsilon^{2}\; \\ \epsilon^{*2} \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix}\)	\(\begin{matrix} 1 \\ 1 \end{matrix}\)	\(\begin{matrix} \epsilon^{2}\; \\ \epsilon^{*2} \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\left.\begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix}\right\}\)		(x²-y², xy)
A"	1	1	1	1	1	-1	-1	-1	-1	-1	z
E₁"	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} \epsilon^{2}\; \\ \epsilon^{*2} \end{matrix}\)	\(\begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\)	\(\begin{matrix} -\epsilon^{2}\; \\ -\epsilon^{*2} \end{matrix}\)	\(\begin{matrix} -\epsilon^{*2} \\ -\epsilon^{2}\; \end{matrix}\)	\(\left.\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\right\}\)	(R_x, R_y)	(xz, yz)
E₂"	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} \epsilon^{2}\; \\ \epsilon^{*2} \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} -\epsilon^{2}\; \\ -\epsilon^{*2} \end{matrix}\)	\(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\)	\(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\)	\(\left.\begin{matrix} -\epsilon^{*2} \\ -\epsilon^{2}\; \end{matrix}\right\}\)

\(C_{6h}\)	E	C₆	C₃	C₂	C₃²	C₆⁵	i	S₃⁵	S₆⁵	σ_h	S₆	S₃		ε=exp(i2π/6)
A_g	1	1	1	1	1	1	1	1	1	1	1	1	R_z	x²+y², z²
B_g	1	-1	1	-1	1	-1	1	-1	1	-1	1	-1
E_1g	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} 1 \\ 1 \end{matrix}\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\)	\(\left.\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\right\}\)	(R_x, R_y)	(xz, yz)
E_2g	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\)	\(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\)	\(\begin{matrix} 1 \\ 1 \end{matrix}\)	\(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\)	\(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\)	\(\begin{matrix} 1 \\ 1 \end{matrix}\)	\(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\)	\(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\)	\(\begin{matrix} 1 \\ 1 \end{matrix}\)	\(\begin{matrix} -\epsilon^* \\ -\epsilon\; \end{matrix}\)	\(\left.\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\right\}\)		(x²-y², xy)
A_u	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	z
B_u	1	-1	1	-1	1	-1	-1	1	-1	1	-1	1
E_1u	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} 1 \\ 1 \end{matrix}\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\left.\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\right\}\)	(x, y)
E_2u	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\)	\(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\)	\(\begin{matrix} 1 \\ 1 \end{matrix}\)	\(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\)	\(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\left.\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\right\}\)

Pyramidal \(C_{nv}\) Groups

These groups are characterized by an n-fold proper rotation axis \(C_n\) and n mirror planes \(σ_v\) which contain \(C_n\)

\(C_{2v}\)	E	C₂	σ(xz)	σ(yz)
A₁	1	1	1	1	z	x², y², z²
A₂	1	1	-1	-1	R_z	xy
B₁	1	-1	1	-1	x, R_y	xz
B₂	1	-1	-1	1	y, R_x	yz

\(C_{3v}\)	E	2C₃	3σ_v
A₁	1	1	1	z	x²+y², z²
A₂	1	1	-1	R_z
E	2	-1	0	(R_x, R_y), (x,y)	(xz, yz) (x²-y², xy)

\(C_{4v}\)	E	2C₄	C₂	2σ_v	2σ_d
A₁	1	1	1	1	1	z	x²+y², z²
A₂	1	1	1	-1	-1	R_z
B₁	1	-1	1	1	-1		x²-y²
B₂	1	-1	1	-1	1		xy
E	2	0	-2	0	0	(R_x, R_y), (x,y)	(xz, yz)

\(C_{5v}\)	E	2C₅	2C₅²	5σ_v
A₁	1	1	1	1	z	x²+y², z²
A₂	1	1	1	-1	R_z
E₁	2	\(2\cos 72^\circ\)	\(2\cos 144^\circ\)	0	(R_x, R_y), (x,y)	(xz, yz)
E₂	2	\(2\cos{144^\circ}\)	\(2\cos 72^\circ\)	0		(x²-y², xy)

\(C_{6v}\)	E	2C₆	2C₃	C₂	3σ_v	3σ_d
A₁	1	1	1	1	1	1	z	x²+y², z²
A₂	1	1	1	1	-1	-1	R_z
B₁	1	-1	1	-1	1	-1
B₂	1	-1	1	-1	-1	1
E₁	2	1	-1	-2	0	0	(R_x, R_y), (x,y)	(xz, yz)
E₂	2	-1	-1	2	0	0		(x²-y², xy)

C_∞v	E	2C_∞	...	∞σ_v
A₁	1	1	...	1	z	x²+y², z²
A₂	1	1	...	-1	R_z
E₁	2	\(2\cos{\phi}\)	...	0	(x,y), (R_x, R_y)	(xz, yz)
E₂	2	\(2\cos{2\phi}\)	...	0		(x²-y², xy)
E₃	2	\(2\cos{3\phi}\)	...	0
...	...	...	...	...

Dihedral \(D_n\) Groups

\(D_2\)	E	C₂(z)	C₂(y)	C₂(x)
A	1	1	1	1		x², y², z²
B₁	1	1	-1	-1	z, R_z	xy
B₂	1	-1	1	-1	y, R_y	zx
B₃	1	-1	-1	1	x, R_x	yz

\(D_3\)	E	2C₃	3C₂
A₁	1	1	1		x²+y², z²
A₂	1	1	-1	z, R_z
E	2	-1	0	(R_x, R_y), (x,y)	(x²-y², xy) (xz, yz)

\(D_4\)	E	2C₄	C₂(C₄²)	2C₂'	2C₂"
A₁	1	1	1	1	1		x²+y², z²
A₂	1	1	1	-1	-1	z, R_z
B₁	1	-1	1	1	-1		x²-y²
B₂	1	-1	1	-1	1		xy
E	2	0	-2	0	0	(R_x, R_y), (x,y)	(xz, yz)

2C₅²

\(D_5\)	E	2C₅		5C₂
A₁	1	1	1	1		x²+y², z²
A₂	1	1	1	-1	z, R_z
E₁	2	\(2\cos{72^\circ}\)	\(2\cos{144^\circ}\)		(R_x, R_y), (x,y)	(xz, yz)
E₂	2	\(2\cos{144^\circ}\)	\(2\cos{72^\circ}\)			(x²-y², xy)

\(D_6\)	E	2C₆	2C₃	C₂	2C₂^'	3C₂^"
A₁	1	1	1	1	1	1		x²+y², z²
A₂	1	1	1	1	-1	-1	z, R_z
B₁	1	-1	1	-1	1	-1
B₂	1	-1	1	-1	-1	1
E₁	2	1	-1	-2	0	0	(R_x, R_y), (x,y)	(xz, yz)
E₂	2	-1	-1	2	0	0		(x²-y², xy)

Prismatic \(D_{nh}\) Groups

These groups are characterized by

an n-fold proper rotation axis \(C_n\)
n 2-fold proper rotation axes \(C_2\) normal to \(C_n\)
a mirror plane \(\sigma_h\) normal to \(C_n\) and containing the \(C_2\) axes.

\(D_{2h}\)	E	C₂(z)	C₂(y)	C₂(x)	i	σ(xy)	σ(xz)	σ(yz)
A_g	1	1	1	1	1	1	1	1		x², y², z²
B_1g	1	1	-1	-1	1	1	-1	-1	R_z	xy
B_2g	1	-1	1	-1	1	-1	1	-1	R_y	xz
B_3g	1	-1	-1	1	1	-1	-1	1	R_x	yz
A_u	1	1	1	1	-1	-1	-1	-1
B_1u	1	1	-1	-1	-1	-1	1	1	z
B_2u	1	-1	1	-1	-1	1	-1	1	y
B_3u	1	-1	-1	1	-1	1	1	-1	x

\(D_{3h}\)	E	2C₃	3C₂	σ_h	2S₃	3σ_v
A₁'	1	1	1	1	1	1		x²+y², z²
A₂'	1	1	-1	1	1	-1	R_z
E'	2	-1	0	2	-1	0	(x,y)	(x²-y², xy)
A₁"	1	1	1	-1	-1	-1
A₂"	1	1	-1	-1	-1	1	z
E"	2	-1	0	-2	1	0	(R_x, R_y)	(xz, yz)

\(D_{4h}\)	E	2C₄	C₂	2C₂'	2C₂"	i	2S₄	σ_h	2σ_v	σ_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	R_z
B_1g	1	-1	1	1	-1	1	-1	1	1	-1		x²-y²
B₂_g	1	-1	1	-1	1	1	-1	1	-1	1		xy
E_g	2	0	-2	0	0	2	0	-2	0	0	(R_x, R_y)	(xz, yz)
A_1u	1	1	1	1	1	-1	-1	-1	-1	-1
A_2u	1	1	1	-1	-1	-1	-1	-1	1	1	z
B_1u	1	-1	1	1	-1	-1	1	-1	-1	1
B_2u	1	-1	1	-1	1	-1	1	-1	1	-1
E_u	2	0	-2	0	0	-2	0	2	0	0	(x,y)

\(D_{5h}\)	E	2C₅	2C₅²	5C₂	σ_h	2S₅	2S₅³	5σ_v
A₁'	1	1	1	1	1	1	1	1		x²+y², z²
A₂'	1	1	1	-1	1	1	1	-1	R_z
E₁'	2	\(2\cos{72^\circ}\)	\(2\cos{144^\circ}\)	0	2	\(2\cos{72^\circ}\)	\(2\cos{144^\circ}\)		(x,y)
E₂'	2	\(2\cos{144^\circ}\)	\(2\cos{72^\circ}\)	0	2	\(2\cos{144^\circ}\)	\(2\cos{72^\circ}\)			(x²-y², xy)
A₁"	1	1	1	1	-1	-1	-1	-1
A₂"	1	1	1	-1	-1	-1	-1	1	z
E₁"	2	\(2\cos{72^\circ}\)	\(2\cos{144^\circ}\)	0	-2	\(-2\cos{72^\circ}\)	\(-2\cos{144^\circ}\)	0	(R_x, R_y)	(xz, yz)
E₂"	2	\(2\cos{144^\circ}\)	\(2\cos{72^\circ}\)	0	-2	\(-2\cos{144^\circ}\)	\(-2\cos{72^\circ}\)	0

D₆_h	E	2C₆	2C₃	C₂	3C₂'	3C₂"	i	2S₃	2S₆	σ_h	3σ_d	3σ_v
A_1g	1	1	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	-1	-1	R_z
B_1g	1	-1	1	-1	1	-1	1	-1	1	-1	1	-1
B_2g	1	-1	1	-1	-1	1	1	-1	1	-1	-1	1
E_1g	2	1	-1	-2	0	0	2	1	-1	-2	0	0	(R_x, R_y)	(xz, yz)
E_2g	2	-1	-1	2	0	0	2	-1	-1	2	0	0		(x²-y², xy)
A_1u	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1
A_2u	1	1	1	1	-1	-1	-1	-1	-1	-1	1	1	z
B_1u	1	-1	1	-1	1	-1	-1	1	-1	1	-1	1
B_2u	1	-1	1	-1	-1	1	-1	1	-1	1	1	-1
E_1u	2	1	-1	-2	0	0	-2	-1	1	2	0	0	(x,y)
E_2u	2	-1	-1	2	0	0	-2	1	1	-2	0	0

D_∞h	E	2C_∞	...	∞σ_v	i	2S_∞	...	∞ C₂
S_g⁺	1	1	...	1	1	1	...	1		x²+y², z²
S_g^-	1	1	...	-1	1	1	...	-1	R_z
π_g	2	\(2\cos{\phi}\)	...	0	2	\(-2\cos{\phi}\)	...	0	(R_x, R_y)	(xz, yz)
D_g	2	\(2\cos{2\phi}\)	...	0	2	\(2\cos{2\phi}\)	...	0		(x²-y², xy)
...	...	...	...	......	...	...	...	...
S_u⁺	1	1	...	1	-1	-1	...	-1	z
S_u^-	1	1	...	-1	-1	-1	...	1
π_u	2	\(2\cos{\phi}\)	...	0	-2	\(2\cos{\phi}\)	...	0	(x, y)
D_u	2	\(2\cos{2\phi}\)	...	0	-2	\(-2\cos{2\phi}\)	...	0
...	...	...	...	...	...	...	...	...

Antiprismatic \(D_{nd}\) Groups

These groups are characterized by

an n-fold proper rotation axis C_n
n 2-fold proper rotation axes C₂ normal to C_n
n mirror planes σ_d which contain C_n.

D₂d	E	2S₄	C₂	2C₂'	2σ_d
A₁	1	1	1	1	1		x²+y², z²
A₂	1	1	1	-1	-1	R_z
B₁	1	-1	1	1	-1		x²-y²
B₂	1	-1	1	-1	1	z	xy
E	2	0	-2	0	0	(x, y), (R_x, R_y)	(xz, yz)

D₃d	E	2C₃	3C₂	i	2S₆	3σ_d
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)

D₄d	E	2S₈	2C₄	2S₈³	C₂	4C₂'	4σ_d
A₁	1	1	1	1	1	1	1		x²+y², z²
A₂	1	1	1	1	1	-1	-1	R_z
B₁	1	-1	1	-1	1	1	-1
B₂	1	-1	1	-1	1	-1	1	z
E₁	2	\(\sqrt{2}\)	0	\(-\sqrt{2}\)	-2	0	0	(x, y)
E₂	2	0	-2	0	2	0	0		(x²-y², xy)
E₃	2	\(-\sqrt{2}\)	0	\(\sqrt{2}\)	-2	0	0	(R_x, R_y)	(xz, yz)

D₅d	E	2C₅	2C₅²	5C₂	i	2S₁₀³	2S₁₀	5σ_d
A_1g	1	1	1	1	1	1	1	1		x²+y², z²
A_2g	1	1	1	-1	1	1	1	-1	R_z
E_1g	2	\(2\cos 72^\circ\)	\(2\cos 144^\circ\)	0	2	\(2\cos 72^\circ\)	\(2\cos 144^\circ\)	0	(R_x, R_y)	(xz, yz)
E_2g	2	\(2\cos 144^\circ\)	\(2\cos 72^\circ\)	0	2	\(2\cos 144^\circ\)	\(2\cos 72^\circ\)	0		(x²-y², xy)
A_1u	1	1	1	1	-1	-1	-1	-1
A_2u	1	1	1	-1	-1	1	-1	1	z
E_1u	2	\(2\cos 72^\circ\)	\(2\cos 144^\circ\)	0	-2	\(-2\cos 72^\circ\)	\(-2\cos 144^\circ\)	0	(x, y)
E_2u	2	\(2\cos 144^\circ\)	\(2\cos 72^\circ\)	0	-2	\(-2\cos 144^\circ\)	\(-2\cos 72^\circ\)	0

D₆d	E	2S₁₂	2C₆	2S₄	2C₃	2S₁₂⁵	C₂	6C₂_^'	6σ_d
A₁	1	1	1	1	1	1	1	1	1		x²+y², z²
A₂	1	1	1	1	1	1	1	-1	-1	R_z
B₁	1	-1	1	-1	1	-1	1	1	-1
B₂	1	-1	1	-1	1	-1	1	-1	1	z
E₁	2	\(\sqrt{3}\)	1	0	-1	\(-\sqrt{3}\)	-2	0	0	(x, y)
E₂	2	1	-1	-2	-1	1	2	0	0		(x²-y², xy)
E₃	2	0	-2	0	2	0	-2	0	0
E₄	2	-1	-1	2	-1	-1	2	0	0
E₅	2	\(-\sqrt{3}\)	1	0	-1	\(\sqrt{3}\)	-2	0	0	(R_x, R_y)	(xz, yz)

Improper Rotation \(S_n\) Groups

These groups are characterized by an n-fold improper rotation axis \(S_n\), where \(n\) is necessarily even

\(S_4\)	E	S₄	C₂	S₄³
A	1	1	1	1	R_z	x²+y², z²
B	1	-1	1	-1	z	x²-y², xy
E	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} \; i \\ -i \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\left.\begin{matrix} -i \\ \; i \end{matrix}\right\}\)	(x, y); (R_x, R_y)	(xz, yz)

S₆	E	C₃	C₃²	i	S₆⁵	S₆
A_g	1	1	1	1	1	1	R_z	x²+y², z²
E_g	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} 1 \\ 1 \end{matrix}\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\left.\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\right\}\)	(R_x, R_y)	(x²-y², xy), (xz, yz)
A_u	1	1	1	-1	-1	-1	z
E_u	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\)	\(\left.\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\right\}\)	(x, y)

S₈	E	S₈	C₄	S₈³	C₂	S₈⁵	C₄³	S₈⁷		ε=exp(i2π/8)
A	1	1	1	1	1	1	1	1	R_z	x²+y², z²
B	1	-1	1	-1	1	-1	1	-1	z
E₁	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} \; i \\ -i \end{matrix}\)	\(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\)	\(\begin{matrix} -i \\ \; i \end{matrix}\)	\(\left.\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\right\}\)	(R_x, R_y), (x, y)
E₂	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} \; i \\ -i \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} -i \\ \; i \end{matrix}\)	\(\begin{matrix} 1 \\ 1 \end{matrix}\)	\(\begin{matrix} \; i \\ -i \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\left.\begin{matrix} -i \\ \; i \end{matrix}\right\}\)		(x²-y², xy)
E₃	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\)	\(\begin{matrix} -i \\ \; i \end{matrix}\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} \; i \\ -i \end{matrix}\)	\(\left.\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\right\}\)		(xz, yz)

Cubic Groups

These polyhedral groups are characterized by not having a \(C_5\) proper rotation axis.

\(T\)	E	4C₃	4C₃²	3C₂
A	1	1	1	1		x²+y²+z²
E	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\left.\begin{matrix} 1 \\ 1 \end{matrix}\right\}\)		(2z²-x²-y², x²-y²)
T	3	0	0		(R_x, R_y, R_z), (x, y, z)	(xz, yz, xy)

T_h	E	4C₃	4C₃²	3C₂	i	4S₆	4S₆⁵	3σ_h		ε=exp(i2π/3)
A_g	1	1	1	1	1	1	1	1		x²+y²+z²
E_g	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} 1 \\ 1 \end{matrix}\)	\(\begin{matrix} 1 \\ 1 \end{matrix}\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\left.\begin{matrix} 1 \\ 1 \end{matrix}\right\}\)		(2z²-x²-y², x²-y²)
T_g	3	0	0	-1	1	0	0	-1	(R_x, R_y, R_z)	(xz, yz, xy)
A_u	1	1	1	1	-1	-1	-1	-1
E_u	\(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\)	\(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\)	\(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\)	\(\begin{matrix} 1 \\ 1 \end{matrix}\)	\(\begin{matrix} -1 \\ -1 \end{matrix}\)	\(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\)	\(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\)	\(\left.\begin{matrix} -1 \\ -1 \end{matrix}\right\}\)
T_u	3	0	0	-1	-1	0	0	1	(x, y, z)

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)	(xz, yz, xy)

O	E	8C₃	3C₂	6C₄	6C₂
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), (x, y, z)
T₂	3	0	-1	-1	1		(xz, yz, xy)

O_h	E	8C₂	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-x²-y², 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