Skip to main content
Chemistry LibreTexts

Character Tables

  • Page ID
    2181
  • 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*
    *
    -1
    -1
    *
    ε*
    ε
    (Rx,Ry) (x,y) (xz, yz)
    E2
    1
    1
    *
    *
    1
    1
    *
    *
    (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
    *
    -1
    -1
    *
    -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*
    -1
    -1
    ε
    ε*
    -i
    i
    *

    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 -1 z
    E"

    1
    1

    ε
    ε*

    ε*
    ε

    -1
    -1

    *

    *
    (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

    *

    2
    2*

    2*
    2

    *
    (Rx, Ry) (xz, yz)
    E2"

    1
    1

    ε2
    ε2*

    ε*
    ε

    ε
    ε*

    ε2*
    ε2

    -1
    -1

    2
    2*

    *

    *

    2*
    2
    \(C_{6h}\) 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*

    *

    -1
    -1

    *

    ε*
    ε

    1
    1

    ε
    ε*

    *

    -1
    -1

    *

    ε*
    ε
    (Rx, Ry) (xz, yz)
    E2g

    1
    1

    *

    *

    1
    1

    *

    *

    1
    1

    *

    *

    1
    1

    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*

    *

    -1
    -1

    *

    ε*
    ε

    -1
    -1

    *

    ε*
    ε

    1
    1

    ε
    ε*

    *
    (x, y)
    E2u

    1
    1

    *

    *

    1
    1

    *

    *

    -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
    *
    *
    (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
    *
    -1
    -1
    *
    -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
    *
    -i
    i
    ε
    ε*
    -1
    -1
    ε*
    ε
    i
    -i
    *
    (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
    *
    *
    -1
    -1
    Tu 3 0 0 -1 -1 0 0 1 (x, y, z)
    Td E 8C3 3C2 6S4 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 (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