Character Tables

 

Nonaxial Groups

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

$C_1$	E
A	1

 

$C_s$	E	σh
A'	1	1	x, y, Rz	x^2, y^2, z^2, xy
A"	1	-1	z, Rx, Ry	yz, xz

 

$C_i$	E	i
Ag	1	1	Rx, Ry, Rz	x^2, y^2, z^2, xy, yz, zx
Au	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, Rz	x^2, y^2, z^2, xy
B	1	-1	x, y, Rx, Ry	yz,xz

 

C3	E	C3	C3^2		ε=exp(2π/3)
A	1	1	1	z, Rz	x^2+y^2, z^2
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), (Rx,Ry)	(x^2-y^2, xy), (xz, yz)

 

C4	E	C4	C2	C4^3
A	1	1	1	1	z, Rz	x^2+y^2, z^2
B	1	-1	1	-1		x^2-y^2, 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), (Rx,Ry)	(xz, yz)

 

C5	E	C5	C5^2	C5^3	C5^4		ε=exp(i2π/5)
A	1	1	1	1	1	Z, Rz	x^2+y^2, z^2
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), (Rx,Ry)	(xz, yz)
E2	$\left\{ \begin{matrix}\sf 1 \\ 1 \end{matrix} \right.$	$\begin{matrix} \epsilon^2\; \\ \epsilon^{*2} \end{matrix}$	$\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\left. \begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix} \right\}$		(x^2-y^2, xy)

 

C6	E	C6	C3	C2	C3^2	C6^5		ε=exp(i2π/6)
A	1	1	1	1	1	1	z, Rz	x^2+y^2, z^2
B	1	-1	1	-1	1	-1
E1	$\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\begin{matrix} -\epsilon^* \\ -\epsilon\;\end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}$	$\left. \begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix} \right\}$	(Rx,Ry), (x,y)	(xz, yz)
E2	$\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.$	$\begin{matrix} -\epsilon^*\; \\ -\epsilon\; \end{matrix}$	$\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}$	$\begin{matrix} 1 \\ 1 \end{matrix}$	$\begin{matrix} -\epsilon^*\; \\ -\epsilon\; \end{matrix}$	$\left. \begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix} \right\}$		(x^2-y^2, xy)

 

C7	E	C7	C7^2	C7^3	C7^4	C7^5	C7^6		ε=exp(i2π/7)
A	1	1	1	1	1	1	1	z, Rz	x^2+y^2, z^2
E1	$\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\begin{matrix} \epsilon^2\; \\ \epsilon^{*2} \end{matrix}$	$\begin{matrix} \epsilon^3\; \\ \epsilon^{*3} \end{matrix}$	$\begin{matrix} \epsilon^{*3}\; \\ \epsilon^3\; \end{matrix}$	$\begin{matrix} \epsilon^{*2}\; \\ \epsilon^2\; \end{matrix}$	$\left. \begin{matrix} \epsilon^{*} \\ \epsilon\; \end{matrix} \right\}$	(Rx,Ry), (x,y)	(xz, yz)
E2	$\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.$	$\begin{matrix} \epsilon^2\; \\ \epsilon^{*2} \end{matrix}$	$\begin{matrix} \epsilon^{*3}\; \\ \epsilon^3\; \end{matrix}$	$\begin{matrix} \epsilon^{*} \\ \epsilon\; \end{matrix}$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\begin{matrix} \epsilon^3\; \\ \epsilon^{*3} \end{matrix}$	$\left. \begin{matrix} \epsilon^{*2}\; \\ \epsilon^2\; \end{matrix} \right\}$		(x^2-y^2, 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	C8^3	C2	C8^5	C4^3	C8^7		ε=exp(i2π/8)
A	1	1	1	1	1	1	1	1	z, Rz	x^2+y^2, z^2
B	1	-1	1	-1	1	-1	1	-1
E1	$\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.$	$\begin{matrix} \epsilon\; \\ \epsilon^*\end{matrix}$	$\begin{matrix} \;i \\ -i \end{matrix}$	$\begin{matrix} -\epsilon^* \\ -\epsilon\; \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}$	$\begin{matrix} -i \\ \;i \end{matrix}$	$\left. \begin{matrix} \epsilon^{*} \\ \epsilon\; \end{matrix} \right\}$	(Rx,Ry), (x,y)	(xz, yz)
E2	$\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.$	$\begin{matrix} \;i \\ -i \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\begin{matrix} -i \\ \;i \end{matrix}$	$\begin{matrix} 1 \\ 1 \end{matrix}$	$\begin{matrix} \;i \\ -i \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\left. \begin{matrix} -i \\ \;i \end{matrix} \right\}$		(x^2-y^2, 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	σh
Ag	1	1	1	1	Rz	x^2, y^2, z^2
Bg	1	-1	1	-1	Rx, Ry	xz, yz
Au	1	1	-1	-1	z
Bu	1	-1	-1	1	x,y

 

$C_{3h}$	E	C3	C3^2	σh	S3	S3^5		ε=exp(i2π/3)
A'	1	1	1	1	1	1	Rz	x^2+y^2, z^2
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)	(x^2-y^2, xy)
A"	1	1	1	-1	-1	-1	z
E"	$\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}$	$\left.\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\right\}$	(Rx, Ry)	(xz, yz)

 

$C_{4h}$	E	C4	C2	C4^3	i	S4^3	σh	S4
Ag	1	1	1	1	1	1	1	1	Rz	x^2+y^2, z^2
Bg	1	-1	1	-1	1	-1	1	-1		x^2-y^2, xy
Eg	$\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.$	$\begin{matrix} \; i \\ -i \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\begin{matrix} -i \\ \; i \end{matrix}$	$\begin{matrix} 1 \\ 1 \end{matrix}$	$\begin{matrix} \; i \\ -i \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\left.\begin{matrix} -i \\ \; i \end{matrix}\right\}$	(Rx, Ry)	(xz, yz)
Au	1	1	1	1	-1	-1	-1	-1	z
Bu	1	-1	1	-1	-1	1	-1	1
Eu	$\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.$	$\begin{matrix} \; i \\ -i \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\begin{matrix} -i \\ \; i \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\begin{matrix} -i \\ \; i \end{matrix}$	$\begin{matrix} 1 \\ 1 \end{matrix}$	$\left.\begin{matrix} \; i \\ -i \end{matrix}\right\}$	(x,y)

 

$C_{5h}$	E	C5	C5^2	C5^3	C5^4	σh	S5	S5^7	S5^3	S5^9		ε=exp(i2π/5)
A'	1	1	1	1	1	1	1	1	1	1	Rz	x^2+y^2, z^2
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'	$\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.$	$\begin{matrix} \epsilon^{2}\; \\ \epsilon^{*2} \end{matrix}$	$\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix}$	$\begin{matrix} 1 \\ 1 \end{matrix}$	$\begin{matrix} \epsilon^{2}\; \\ \epsilon^{*2} \end{matrix}$	$\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\left.\begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix}\right\}$		(x^2-y^2, xy)
A"	1	1	1	1	1	-1	-1	-1	-1	-1	z
E1"	$\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\begin{matrix} \epsilon^{2}\; \\ \epsilon^{*2} \end{matrix}$	$\begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix}$	$\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}$	$\begin{matrix} -\epsilon^{2}\; \\ -\epsilon^{*2} \end{matrix}$	$\begin{matrix} -\epsilon^{*2} \\ -\epsilon^{2}\; \end{matrix}$	$\left.\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\right\}$	(Rx, Ry)	(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	C3^2	C6^5	i	S3^5	S6^5	σh	S6	S3		ε=exp(i2π/6)
Ag	1	1	1	1	1	1	1	1	1	1	1	1	Rz	x^2+y^2, z^2
Bg	1	-1	1	-1	1	-1	1	-1	1	-1	1	-1
E1g	$\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}$	$\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}$	$\begin{matrix} 1 \\ 1 \end{matrix}$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}$	$\left.\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\right\}$	(Rx, Ry)	(xz, yz)
E2g	$\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.$	$\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}$	$\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}$	$\begin{matrix} 1 \\ 1 \end{matrix}$	$\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}$	$\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}$	$\begin{matrix} 1 \\ 1 \end{matrix}$	$\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}$	$\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}$	$\begin{matrix} 1 \\ 1 \end{matrix}$	$\begin{matrix} -\epsilon^* \\ -\epsilon\; \end{matrix}$	$\left.\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\right\}$		(x^2-y^2, xy)
Au	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	z
Bu	1	-1	1	-1	1	-1	-1	1	-1	1	-1	1
E1u	$\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}$	$\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}$	$\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}$	$\begin{matrix} 1 \\ 1 \end{matrix}$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\left.\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\right\}$	(x, y)
E2u	$\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.$	$\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}$	$\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}$	$\begin{matrix} 1 \\ 1 \end{matrix}$	$\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}$	$\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}$	$\left.\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\right\}$

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	x^2, y^2, z^2
A2	1	1	-1	-1	Rz	xy
B1	1	-1	1	-1	x, Ry	xz
B2	1	-1	-1	1	y, Rx	yz

 

$C_{3v}$	E	2C3	3σv
A1	1	1	1	z	x^2+y^2, z^2
A2	1	1	-1	Rz
E	2	-1	0	(Rx, Ry), (x,y)	(xz, yz) (x^2-y^2, xy)

 

$C_{4v}$	E	2C4	C2	2σv	2σd
A1	1	1	1	1	1	z	x^2+y^2, z^2
A2	1	1	1	-1	-1	Rz
B1	1	-1	1	1	-1		x^2-y^2
B2	1	-1	1	-1	1		xy
E	2	0	-2	0	0	(Rx, Ry), (x,y)	(xz, yz)

 

$C_{5v}$	E	2C5	2C5^2	5σv
A1	1	1	1	1	z	x^2+y^2, z^2
A2	1	1	1	-1	Rz
E1	2	$2\cos 72^\circ$	$2\cos 144^\circ$	0	(Rx, Ry), (x,y)	(xz, yz)
E2	2	$2\cos{144^\circ}$	$2\cos 72^\circ$	0		(x^2-y^2, xy)

 

$C_{6v}$	E	2C6	2C3	C2	3σv	3σd
A1	1	1	1	1	1	1	z	x^2+y^2, z^2
A2	1	1	1	1	-1	-1	Rz
B1	1	-1	1	-1	1	-1
B2	1	-1	1	-1	-1	1
E1	2	1	-1	-2	0	0	(Rx, Ry), (x,y)	(xz, yz)
E2	2	-1	-1	2	0	0		(x^2-y^2, xy)

 

C∞v	E	2C∞	...	∞σv
A1	1	1	...	1	z	x^2+y^2, z^2
A2	1	1	...	-1	Rz
E1	2	$2\cos{\phi}$	...	0	(x,y), (Rx, Ry)	(xz, yz)
E2	2	$2\cos{2\phi}$	...	0		(x^2-y^2, 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		x^2, y^2, z^2
B1	1	1	-1	-1	z, Rz	xy
B2	1	-1	1	-1	y, Ry	zx
B3	1	-1	-1	1	x, Rx	yz

 

$D_3$	E	2C3	3C2
A1	1	1	1		x^2+y^2, z^2
A2	1	1	-1	z, Rz
E	2	-1	0	(Rx, Ry), (x,y)	(x^2-y^2, xy) (xz, yz)

 

$D_4$	E	2C4	C2(C4^2)	2C2'	2C2"
A1	1	1	1	1	1		x^2+y^2, z^2
A2	1	1	1	-1	-1	z, Rz
B1	1	-1	1	1	-1		x^2-y^2
B2	1	-1	1	-1	1		xy
E	2	0	-2	0	0	(Rx, Ry), (x,y)	(xz, yz)

 

2C₅²

$D_5$	E	2C5		5C2
A1	1	1	1	1		x^2+y^2, z^2
A2	1	1	1	-1	z, Rz
E1	2	$2\cos{72^\circ}$	$2\cos{144^\circ}$		(Rx, Ry), (x,y)	(xz, yz)
E2	2	$2\cos{144^\circ}$	$2\cos{72^\circ}$			(x^2-y^2, xy)

 

$D_6$	E	2C6	2C3	C2	2C2^'	3C2^"
A1	1	1	1	1	1	1		x^2+y^2, z^2
A2	1	1	1	1	-1	-1	z, Rz
B1	1	-1	1	-1	1	-1
B2	1	-1	1	-1	-1	1
E1	2	1	-1	-2	0	0	(Rx, Ry), (x,y)	(xz, yz)
E2	2	-1	-1	2	0	0		(x^2-y^2, xy)

Prismatic $D_{nh}$ Groups

These groups are characterized by

1. an n-fold proper rotation axis $C_n$
2. n 2-fold proper rotation axes $C_2$ normal to $C_n$
3. 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)
Ag	1	1	1	1	1	1	1	1		x^2, y^2, z^2
B1g	1	1	-1	-1	1	1	-1	-1	Rz	xy
B2g	1	-1	1	-1	1	-1	1	-1	Ry	xz
B3g	1	-1	-1	1	1	-1	-1	1	Rx	yz
Au	1	1	1	1	-1	-1	-1	-1
B1u	1	1	-1	-1	-1	-1	1	1	z
B2u	1	-1	1	-1	-1	1	-1	1	y
B3u	1	-1	-1	1	-1	1	1	-1	x

 

$D_{3h}$	E	2C3	3C2	σh	2S3	3σv
A1'	1	1	1	1	1	1		x^2+y^2, z^2
A2'	1	1	-1	1	1	-1	Rz
E'	2	-1	0	2	-1	0	(x,y)	(x^2-y^2, xy)
A1"	1	1	1	-1	-1	-1
A2"	1	1	-1	-1	-1	1	z
E"	2	-1	0	-2	1	0	(Rx, Ry)	(xz, yz)

 

$D_{4h}$	E	2C4	C2	2C2'	2C2"	i	2S4	σh	2σv	σd
A1g	1	1	1	1	1	1	1	1	1	1		x^2+y^2, z^2
A2g	1	1	1	-1	-1	1	1	1	-1	-1	Rz
B1g	1	-1	1	1	-1	1	-1	1	1	-1		x^2-y^2
B2g	1	-1	1	-1	1	1	-1	1	-1	1		xy
Eg	2	0	-2	0	0	2	0	-2	0	0	(Rx, Ry)	(xz, yz)
A1u	1	1	1	1	1	-1	-1	-1	-1	-1
A2u	1	1	1	-1	-1	-1	-1	-1	1	1	z
B1u	1	-1	1	1	-1	-1	1	-1	-1	1
B2u	1	-1	1	-1	1	-1	1	-1	1	-1
Eu	2	0	-2	0	0	-2	0	2	0	0	(x,y)

 

$D_{5h}$	E	2C5	2C5^2	5C2	σh	2S5	2S5^3	5σv
A1'	1	1	1	1	1	1	1	1		x^2+y^2, z^2
A2'	1	1	1	-1	1	1	1	-1	Rz
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	$2\cos{144^\circ}$	$2\cos{72^\circ}$			(x^2-y^2, 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	(Rx, Ry)	(xz, yz)
E2"	2	$2\cos{144^\circ}$	$2\cos{72^\circ}$	0	-2	$-2\cos{144^\circ}$	$-2\cos{72^\circ}$	0

 

D6h	E	2C6	2C3	C2	3C2'	3C2"	i	2S3	2S6	σh	3σd	3σv
A1g	1	1	1	1	1	1	1	1	1	1	1	1		x^2+y^2, z^2
A2g	1	1	1	1	-1	-1	1	1	1	1	-1	-1	Rz
B1g	1	-1	1	-1	1	-1	1	-1	1	-1	1	-1
B2g	1	-1	1	-1	-1	1	1	-1	1	-1	-1	1
E1g	2	1	-1	-2	0	0	2	1	-1	-2	0	0	(Rx, Ry)	(xz, yz)
E2g	2	-1	-1	2	0	0	2	-1	-1	2	0	0		(x^2-y^2, xy)
A1u	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1
A2u	1	1	1	1	-1	-1	-1	-1	-1	-1	1	1	z
B1u	1	-1	1	-1	1	-1	-1	1	-1	1	-1	1
B2u	1	-1	1	-1	-1	1	-1	1	-1	1	1	-1
E1u	2	1	-1	-2	0	0	-2	-1	1	2	0	0	(x,y)
E2u	2	-1	-1	2	0	0	-2	1	1	-2	0	0

 

D∞h	E	2C∞	...	∞σv	i	2S∞	...	∞ C2
Sg^+	1	1	...	1	1	1	...	1		x^2+y^2, z^2
Sg^-	1	1	...	-1	1	1	...	-1	Rz
πg	2	$2\cos{\phi}$	...	0	2	$-2\cos{\phi}$	...	0	(Rx, Ry)	(xz, yz)
Dg	2	$2\cos{2\phi}$	...	0	2	$2\cos{2\phi}$	...	0		(x^2-y^2, xy)
...	...	...	...	......	...	...	...	...
Su^+	1	1	...	1	-1	-1	...	-1	z
Su^-	1	1	...	-1	-1	-1	...	1
πu	2	$2\cos{\phi}$	...	0	-2	$2\cos{\phi}$	...	0	(x, y)
Du	2	$2\cos{2\phi}$	...	0	-2	$-2\cos{2\phi}$	...	0
...	...	...	...	...	...	...	...	...

Antiprismatic $D_{nd}$ Groups

These groups are characterized by

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

D2d	E	2S4	C2	2C2'	2σd
A1	1	1	1	1	1		x^2+y^2, z^2
A2	1	1	1	-1	-1	Rz
B1	1	-1	1	1	-1		x^2-y^2
B2	1	-1	1	-1	1	z	xy
E	2	0	-2	0	0	(x, y), (Rx, Ry)	(xz, yz)

 

D3d	E	2C3	3C2	i	2S6	3σd
A1g	1	1	1	1	1	1		x^2+y^2, z^2
A2g	1	1	-1	1	1	-1	Rz
Eg	2	-1	0	2	-1	0	(Rx, Ry)	(x^2-y^2, xy),(xz, yz)
A1u	1	1	1	-1	-1	-1
A2u	1	1	-1	-1	-1	1	z
Eu	2	-1	0	-2	1	0	(x, y)

 

D4d	E	2S8	2C4	2S8^3	C2	4C2'	4σd
A1	1	1	1	1	1	1	1		x^2+y^2, z^2
A2	1	1	1	1	1	-1	-1	Rz
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		(x^2-y^2, xy)
E3	2	$-\sqrt{2}$	0	$\sqrt{2}$	-2	0	0	(Rx, Ry)	(xz, yz)

 

D5d	E	2C5	2C5^2	5C2	i	2S10^3	2S10	5σd
A1g	1	1	1	1	1	1	1	1		x^2+y^2, z^2
A2g	1	1	1	-1	1	1	1	-1	Rz
E1g	2	$2\cos 72^\circ$	$2\cos 144^\circ$	0	2	$2\cos 72^\circ$	$2\cos 144^\circ$	0	(Rx, Ry)	(xz, yz)
E2g	2	$2\cos 144^\circ$	$2\cos 72^\circ$	0	2	$2\cos 144^\circ$	$2\cos 72^\circ$	0		(x^2-y^2, xy)
A1u	1	1	1	1	-1	-1	-1	-1
A2u	1	1	1	-1	-1	1	-1	1	z
E1u	2	$2\cos 72^\circ$	$2\cos 144^\circ$	0	-2	$-2\cos 72^\circ$	$-2\cos 144^\circ$	0	(x, y)
E2u	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	2S12^5	C2	6C2^'	6σd
A1	1	1	1	1	1	1	1	1	1		x^2+y^2, z^2
A2	1	1	1	1	1	1	1	-1	-1	Rz
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		(x^2-y^2, 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	(Rx, Ry)	(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	S4^3
A	1	1	1	1	Rz	x^2+y^2, z^2
B	1	-1	1	-1	z	x^2-y^2, 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); (Rx, Ry)	(xz, yz)

 

S6	E	C3	C3^2	i	S6^5	S6
Ag	1	1	1	1	1	1	Rz	x^2+y^2, z^2
Eg	$\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}$	$\begin{matrix} 1 \\ 1 \end{matrix}$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\left.\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\right\}$	(Rx, Ry)	(x^2-y^2, xy), (xz, yz)
Au	1	1	1	-1	-1	-1	z
Eu	$\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}$	$\left.\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\right\}$	(x, y)

 

S8	E	S8	C4	S8^3	C2	S8^5	C4^3	S8^7		ε=exp(i2π/8)
A	1	1	1	1	1	1	1	1	Rz	x^2+y^2, z^2
B	1	-1	1	-1	1	-1	1	-1	z
E1	$\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\begin{matrix} \; i \\ -i \end{matrix}$	$\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}$	$\begin{matrix} -i \\ \; i \end{matrix}$	$\left.\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\right\}$	(Rx, Ry), (x, y)
E2	$\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.$	$\begin{matrix} \; i \\ -i \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\begin{matrix} -i \\ \; i \end{matrix}$	$\begin{matrix} 1 \\ 1 \end{matrix}$	$\begin{matrix} \; i \\ -i \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\left.\begin{matrix} -i \\ \; i \end{matrix}\right\}$		(x^2-y^2, 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	4C3^2	3C2
A	1	1	1	1		x^2+y^2+z^2
E	$\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}$	$\left.\begin{matrix} 1 \\ 1 \end{matrix}\right\}$		(2z^2-x^2-y^2, x^2-y^2)
T	3	0	0		(Rx, Ry, Rz), (x, y, z)	(xz, yz, xy)

 

Th	E	4C3	4C3^2	3C2	i	4S6	4S6^5	3σh		ε=exp(i2π/3)
Ag	1	1	1	1	1	1	1	1		x^2+y^2+z^2
Eg	$\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}$	$\begin{matrix} 1 \\ 1 \end{matrix}$	$\begin{matrix} 1 \\ 1 \end{matrix}$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}$	$\left.\begin{matrix} 1 \\ 1 \end{matrix}\right\}$		(2z^2-x^2-y^2, x^2-y^2)
Tg	3	0	0	-1	1	0	0	-1	(Rx, Ry, Rz)	(xz, yz, xy)
Au	1	1	1	1	-1	-1	-1	-1
Eu	$\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.$	$\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}$	$\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}$	$\begin{matrix} 1 \\ 1 \end{matrix}$	$\begin{matrix} -1 \\ -1 \end{matrix}$	$\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}$	$\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}$	$\left.\begin{matrix} -1 \\ -1 \end{matrix}\right\}$
Tu	3	0	0	-1	-1	0	0	1	(x, y, z)

 

Td	E	8C3	3C2	6S4	6σd
A1	1	1	1	1	1		x^2+y^2+z^2
A2	1	1	1	-1	-1
E	2	-1	2	0	0		(2z^2-x^2-y^2, x^2-y^2)
T1	3	0	-1	1	-1	(Rx, Ry, Rz)
T2	3	0	-1	-1	1	(x, y, z)	(xz, yz, xy)

 

O	E	8C3	3C2	6C4	6C2
A1	1	1	1	1	1		x^2+y^2+z^2
A2	1	1	1	-1	-1
E	2	-1	2	0	0		(2z^2-x^2-y^2, x^2-y^2)
T1	3	0	-1	1	-1	(Rx, Ry, Rz), (x, y, z)
T2	3	0	-1	-1	1		(xz, yz, xy)

 

Oh	E	8C2	6C2	6C4	3C2(C4^2)	i	6S4	8S6	3σh	6σd
A1g	1	1	1	1	1	1	1	1	1	1		x^2+y^2+z^2
A2g	1	1	-1	-1	1	1	-1	1	1	-1
Eg	2	-1	0	0	2	2	0	-1	2	0		(2z^2-x^2-y^2, x^2-y^2)
T1g	3	0	-1	1	-1	3	1	0	-1	-1	(Rx, Ry, Rz)
T2g	3	0	1	-1	-1	3	-1	0	-1	1		(xz, yz, xy)
A1u	1	1	1	1	1	-1	-1	-1	-1	-1
A2u	1	1	-1	-1	1	-1	1	-1	-1	1
Eu	2	-1	0	0	2	-2	0	1	-2	0
T1u	3	0	-1	1	-1	-3	-1	0	1	1	(x, y, z)
T2u	3	0	1	-1	-1	-3	1	0	1	-1