# 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 x2, y2, z2, xy A" 1 -1 z, Rx, Ry yz, xz
 $$C_i$$ E i Au 1 1 Rx, Ry, Rz x2, y2, z2, 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 x2, y2, z2, xy B 1 -1 x, y, Rx, Ry yz,xz
C3 E C3 C32    ε=exp (2πi/3)
A  1  1  1  z, Rz  x2+y2, z2
E

 1 1

 ε ε*

 ε* ε
(x,y) (Rx,Ry)  (x2-y2, xy), (xz, yz)
C4  E  C4  C2  C43
A  1  1  1  1  z, Rz  x2+y2, z2
B  1  -1  1  -1    x2-y2, xy
E

 1 1

 i -i

 -1 -1

 -i i
(x,y) (Rx,Ry)  (xz, yz)
C5 E C5 C52 C53 C54
A 1 1 1 1 1 Z, Rz x2+y2, z2
E1
 1 1
 ε ε*
 ε2 ε2*
 ε2* ε2
 ε* ε
(x, y)(Rx,Ry) (xz, yz)
E2
 1 1
 ε2 ε2*
 ε* ε
 ε ε*
 ε2* ε2
(x2-y2, xy)
C6  E  C6  C3  C2  C32  C65     ε=exp (2πi/6)
A  1  1  1  1  1  1  z, Rz  x2+y2, z2
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*
(x2-y2, xy)
C7  E  C7  C72  C73  C74  C75  C76     ε=exp (2πi/7)
A  1  1  1  1  1  1  1  z, Rz  x2+y2, z2
E1
 1 1

 ε ε*

 ε2 ε2*

 ε3 ε3*

 ε3* ε3

 ε2* ε2

 ε* ε
(Rx,Ry) (x,y)  (xz, yz)
E2
 1 1

 ε2 ε2*

 ε3* ε3

 ε* ε

 ε ε*

 ε3 ε3*

 ε2* ε2
(x2-y2, xy)
E3
 1 1

 ε3 ε3*

 ε* ε

 ε2 ε2*

 ε2* ε2

 ε ε*

 ε3* ε3

C8  E  C8  C4  C83  C2  C85  C43  C87     ε=exp (2πi/8)
A  1  1  1  1  1  1  1  1  z, Rz  x2+y2, z2
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
(x2-y2, 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 x2, y2, z2 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  C32  σh  S3  S35     ε=exp (2πi/3)
A'  1  1  1  1  1  1  Rz   x2+y2, z2
E'

 1 1

 ε ε*

 ε* ε

 1 1

 ε ε*

 ε* ε
(x,y)  (x2-y2, xy)
A"  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  C43  i  S43  σh  S4
Ag  1  1  1  1  1  1  1  1  Rz  x2+y2, z2
Bg  1  -1  1  -1  1  -1  1  -1    x2-y2, 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  C52  C53  C54  σh  S5  S57  S53  S59     ε=exp (2πi/5)
A'  1  1  1  1  1  1  1  1  1  1  Rz  x2+y2, z2
E1'

 1 1

 ε ε*

 ε2 ε2*

 ε2* ε2

 ε* ε

 1 1

 ε ε*

 ε2 ε2*

 ε2* ε2

 ε* ε
(x, y)
E2'

 1 1

 ε2 ε2*

 ε* ε

 ε ε*

 ε2* ε2

 1 1

 ε2 ε2*

 ε* ε

 ε ε*

 ε2* ε2
(x2-y2, xy)
A"  1  1  1  1  1  -1  -1  -1  -1  -1  z
E1"

 1 1

 ε ε*

 ε2 ε2*

 ε2* ε2

 ε* ε

 -1 -1

 -ε -ε*

 -ε2 -ε2*

 -ε2* -ε2

 -ε* -ε
(Rx, Ry)  (xz, yz)
E2"

 1 1

 ε2 ε2*

 ε* ε

 ε ε*

 ε2* ε2

 -1 -1

 -ε2 -ε2*

 -ε* -ε

 -ε -ε*

 -ε2* -ε2

$$C_{6h}$$  E  C6  C3  C2  C32  C65  i  S35  S65  σh  S6  S3     ε=exp (2πi/6)
Ag  1  1  1  1  1  1  1  1  1  1  1  1  Rz  x2+y2, z2
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*
(x2-y2, 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 x2, y2, z2 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 x2+y2, z2 A2 1 1 -1 Rz E 2 -1 0 (Rx, Ry), (x,y) (xz, yz) (x2-y2, xy)
 $$C_{4v}$$ E 2C4 C2 2σv 2σd A1 1 1 1 1 1 z x2+y2, z2 A2 1 1 1 -1 -1 Rz B1 1 -1 1 1 -1 x2-y2 B2 1 -1 1 -1 1 xy E 2 0 -2 0 0 (Rx, Ry)(x,y) (xz, yz)
 $$C_{5v}$$ E 2C5 2C52 5σv A1 1 1 1 1 z x2+y2, z2 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 (x2-y2, xy)
 $$C_{6v}$$ E 2C6 2C3 C2 3σv 3σd A1 1 1 1 1 1 1 z x2+y2, z2 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 (x2-y2, xy)
 C∞v E 2C∞ ... ∞σv A1 1 1 ... 1 z x2+y2, z2 A2 1 1 ... -1 Rz E1 2 2cos θ ... 0 (x,y);(Rx, Ry) (xz, yz) E2 2 2cos 2θ ... 0 (x2-y2, xy) E3 2 2cos 3θ ... 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, 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 x2+y2, z2 A2 1 1 -1 z, Rz E 2 -1 0 (Rx, Ry)(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, Rz B1 1 -1 1 1 -1 x2-y2 B2 1 -1 1 -1 1 xy E 2 0 -2 0 0 (Rx, Ry)(x,y) (xz, yz)
 $$D_5$$ E 2C5 2C52 5C2 A1 1 1 1 1 x2+y2, z2 A2 1 1 1 -1 z, Rz E1 2 2cos72 2cos144 (Rx, Ry)(x,y) (xz, yz) E2 2 2cos144 2cos72 (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, 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 (x2-y2, 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 x2, y2, z2 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 x2+y2, z2 A2' 1 1 -1 1 1 -1 Rz 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 (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 x2+y2, z2 A2g 1 1 1 -1 -1 1 1 1 -1 -1 Rz B1g 1 -1 1 1 -1 1 -1 1 1 -1 x2-y2 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 2C52 5C2 σh 2S5 2S53 5σv A1' 1 1 1 1 1 1 1 1 x2+y2, z2 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 (x2-y2, 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 x2+y2, z2 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 (x2-y2, 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 x2+y2, z2 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 (x2-y2, 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 Cn
2. n 2-fold proper rotation axes C2 normal to Cn
3. n mirror planes σd which contain Cn.

 D2d E 2S4 C2 2C2' 2σd A1 1 1 1 1 1 x2+y2, z2 A2 1 1 1 -1 -1 Rz B1 1 -1 1 1 -1 x2-y2 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 x2+y2, z2 A2g 1 1 -1 1 1 -1 Rz Eg 2 -1 0 2 -1 0 (Rx, Ry) (x2-y2, 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 2S83 C2 4C2' 4σd A1 1 1 1 1 1 1 1 x2+y2, z2 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 (x2-y2, xy) E3 2 - 1.414 0 1.414 -2 0 0 (Rx, Ry) (xz, yz)
 D5d E 2C5 2C52 5C2 i 2S103 2S10 5σd A1g 1 1 1 1 1 1 1 1 x2+y2, z2 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 (x2-y2, 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 2S125 C2 6C2' 6σd A1 1 1 1 1 1 1 1 1 1 x2+y2, z2 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 (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 -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  S43
A  1  1  1  1  Rz  x2+y2, z2
B  1  -1  1  -1  z  x2-y2, xy
E

 1 1

 i -i

 -1 -1

 -i i
(x, y); (Rx, Ry)  (xz, yz)
S6  E  C3  C32  i  S65  S6
Ag  1  1  1  1  1  1  Rz  x2+y2, z2
Eg
 1 1

 ε ε*

 ε* ε

 1 1

 ε ε*

 ε* ε
(Rx, Ry)  (x2-y2, 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  S83  C2  S85  C43  S87     ε=exp (2πi/8)
A  1  1  1  1  1  1  1  1  Rz  x2+y2, z2
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
(x2-y2, 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  4C32  3C2
A  1  1  1  1     x2+y2+z2
E

 1 1

 ε ε*

 ε* ε

 1 1
(2z2-x2-y2, x2-y2)
T  3  0  0    (Rx, Ry, Rz) (x, y, z)  (xz, yz, xy)
Th E 4C3 4C32 3C2 i 4S6 4S65 h     ε=exp (2πi/3)
Ag 1 1 1 1 1 1 1 1   x2+y2+z2
Eg
 1 1

 ε ε*

 ε* ε

 1 1

 1 1

 ε ε*

 ε* ε

 1 1
(2z2-x2-y2, x2-y2)
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 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 (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 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 (Rx, Ry, Rz)(x, y, z) T2 3 0 -1 -1 1 (xz, yz, xy)
 Oh E 8C2 6C2 6C4 3C2(C42) i 6S4 8S6 3σh 6σd A1g 1 1 1 1 1 1 1 1 1 1 x2+y2+z2 A2g 1 1 -1 -1 1 1 -1 1 1 -1 Eg 2 -1 0 0 2 2 0 -1 2 0 (2z2-x2-y2, x2-y2) 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