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
Au	1	1	Rx, Ry, Rz	x^2, y^2, z^2, xy, yz, zx
Ag	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 (2πi/3)
A	1	1	1	z, Rz	x^2+y^2, z^2
E	1 1	ε ε^*	ε* ε	(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	1 1	i -i	-1 -1	-i i	(x,y) (Rx,Ry)	(xz, yz)

C5	E	C5	C5^2	C5^3	C5^4
A	1	1	1	1	1	Z, Rz	x^2+y^2, z^2
E1	1 1	ε ε^*	ε^2 ε^2^*	ε^2^* ε^2	ε* ε	(x, y)(Rx,Ry)	(xz, yz)
E2	1 1	ε^2 ε^2^*	ε* ε	ε ε^*	ε^2^* ε^2		(x^2-y^2, xy)

C6	E	C6	C3	C2	C3^2	C6^5		ε=exp (2πi/6)
A	1	1	1	1	1	1	z, Rz	x^2+y^2, z^2
B	1	-1	1	-1	1	-1
E1	1 1	ε ε^*	-ε^* -ε	-1 -1	-ε -ε^*	ε^* ε	(Rx,Ry) (x,y)	(xz, yz)
E2	1 1	-ε^* -ε	-ε -ε^*	1 1	-ε^* -ε	-ε -ε^*		(x^2-y^2, xy)

C7	E	C7	C7^2	C7^3	C7^4	C7^5	C7^6		ε=exp (2πi/7)
A	1	1	1	1	1	1	1	z, Rz	x^2+y^2, z^2
E1	1 1	ε ε^*	ε^2 ε^2^*	ε^3 ε^3^*	ε^3^* ε^3	ε^2^* ε^2	ε^* ε	(Rx,Ry) (x,y)	(xz, yz)
E2	1 1	ε^2 ε^2^*	ε^3^* ε^3	ε^* ε	ε ε^*	ε^3 ε^3^*	ε^2^* ε^2		(x^2-y^2, xy)
E3	1 1	ε^3 ε^3^*	ε^* ε	ε^2 ε^2^*	ε^2^* ε^2	ε ε^*	ε^3^* ε^3

C8	E	C8	C4	C8^3	C2	C8^5	C4^3	C8^7		ε=exp (2πi/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	1 1	ε ε^*	i -i	-ε^* -ε	-1 -1	-ε -ε^*	-i i	ε^* ε	(Rx,Ry) (x,y)	(xz, yz)
E2	1 1	i -i	-1 -1	-i i	1 1	i -i	-1 -1	-i i		(x^2-y^2, xy)
E3	1 1	-ε^* -ε	ε^* ε	ε ε^*	-1 -1	ε ε^*	-i i	-ε^* -ε

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 (2πi/3)
A'	1	1	1	1	1	1	Rz	x^2+y^2, z^2
E'	1 1	ε ε^*	ε^* ε	1 1	ε ε^*	ε^* ε	(x,y)	(x^2-y^2, xy)
A"	1	1	1	-1	-1	-1	z
E"	1 1	ε ε^*	ε^* ε	-1 -1	-ε -ε^*	-ε^* -ε	(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	1 1	i -i	-1 -1	-i i	1 1	i -i	-1 -1	-i i	(Rx, Ry)	(xz, yz)
Au	1	1	1	1	-1	-1	-1	-1	z
Bu	1	-1	1	-1	-1	1	-1	1
Eu	1 1	i -i	-1 -1	-i i	-1 -1	-i i	1 1	i -i	(x,y)

\(C_{5h}\)

E

C5

C5^2

C5^3

C5^4

σh

S5

S5^7

S5^3

S5^9

ε=exp (2πi/5)

A'

1

1

1

1

1

1

1

1

1

1

Rz

x^2+y^2, z^2

E1'

1 1

ε ε^*

ε^2 ε^2^*

ε^2^* ε^2

ε^* ε

1 1

ε ε^*

ε^2 ε^2^*

ε^2^* ε^2

ε^* ε

(x, y)

E2'

1 1

ε^2 ε^2^*

ε^* ε

ε ε^*

ε^2^* ε^2

1 1

ε^2 ε^2^*

ε^* ε

ε ε^*

ε^2^* ε^2

(x^2-y^2, xy)

A"

1

1

1

1

1

-1

-1

-1

-1

-1

z

E1"

1 1

ε ε^*

ε^2 ε^2^*

ε^2^* ε^2

ε^* ε

-1 -1

-ε -ε^*

-ε^2 -ε^2^*

-ε^2^* -ε^2

-ε^* -ε

(Rx, Ry)

(xz, yz)

E2"

1 1

ε^2 ε^2^*

ε^* ε

ε ε^*

ε^2^* ε^2

-1 -1

-ε^2 -ε^2^*

-ε^* -ε

-ε -ε^*

-ε^2^* -ε^2

\(C_{6h}\)	E	C6	C3	C2	C3^2	C6^5	i	S3^5	S6^5	σh	S6	S3		ε=exp (2πi/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	1 1	ε ε^*	-ε^* -ε	-1 -1	-ε -ε^*	ε^* ε	1 1	ε ε^*	-ε^* -ε	-1 -1	-ε -ε^*	ε^* ε	(Rx, Ry)	(xz, yz)
E2g	1 1	-ε^* -ε	-ε -ε^*	1 1	-ε^* -ε	-ε -ε^*	1 1	-ε^* -ε	-ε -ε^*	1 1	ε^* ε	-ε -ε^*		(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	1 1	ε ε^*	-ε^* -ε	-1 -1	-ε -ε^*	ε^* ε	-1 -1	-ε -ε^*	ε^* ε	1 1	ε ε^*	-ε^* -ε	(x, y)
E2u	1 1	-ε^* -ε	-ε -ε^*	1 1	-ε^* -ε	-ε -ε^*	-1 -1	ε^* ε	ε ε^*	-1 -1	ε^* ε	ε ε^*

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	σV	σh'
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	2cos 72	2cos 144	0	(Rx, Ry)(x,y)	(xz, yz)
E2	2	2cos144	2cos 72	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	2cos θ	...	0	(x,y);(Rx, Ry)	(xz, yz)
E2	2	2cos 2θ	...	0		(x^2-y^2, xy)
E3	2	2cos 3θ	...	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)

\(D_5\)	E	2C5	2C5^2	5C2
A1	1	1	1	1		x^2+y^2, z^2
A2	1	1	1	-1	z, Rz
E1	2	2cos72	2cos144		(Rx, Ry)(x,y)	(xz, yz)
E2	2	2cos144	2cos72			(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	2cos72	2cos144	0	2	2cos72	2cos144		(x,y)
E2'	2	2cos144	2cos72	0	2	2cos144	2cos72			(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	2cos72	2cos144	0	-2	-2cos72	-2cos144	0	(Rx, Ry)	(xz, yz)
E2"	2	2cos144	2cos72	0	-2	-2cos144	-2cos72	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	2cos	...	0	2	-2cos	...	0	(Rx, Ry)	(xz, yz)
Dg	2	2cos2	...	0	2	2cos2	...	0		(x^2-y^2, xy)
...	...	...	...	......	...	...	...	...
Su^+	1	1	...	1	-1	-1	...	-1	z
Su^-	1	1	...	-1	-1	-1	...	1
πu	2	2cos	...	0	-2	2cos	...	0	(x, y)
Du	2	2cos2	...	0	-2	-2cos	...	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	1.414	0	- 1.414	-2	0	0	(x, y)
E2	2	0	-2	0	2	0	0		(x^2-y^2, xy)
E3	2	- 1.414	0	1.414	-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	2cos 72	2cos 144	0	2	2cos 72	2cos 144	0	(Rx, Ry)	(xz, yz)
E2g	2	2cos 144	2cos 72	0	2	2cos 144	2cos 72	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	2cos 72	2cos 144	0	-2	-2cos 72	-2cos 144	0	(x, y)
E2u	2	2cos 144	2cos 72	0	-2	-2cos 144	-2cos 72	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	1.732	1	0	-1	-1.732	-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	-1.732	1	0	-1	1.732	-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	1 1	i -i	-1 -1	-i i	(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	1 1	ε ε^*	ε^* ε	1 1	ε ε^*	ε^* ε	(Rx, Ry)	(x^2-y^2, xy)(xz, yz)
Au	1	1	1	-1	-1	-1	z
Eu	1 1	ε ε^*	ε^* ε	-1 -1	-ε -ε^*	-ε^* -ε	(x, y)

S8	E	S8	C4	S8^3	C2	S8^5	C4^3	S8^7		ε=exp (2πi/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	1 1	ε ε^*	i -i	-ε^* -ε	-1 -1	-ε -ε^*	-i i	ε^* ε	(Rx, Ry), (x, y)
E2	1 1	i -i	-1 -1	-i i	1 1	i -i	-1 -1	-i i		(x^2-y^2, xy)
E3	1 1	-ε^* -ε	-i i	ε ε^*	-1 -1	ε^* ε	i -i	-ε -ε^*		(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	1 1	ε ε^*	ε^* ε	1 1		(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 (2πi/3)
Ag	1	1	1	1	1	1	1	1		x^2+y^2+z^2
Eg	1 1	ε ε^*	ε^* ε	1 1	1 1	ε ε^*	ε^* ε	1 1		(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	1 1	ε ε^*	ε^* ε	1 1	-1 -1	-ε -ε^*	-ε^* -ε	-1 -1
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