This lists the character table

Character table

In group theory, a character table is a two-dimensional table whose rows correspond to irreducible group representations, and whose columns correspond to classes of group elements...

s for the more common molecular point groups

Point groups in three dimensions

In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O, the group of all isometries that leave the origin fixed, or correspondingly, the group...

 used in the study of molecular symmetry

Molecular symmetry

Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can predict or explain many of a molecule's chemical properties, such as its dipole moment...

. These tables are based on the group-theoretical

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

 treatment of the symmetry

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

 operations present in common molecule

Molecule

A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...

s, and are useful in molecular spectroscopy

Spectroscopy

Spectroscopy is the study of the interaction between matter and radiated energy. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative...

 and quantum chemistry

Quantum chemistry

Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...

. Information regarding the use of the tables, as well as more extensive lists of them, can be found in the references.

Notation

For each non-linear group, the tables give the most standard notation of the finite group isomorphic to the point group, followed by the order of the group

Order (group theory)

In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....

 (number of invariant symmetry operations). The finite group notation used is: Z_n: cyclic group

Cyclic group

In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

 of order n, D_n: dihedral group

Dihedral group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...

 isomorphic to the symmetry group of an n–sided regular polygon, S_n: symmetric group

Symmetric group

In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

 on n letters, and A_n: alternating group on n letters.

The character tables then follow for all groups. The rows of the character tables correspond to the irreducible representations of the group, with their conventional names in the left margin. The naming conventions are as follows:

• A and B are singly degenerate representations, with the former transforming symmetrically around the principal axis of the group, and the latter asymmetrically. E, T, G, H, ... are doubly, triply, quadruply, quintuply, ... degenerate representations.
• g and u subscripts denote symmetry and antisymmetry, respectively, with respect to a center of inversion. Subscripts "1" and "2" denote symmetry and antisymmetry, respectively, with respect to a nonprincipal rotation axis. Higher numbers denote additional representations with such asymmetry.
• Single prime ( ' ) and double prime ( ) superscripts denote symmetry and antisymmetry, respectively, with respect to a horizontal mirror plane σ_h, one perpendicular to the principal rotation axis.

All but the two rightmost columns correspond to the symmetry operation

Symmetry operation

In the context of molecular symmetry, a symmetry operation may be defined as a permutation of atoms such that the molecule or crystal is transformed into a state indistinguishable from the starting state....

s which are invariant in the group. In the case of sets of similar operations with the same characters for all representations, they are presented as one column, with the number of such similar operations noted in the heading.

The body of the tables contain the characters in the respective irreducible representations for each respective symmetry operation, or set of symmetry operations.

The two rightmost columns indicate which irreducible representations describe the symmetry transformations of the three Cartesian coordinates (x, y and z), rotations about those three coordinates (R_x, R_y and R_z), and functions of the quadratic terms of the coordinates(x², y², z², xy, xz, and yz).

The symbol i used in the body of the table denotes the imaginary unit

Imaginary unit

In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...

: i ² = −1. Used in a column heading, it denotes the operation of inversion. A superscripted uppercase "C" denotes complex conjugation

Complex conjugate

In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...

.

Nonaxial symmetries

These groups are characterized by a lack of a proper rotation axis, noting that a C₁ rotation is considered the identity operation. These groups have involutional symmetry: the only nonidentity operation, if any, is its own inverse.

In the group C₁, all functions of the Cartesian coordinates and rotations about them transform as the A irreducible representation.
! Point
Group !! Canonical
Group !! Order !! Character Table
|- style="background:#fafafa;"
| C₁  >

Z1

{> border="1" style="text-align:center" |

>- | A

|- style="background:#fafafa;" | Ci

Z2

{> border="1" style="text-align:center" |

E

i

|- | Ag

1

> Rx, Ry, Rz | x2, y2, z2, xy, xz, yz |- | Au

1

−1

x, y, z

|- style="background:#fafafa;" | Cs

Z2

{|>border="1" style="text-align:center" |

E

σh

|- | A'

1

> x, y, Rz | x2, y2, z2, xy |- | A

1

> z, Rx, Ry

>} |-

Cyclic groups (C_n)

The cyclic groups are denoted by C_n. These groups are characterized by an n-fold proper rotation axis C_n. The C₁ group is covered in the nonaxial groups section.
! Point
Group !! Canonical
Group !! Order !! Character Table
|-
| C₂ >

Z2

{> style="text-align:center" |

E

C2

|- | A

1

1

z, z > x2, y2, z2, xy |- | B

1

−1

x, Ry, x, y > xz, yz |} |- | C3

Z3

{|>style="text-align:center" |

E

C3

32 > colspan="2" | θ = e2πi /3 |- | A

1

1

1

z, z > x2 + y2 |- | E

1 > θ  θC | θC θ  | (Rx, Ry), (x, y) | (x2 + y2, xy), (xz, yz) |- |} |- | C4

Z4

{|>style="text-align:center" |

E

4  > C2

43 > colspan="2" |   |- | A

1

1

1

1

z, z > x2 + y2, z2 |- | B

1

−1

1

−1

> x2 − y2, xy |- | E

1 1

i −i

−1 −1

i > (Rx, Ry), (x, y)

>- |} |- | C5

Z5

{|>style="text-align:center" |

> C5

52 > C53

54 > colspan="2" | θ = e2πi /5 |- | A

1

1

1

1

1

z, z > x2 + y2, z2 |- | E1 | 1 1 | θ  θC | θ2 (θ2)C | (θ2)C θ2 | θC θ  | (Rx, Ry), (x, y)

>- | E2 | 1 1 | θ2 (θ2)C | θC θ  | θ  θC | (θ2)C θ2 |

2 - y2, xy) >- |} |- | C6

Z6

{|>style="text-align:center" |

> C6

3  > C2

32 > C65 | colspan="2" | θ = e2πi /6 |- | A

1

1

1

1

1

1

z, z > x2 + y2, z2 |- | B

1

−1

1

−1

1

−1

>- | E1 | 1 1 | θ  θC | −θC −θ  | −1 −1 | −θ  −θC | θC −θ  | (Rx, Ry), (x, y) | (xz, yz) |- | E2 | 1 1 | −θC −θ  | −θ  −θC | 1 1 | −θC −θ  | −θ  −θC |

2 − y2, xy) >- |} |- | C8

Z8

{|>style="text-align:center" |

> C8

4  > C83

2  > C85

42 > C87 | colspan="2" | θ = e2πi /8 |- | A

1

1

1

1

1

1

1

1

z, z > x2 + y2, z2 |- | B

1

−1

1

−1

1

−1

1

−1

>- | E1 | 1 1 | θ  θC | i −i | −θC −θ  | −1 −1 | −θ  −θC | −i i | θC θ  | (Rx, Ry), (x, y) | (xz, yz) |- | E2 | 1 1 | i −i | −1 −1 | −i i | 1 1 | i −i | −1 −1 | −i i |

2 − y2, xy) >- | E3 | 1 1 | −θ  −θC | i −i | θC θ  | −1 −1 | θ  θC | −i i | −θC −θ  |

>- |} |-

Reflection groups (C_nh)

The reflection groups are denoted by C_nh. These groups are characterized by i) an n-fold proper rotation axis C_n; ii) a mirror plane σ_h normal to C_n. The C_1h group is the same as the C_s group in the nonaxial groups section.
! Point
Group !! Canonical
group !! Order !! Character Table
|-
| C_2h >

Z2 × Z2

{|> style="text-align:center" |

E

2  > i

σh

|- | Ag

1

1

1

1

z > x2, y2, z2, xy |- | Bg

1

−1

1

−1

x, Ry > xz, yz |- | Au

1

1

−1

−1

z

>- | Bu

1

−1

−1

1

x, y

>- |} |- | C3h

Z6

{> style="text-align:center" |

E

3  > C32

h  > S3

35 > colspan="2" | θ = e2πi /3 |- | A'

1

1

1

1

1

1

z > x2 + y2, z2 |- | E'

1 1 | θ  θC | θC θ  | 1 1 | θ  θC | θC θ  | (x, y)

2 − y2, xy) >- | A

1

1

1

−1

−1

−1

z

>- | E

1 1 | θ  θC | θC θ  | −1 −1 | −θ  −θC | −θC −θ  | (Rx, Ry)

>- |} |- | C4h

Z2 × Z4

{> style="text-align:center" |

E

4  > C2

43 > i

S43

h  > S4

|- | Ag

1

1

1

1

1

1

1

> Rz

2 + y2, z2 >- | Bg

1

−1

1

−1

1

−1

1

>

2 − y2, xy >- | Eg

1 1

i −i

−1 > −i i

1 1

−i > −1 −1

i > (Rx, Ry)

>- | 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 > −i i

−1 −1

i > 1 1

−i > (x, y)

>- |} |- | C5h

Z10

{|>style="text-align:center" |

> C5

52 > C53

54 > σh

5  > S57

53 > S59 | colspan="2" | θ = e2πi /5 |- | A'

1

1

1

1

1

1

1

1

1

1

z > x2 + y2, z2 |- | E1' | 1 1 | θ  θC | θ2 (θ2)C | (θ2)C θ2 | θC θ  | 1 1 | θ  θC | θ2 (θ2)C | (θ2)C θ2 | θC θ  | (x, y)

>- | E2' | 1 1 | θ2 (θ2)C | θC θ  | θ  θC | (θ2)C θ2 | 1 1 | θ2 (θ2)C | θC θ  | θ  θC | (θ2)C θ2 |

2 - y2, xy) >- | A

1

1

1

1

> −1

−1

−1

−1

> z

>- | E1 | 1 1 | θ  θC | θ2 (θ2)C | (θ2)C θ2 | θC θ  | −1 −1 | −θ  -θC | −θ2 −(θ2)C | −(θ2)C −θ2 | −θC −θ  | (Rx, Ry)

>- | E2 | 1 1 | θ2 (θ2)C | θC θ  | θ  θC | (θ2)C θ2 | −1 −1 | −θ2 −(θ2)C | −θC −θ  | −θ  −θC | −(θ2)C −θ2 |

>- |} |- | C6h

Z2 × Z6

{|>style="text-align:center" |

> C6

3  > C2

32 > C65

i

35 > S65

h  > S6

3  > colspan="2" | θ = e2πi /6 |- | Ag

1

1

1

1

1

1

1

1

1

1

1

> Rz

2 + y2, z2 >- | Bg

1

−1

1

−1

1

> 1

−1

1

−1

1

>

>- | E1g | 1 1 | θ  θC | −θC −θ  | −1 −1 | −θ  −θC | θC θ  | 1 1 | θ  θC | −θC −θ  | −1 −1 | −θ  −θC | θC θ  | (Rx, Ry)

>- | E2g | 1 1 | −θC −θ  | −θ  −θC | 1 1 | −θC −θ  | −θ  −θC | 1 1 | −θC −θ  | −θ  −θC | 1 1 | −θC −θ  | −θ  −θC |

2 − y2, xy) >- | Au

1

1

1

1

1

> −1

−1

−1

−1

−1

> z

>- | Bu

1

−1

1

−1

1

> −1

1

−1

1

−1

>

>- | E1u | 1 1 | θ  θC | −θC −θ  | −1 −1 | −θ  −θC | θC θ  | −1 −1 | −θ  −θC | θC θ  | 1 1 | θ  θC | −θC −θ  | (x, y)

>- | E2u | 1 1 | −θC −θ  | −θ  −θC | 1 1 | −θC −θ  | −θ  −θC | −1 −1 | θC θ  | θ  θC | −1 −1 | θC θ  | θ  θC |

>} |-

Pyramidal groups (C_nv)

The pyramidal groups are denoted by C_nv. These groups are characterized by i) an n-fold proper rotation axis C_n; ii) n mirror planes σ_v which contain C_n. The C_1v group is the same as the C_s group in the nonaxial groups section.
! Point
Group !! Canonical
group !!Order !! Character Table
|-
| C_2v >

Z2 × Z2 (=D2)

{> style="text-align:center" |

E

2  > σv  | σv'  | colspan="2" |   |- | A1

1

1

1

1

> x2, y2, z2 |- | A2

1

1

−1

−1

Rz

>- | B1

1

−1

1

−1

Ry, x

>- | B2

1

−1

−1

1

Rx, y

>- |} |- | C3v

D3

{> style="text-align:center" |

E

3  > 3 σv  | colspan="2" |   |- | A1

1

1

1

> x2 + y2, z2 |- | A2

1

1

−1

Rz

>- | E

2

−1

0

x, Ry), (x, y) > (x2 − y2, xy), (xz, yz) |- |} |- | C4v

D4

{> style="text-align:center" |

E

4  > C2

v  > 2 σd

|- | A1

1

1

1

1

> z

2 + y2, z2 >- | A2

1

1

1

−1

−1

Rz

>- | B1

1

−1

1

1

>

2 − y2 >- | B2

1

−1

1

−1

1

>- | E

2

0

−2

0

> (Rx, Ry), (x, y)

>- |} |- | C5v

D5

{|>style="text-align:center" |

> 2 C5

52 > 5 σv

θ = 2π/5 |- | A1

1

1

1

1

> x2 + y2, z2 |- | A2

1

1

1

−1

Rz

>- | E1

2

2 cos(θ)

2 cos(2θ)

> (Rx, Ry), (x, y)

>- | E2

2

2 cos(2θ)

2 cos(θ)

>

2 − y2, xy) >- |} |- | C6v

D6

{|>style="text-align:center" |

> 2 C6

3  > C2

v  > 3 σd

|- | A1

1

1

1

1

1

> z

2 + 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

> (Rx, Ry), (x, y)

>- | E2

2

−1

−1

2

0

0

> (x2 − y2, xy) |} |-

Improper rotation groups (S_n)

The improper rotation groups are denoted by S_n. These groups are characterized by an n-fold improper rotation axis S_n, where n is necessarily even. The S₂ group is the same as the C_s group in the nonaxial groups section.

The S₈ table reflects the 2007 discovery of errors in older references. Specifically, (R_x, R_y) transform not as E₁ but rather as E₃.
! Point
Group !! Canonical
group !! Order !! Character Table
|-
| S₄ >

Z4

{> style="text-align:center" |

E

4  > C2

43 > colspan="2" |   |- | A

1

1

1

1

z,   > x2 + y2, z2 |- | B

1

−1

1

−1

> x2 − y2, xy |- | E

1 1

i −i

−1 > −i i | (Rx, Ry), (x, y)

>- |} |- | S6

Z6

{|>style="text-align:center" |

> S6

3  > i

C32

65 > colspan="2" | θ = e2πi /6 |- | Ag

1

1

1

1

1

1

z > x2 + y2, z2 |- | Eg | 1 1 | θC θ  | θ  θC | 1 1 | θC θ  | θ  θC | (Rx, Ry) | (x2 − y2, xy), (xz, yz) |- | Au

1

−1

1

−1

1

−1

z

>- | Eu | 1 1 | −θC −θ  | θ  θC | −1 −1 | θC θ  | −θ  −θC | (x, y)

>- |} |- | S8

Z8

{|>style="text-align:center" |

> S8

4  > S83

> S85

42 > S87 | colspan="2" | θ = e2πi /8 |- | A

1

1

1

1

1

1

1

1

z > x2 + y2, z2 |- | B

1

−1

1

−1

1

−1

1

−1

z

>- | E1 | 1 1 | θ  θC | i −i | −θC −θ  | −1 −1 | −θ  −θC | −i i | θC θ  | (x, y)

>- | E2 | 1 1

i −i

−1 > −i i

1 1

−i > −1 −1

i >

2 − y2, xy) >- | E3 | 1 1 | −θC −θ  | −i i | θ  θC | −1 −1 | θC θ  | i −i | −θ −θC | (Rx, Ry)

>- |} |-

Dihedral symmetries

The families of groups with these symmetries are characterized by 2-fold proper rotation axes normal to a principal rotation axis.

Dihedral groups (D_n)

The dihedral groups are denoted by D_n. These groups are characterized by i) an n-fold proper rotation axis C_n; ii) n 2-fold proper rotation axes C₂ normal to C_n. The D₁ group is the same as the C₂ group in the cyclic groups section.
! Point
Group !! Canonical
group !!Order !! Character Table
|-
| D₂ >

Z2 × Z2 (=D2)

{|> style="text-align:center" |

E

2 (z) > C2 (x) | C2 (y)

|- | A

1

1

1

1

> x2, y2, z2 |- | B1

1

1

−1

−1

Rz, z

>- | B2

1

−1

−1

1

Ry, y

>- | B3

1

−1

1

−1

Rx, x

>- |} |- | D3

D3

{|> style="text-align:center" |

E

3  > 3 C2

|- | A1

1

1

1

> x2 + y2, z2 |- | A2

1

1

−1

Rz, z

>- | E

2

−1

0

x, Ry), (x, y) > (x2 − y2, xy), (xz, yz) |- |} |- | D4

D4

{|>style="text-align:center" |

E

4  > C2

2'  > 2 C2  | colspan="2" |   |- | A1

1

1

1

1

1

> x2 + y2, z2 |- | A2

1

1

1

−1

−1

Rz, z

>- | B1

1

−1

1

1

>

2 − y2 >- | B2

1

−1

1

−1

1

>- | E

2

0

−2

0

> (Rx, Ry), (x, y)

>- |} |- | D5

D5

{| s>yle="text-align:center" |

> 2 C5

52 > 5 C2

θ=2π/5 |- | A1

1

1

1

1

> x2 + y2, z2 |- | A2

1

1

1

−1

Rz, z

>- | E1

2

2 cos(θ)

2 cos(2θ)

> (Rx, Ry), (x, y)

>- | E2

2

2 cos(2θ)

2 cos(θ)

>

2 − y2, xy) >- |} |- | D6

D6

{| s>yle="text-align:center" |

> 2 C6

3  > C2

2'  > 3 C2  | colspan="2" |   |- | A1

1

1

1

1

1

1

> x2 + y2, z2 |- | A2

1

1

1

1

−1

> Rz, z

>- | B1

1

−1

1

−1

1

−1

>- | B2

1

−1

1

−1

−1

1

>- | E1

2

1

−1

−2

0

> (Rx, Ry), (x, y)

>- | E2

2

−1

−1

2

0

0

> (x2 − y2, xy) |} |-

Prismatic groups (D_nh)

The prismatic groups are denoted by D_nh. These groups are characterized by i) an n-fold proper rotation axis C_n; ii) n 2-fold proper rotation axes C₂ normal to C_n; iii) a mirror plane σ_h normal to C_n and containing the C₂s. The D_1h group is the same as the C_2v group in the pyramidal groups section.

The D_8h table reflects the 2007 discovery of errors in older references. Specifically, symmetry operation column headers 2S₈ and 2S₈³ were reversed in the older references.
! Point
Group !! Canonical
group !!Order !! Character Table
|-
| D_2h
| Z₂×Z₂×Z₂
(=Z₂×D₂) >

{| >style="text-align:center" |

E

2  > C2 (x) | C2 (y)

> 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

> Rz

>- | B2g

1

−1

−1

1

1

−1

1

> Ry

>- | B3g

1

−1

1

−1

1

−1

−1

> Rx

>- | Au

1

1

1

> −1

−1

−1

−1

>- | B1u

1

1

−1

> −1

−1

1

1

z

>- | B2u

1

−1

−1

> −1

1

−1

1

y

>- | B3u

1

−1

1

> −1

1

1

−1

x

>- |} |- | D3h

D6

{|> style="text-align:center" |

E

3  > 3 C2

h  > 2 S3

v  > colspan="2" |   |- | 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

> (x2 − y2, xy) |- | A1

1

1

1

−1

−1

>

>- | A2

1

1

−1

−1

−1

> z

>- | E

2

−1

0

−2

1

> (Rx, Ry)

>- |} |- | D4h

Z2×D4

{|>style="text-align:center" |

E

4  > C2

2'  > 2 C2

> 2 S4

h  > 2 σv  | 2 σd

|- | A1g

1

1

1

1

1

1

1

1

1

>

2 + y2, z2 >- | A2g

1

1

1

−1

> 1

1

1

−1

> Rz

>- | B1g

1

−1

1

1

> 1

−1

1

1

>

2 − y2 >- | B2g

1

−1

1

−1

> 1

−1

1

−1

>

>- | Eg

2

0

−2

0

0

2

0

−2

0

> (Rx, Ry)

>- | A1u

1

1

1

1

> −1

−1

−1

−1

>

>- | A2u

1

1

1

−1

> −1

−1

−1

1

> z

>- | B1u

1

−1

1

1

> −1

1

−1

−1

>

>- | B2u

1

−1

1

−1

> −1

1

−1

1

>

>- | Eu

2

0

−2

0

0

−2

0

2

0

> (x, y)

>- |} |- | D5h

D10

{|>style="text-align:center" |

> 2 C5

52 > 5 C2  | σh

5  > 2 S53

v  > colspan="2" | θ=2π/5 |- | A1'

1

1

1

1

1

1

1

>

2 + y2, z2 >- | A2'

1

1

1

−1

1

1

1

> Rz

>- | E1'

2

2 cos(θ)

2 cos(2θ)

0

> 2 cos(θ)

2 cos(2θ)

0

(x, y)

>- | E2'

2

2 cos(2θ)

2 cos(θ)

0

> 2 cos(2θ)

2 cos(θ)

>

2 − y2, xy) >- | A1

1

1

1

> −1

−1

−1

>

>- | A2

1

1

1

> −1

−1

−1

> z

>- | E1

2

> 2 cos(2θ)

0

−2

> −2 cos(2θ)

> (Rx, Ry)

>- | E2

2

> 2 cos(θ)

0

−2

> −2 cos(θ)

0

>- |} |- | D6h | Z2×D6

{| s>yle="text-align:center" |

> 2 C6

3  > C2

2'  > 3 C2

> 2 S3

6  > σh

d  > 3 σv

|- | A1g

1

1

1

1

1

1

1

1

1

1

1

>

2 + 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

>

>- | B2g

1

−1

1

−1

−1

> 1

−1

1

−1

−1

>

>- | E1g

2

1

−1

−2

0

> 2

1

−1

−2

0

> (Rx, Ry)

>- | E2g

2

−1

−1

2

0

> 2

−1

−1

2

0

>

2 − y2, xy) >- | 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

>

>- | E1u

2

1

−1

−2

0

> −2

−1

1

2

0

> (x, y)

>- | E2u

2

−1

−1

2

0

> −2

1

1

−2

0

>

>} |- | D8h

Z2×D8

{|>style="text-align:center" |

> 2 C8

83 > 2 C4

2  > 4 C2'  | 4 C2

> 2 S83

8  > 2 S4  | σh  | 4 σd  | 4 σv  | colspan="2" | θ=21/2 |- | A1g

1

1

1

1

1

1

> 1

1

1

1

1

1

>

2 + y2, z2 >- | A2g

1

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

−1

>- | B2g

1

−1

−1

1

1

−1

> 1

−1

−1

1

1

−1

1

>- | E1g

2

θ

−θ

0

−2

0

> 2

θ

−θ

0

−2

0

> (Rx, Ry)

>- | E2g

2

0

0

−2

2

0

> 2

0

0

−2

2

0

>

2 − y2, xy) >- | E3g

2

−θ

θ

0

−2

0

> 2

−θ

θ

0

−2

0

>

>- | A1u

1

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

1

z

>- | B1u

1

−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

θ

−θ

0

−2

0

> −2

−θ

θ

0

2

0

> (x, y)

>- | E2u

2

0

0

−2

2

0

> −2

0

0

2

−2

0

0

>- | E3u

2

−θ

θ

0

−2

0

> −2

θ

−θ

0

2

0

>

>} |-

Antiprismatic groups (D_nd)

The antiprismatic groups are denoted by D_nd. These groups are characterized by i) an n-fold proper rotation axis C_n; ii) n 2-fold proper rotation axes C₂ normal to C_n; iii) n mirror planes σ_d which contain C_n. The D_1d group is the same as the C_2h group in the reflection groups section.
! Point
Group !! Canonical
group !! Order !! Character Table
|-
| D_2d >

D4

{> style="text-align:center" |

E

4  > C2

2'  > 2 σd

|- | 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

>- | E

2

0

−2

0

> (Rx, Ry), (x, y)

>- |} |- | D3d

D6

{|> style="text-align:center" |

E

3  > 3 C2

> 2 S6  | 3 σd  | colspan="2" |   |- | A1g

1

1

1

1

1

1

> x2 + y2, z2 |- | A2g

1

1

−1

1

1

> Rz

>- | Eg

2

−1

0

2

−1

> (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

D8

{|>style="text-align:center" |

E

8  > 2 C4

83 > C2

2'  > 4 σd  | colspan="2" | θ=21/2 |- | A1

1

1

1

1

1

1

>

2 + y2, z2 >- | A2

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

θ

0

−θ

−2

0

> (x, y)

>- | E2

2

0

−2

0

2

0

>

2 − y2, xy) >- | E3

2

−θ

0

θ

−2

0

> (Rx, Ry)

>- |} |- | D5d

D10

{|>style="text-align:center" |

> 2 C5

52 > 5 C2

> 2 S10

103 > 5 σd  | colspan="2" | θ=2π/5 |- | A1g

1

1

1

1

1

1

1

1

> x2 + y2, z2 |- | A2g

1

1

1

−1

1

1

1

> Rz

>- | E1g

2

2 cos(θ)

2 cos(2θ)

> 2

2 cos(2θ)

2 cos(θ)

> (Rx, Ry)

>- | E2g

2

2 cos(2θ)

2 cos(θ)

> 2

2 cos(θ)

2 cos(2θ)

>

2 − y2, xy) >- | A1u

1

1

1

> −1

−1

−1

−1

>- | A2u

1

1

1

> −1

−1

−1

1

z

>- | E1u

2

2 cos(θ)

2 cos(2θ)

> −2

−2 cos(2θ)

−2 cos(θ)

> (x, y)

>- | E2u

2

2 cos(2θ)

2 cos(θ)

> −2

−2 cos(θ)

−2 cos(2θ)

>

>- |} |- | D6d

D12

{|>style="text-align:center" |

> 2 S12

6  > 2 S4

3  > 2 S125

2  > 6 C2'

d  > colspan="2" | θ=31/2 |- | A1

1

1

1

1

1

1

1

1

>

2 + y2, z2 >- | A2

1

1

1

1

1

1

1

−1

> Rz

>- | B1

1

−1

1

−1

1

−1

1

1

>

>- | B2

1

−1

1

−1

1

−1

1

−1

> z

>- | E1

2

θ

1

0

> −θ

−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

>

>- | E4

2

−1

−1

2

−1

−1

2

0

>

>- | E5

2

−θ

1

0

> θ

−2

0

> (Rx, Ry)

>- |} |-

Polyhedral

Polyhedron

In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

 symmetries

These symmetries are characterized by having more than one proper rotation axis of order greater than 2.

Cubic groups

These polyhedral groups are characterized by not having a C₅ proper rotation axis.
! Point
Group !! Canonical
group !! Order !! Character Table
|-
| T >

A4

{|> style="text-align:center" |

E

3  > 4 C32 | 3 C2  | colspan="2" | θ=e2π i/3 |- | A

1

1

1

1

> x2 + y2 + z2 |- | E

1 1

θC > θC θ  | 1 1

> (2 z2 − x2 − y2, x2 − y2) |- | T

3

0

0

> (Rx, Ry, Rz), (x, y, z) | (xy, xz, yz) |- |} |- | Td

S4

{|> style="text-align:center" |

E

3  > 3 C2

4  > 6 σd  | colspan="2" |   |- | A1

1

1

1

1

1

> x2 + y2 + z2 |- | A2

1

1

1

−1

−1

>- | E

2

−1

2

0

0

> (2 z2 − x2 − y2, x2 − y2) |- | T1

3

0

−1

1

> (Rx, Ry, Rz)

>- | T2

3

0

−1

−1

> (x, y, z)

>- |} |- | Th

Z2×A4

{|>style="text-align:center" |

E

3  > 4 C32 | 3 C2

> 4 S6

65 > 3 σh  | colspan="2" | θ=e2π i/3 |- | Ag

1

1

1

1

1

1

1

>

2 + y2 + z2 >- | Au

1

1

1

1

−1

−1

−1

>

>- | Eg

1 > θ  θC | θC θ  | 1 1

1 > θ  θC | θC θ  | 1 1 |   | (2 z2 − x2 − y2, x2 − y2) |- | Eu

1 > θ  θC | θC θ  | 1 1

−1 > −θ  −θC | −θC −θ  | −1 −1 |

>- | Tg

3

0

0

−1

3

0

0

> (Rx, Ry, Rz) | (xy, xz, yz) |- | Tu

3

0

0

−1

−3

0

0

> (x, y, z)

>- |} |- | O

S4

{|>style="text-align:center" |

> 6 C4  | 3 C2  (C42) | 8 C3

2  > colspan="2" |   |- | A1

1

1

1

1

1

> x2 + y2 + z2 |- | A2

1

−1

1

1

−1

>- | E

2

0

2

−1

0

> (2 z2 − x2 − y2, x2 − y2) |- | T1

3

1

−1

0

> (Rx, Ry, Rz), (x, y, z) |   |- | T2

3

−1

−1

0

>

>- |} |- | Oh | Z2×S4

{|>style="text-align:center" |

> 8 C3

2  > 6 C4  | 3 C2  (C42) | i

4  > 8 S6

h  > 6 σd  | colspan="2" |   |- | A1g

1

1

1

1

1

1

1

1

1

>

2 + y2 + z2 >- | A2g

1

1

−1

−1

1

1

−1

1

1

>

>- | Eg

2

−1

0

0

2

2

0

−1

2

>   | (2 z2 − x2 − y2, x2 − y2) |- | T1g

3

0

−1

1

−1

3

1

0

−1

> (Rx, Ry, Rz) |   |- | T2g

3

0

1

−1

−1

3

−1

0

−1

>

>- | A1u

1

1

1

1

> −1

−1

−1

−1

>

>- | A2u

1

1

−1

−1

> −1

1

−1

−1

>

>- | Eu

2

−1

0

0

2

−2

0

1

−2

>

>- | T1u

3

0

−1

1

> −3

−1

0

1

> (x, y, z)

>- | T2u

3

0

1

−1

> −3

1

0

1

>

>} |-

Icosahedral groups

These polyhedral groups are characterized by having a C₅ proper rotation axis.
! Point
Group !! Canonical
group !!Order !! Character Table
|-
| I >

A5

{|> style="text-align:center" |

E

5  > 12 C52 | 20 C3  | 15 C2  | colspan="2" | θ=π/5 |- | A

1

1

1

1

1

> x2 + y2 + z2 |- | T1

3

2 cos(θ)

2 cos(3θ)

0

> (Rx, Ry, Rz), (x, y, z)

>- | T2

3

2 cos(3θ)

2 cos(θ)

0

>

>- | G

4

−1

−1

1

0

>- | H

5

0

0

−1

1

> (2 z2 − x2 − y2, x2 − y2, xy, xz, yz) |- |} |- | Ih

Z2×A5

{| >style="text-align:center" |

E

5  > 12 C52 | 20 C3  | 15 C2

> 12 S10  | 12 S103 | 20 S6  | 15 σ | colspan="2" | θ=π/5 |- | Ag

1

1

1

1

1

1

1

1

1

1

> x2 + y2 + z2 |- | T1g

3

2 cos(θ)

2 cos(3θ)

0

> 3

2 cos(3θ)

2 cos(θ)

0

> (Rx, Ry, Rz)

>- | T2g

3

2 cos(3θ)

2 cos(θ)

0

> 3

2 cos(θ)

2 cos(3θ)

0

−1

>- | Gg

4

−1

−1

1

0

4

−1

−1

1

>

>- | Hg

5

0

0

−1

1

5

0

0

−1

1

> (2 z2 − x2 − y2, x2 − y2, xy, xz, yz) |- | Au

1

1

1

1

> −1

−1

−1

−1

>

>- | T1u

3

2 cos(θ)

2 cos(3θ)

0

> −3

−2 cos(3θ)

−2 cos(θ)

0

> (x, y, z)

>- | T2u

3

2 cos(3θ)

2 cos(θ)

0

> −3

−2 cos(θ)

−2 cos(3θ)

0

>

>- | Gu

4

−1

−1

1

> −4

1

1

−1

>

>- | Hu

5

0

0

−1

1

−5

0

0

1

>

>- |} |-

Linear (cylindrical) groups

These groups are characterized by having a proper rotation axis C_∞ around which the symmetry is invariant to any rotation.
! Point
Group !! Character Table
|-
| C_∞v
| align="left" |
{| style="text-align:center"
|   >

E

∞Φ > ... | ∞ σv  | colspan="2" |   |- | A1=Σ+

1

1

...

1

> x2 + y2, z2 |- | A2=Σ−

1

1

...

−1

z >   |- | E1=Π

2

2 cos(Φ)

...

> (x, y), (Rx, Ry)

>- | E2=Δ

2

2 cos(2Φ)

...

>

2 - y2, xy) >- | E3=Φ

2

2 cos(3Φ)

...

>

>- | ...

...

...

...

...

|} |- | D∞h | align="left" | {| sty>e="text-align:center" |

E

2 C∞Φ

> ∞ σv

> 2 S∞Φ

...

2  > colspan="2" |   |- | Σg+

1

1

...

1

1

1

...

>

2 + y2, z2 >- | Σg−

1

1

> −1

1

1

...

> Rz

>- | Πg

2

2 cos(Φ)

...

0

2

−2 cos(Φ)

..

> (Rx, Ry)

>- | Δg

2

2 cos(2Φ)

...

0

2

2 cos(2Φ)

..

>

2 − y2, xy) >- | ...

...

...

...

...

...

...

...

...

>- | Σu+

1

1

> 1

−1

−1

...

> z

>- | Σu−

1

1

> −1

−1

−1

...

>

>- | Πu

2

2 cos(Φ)

> 0

−2

2 cos(Φ)

..

> (x, y)

>- | Δu

2

2 cos(2Φ)

> 0

−2

−2 cos(2Φ)

..

>

>- | ...

...

...

...

...

...

...

...

...

>} |-

See also

• Linear combination of atomic orbitals molecular orbital method
  Linear combination of atomic orbitals molecular orbital method
  A linear combination of atomic orbitals or LCAO is a quantum superposition of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry. In quantum mechanics, electron configurations of atoms are described as wavefunctions...
  
  
• Raman spectroscopy
  Raman spectroscopy
  Raman spectroscopy is a spectroscopic technique used to study vibrational, rotational, and other low-frequency modes in a system.It relies on inelastic scattering, or Raman scattering, of monochromatic light, usually from a laser in the visible, near infrared, or near ultraviolet range...
  
  
• Vibrational spectroscopy (molecular vibration)
  Molecular vibration
  A molecular vibration occurs when atoms in a molecule are in periodic motion while the molecule as a whole has constant translational and rotational motion...
  
  
• List of small groups
• Cubic harmonic
  Cubic harmonic
  In fields like computational chemistry and solid-state and condensed matter physics the so called atomic orbitals, or spin-orbitals, as they appear in textbooks on quantum physics, are often partially replaced by cubic harmonics for a number of reasons....
  
  s