List of character tables for chemically important 3D point groups

This lists the character tables for the more common molecular point groups used in the study of molecular symmetry. These tables are based on the group-theoretical treatment of the symmetry operations present in common molecules, and are useful in molecular spectroscopy and quantum chemistry. Information regarding the use of the tables, as well as more extensive lists of them, can be found in the references.[1][2][3][4][5]

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 (number of invariant symmetry operations). The finite group notation used is: Z_n: cyclic group of order n, D_n: dihedral group isomorphic to the symmetry group of an n–sided regular polygon, S_n: symmetric group 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, known as Mulliken symbols,[6] 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 operations 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 symbol i used in the body of the table denotes the imaginary unit: i² = −1. Used in a column heading, it denotes the operation of inversion. A superscripted uppercase "C" denotes complex conjugation.

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).

A further column is included in some tables, such as those of Salthouse and Ware[7] For example,

$C_{s}$	$E$	$\sigma _{h}$
$A'$	$1$	$1$	$x$ , $y$ , $R_{z}$	$x^{2}$ , $y^{2}$ , $z^{2}$ , $xy$	$zx^{2}$ , $yz^{2}$ , $x^{2}y$ , $xy^{2}$ , $x^{3}$ , $y^{3}$
$A''$	$1$	$-1$	$z$ , $R_{x}$ , $R_{y}$	$yz$ , $xz$	$z^{3}$ , $xyz$ , $y^{2}z$ , $x^{2}z$

The last column relates to cubic functions which may be used in applications regarding f orbitals in atoms.

Character tables

Nonaxial symmetries

These groups are characterized by a lack of a proper rotation axis, noting that a $C_{1}$ rotation is considered the identity operation. These groups have involutional symmetry: the only nonidentity operation, if any, is its own inverse.

In the group $C_{1}$ , all functions of the Cartesian coordinates and rotations about them transform as the $A$ irreducible representation.

Point Group

Canonical Group

Order

Character Table

C_{1}

Z_{1}

1

	$E$
$A$	$1$

C_{i}

Z_{2}

2

	$E$	$i$
$A_{g}$	$1$	$1$	$R_{x}$ , $R_{y}$ , $R_{z}$	$x^{2}$ , $y^{2}$ , $z^{2}$ , $xy$ , $xz$ , $yz$
$A_{u}$	$1$	$-1$	$x$ , $y$ , $z$

C_{s}

Z_{2}

2

	$E$	$\sigma _{h}$
$A'$	$1$	$1$	$x$ , $y$ , $R_{z}$	$x^{2}$ , $y^{2}$ , $z^{2}$ , $xy$
$A''$	$1$	$-1$	$z$ , $R_{x}$ , $R_{y}$	$yz$ , $xz$

Cyclic symmetries

The families of groups with these symmetries have only one rotation axis.

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

2

	E	C₂
A	1	1	R_z, z	x², y², z², xy
B	1	−1	R_x, R_y, x, y	xz, yz

C₃

Z₃

3

	E	C₃	C₃²	θ = e^{2πi /3}
A	1	1	1	R_z, z	x² + y²
E	1 1	θ θ^C	θ^C θ	(R_x, R_y), (x, y)	(x² - y², xy), (xz, yz)

C₄

Z₄

4

	E	C₄	C₂	C₄³
A	1	1	1	1	R_z, z	x² + y², z²
B	1	−1	1	−1		x² − y², xy
E	1 1	i −i	−1 −1	−i i	(R_x, R_y), (x, y)	(xz, yz)

C₅

Z₅

5

	E	C₅	C₅²	C₅³	C₅⁴	θ = e^{2πi /5}
A	1	1	1	1	1	R_z, z	x² + y², z²
E₁	1 1	θ θ^C	θ² (θ²)^C	(θ²)^C θ²	θ^C θ	(R_x, R_y), (x, y)	(xz, yz)
E₂	1 1	θ² (θ²)^C	θ^C θ	θ θ^C	(θ²)^C θ²		(x² - y², xy)

C₆

Z₆

6

	E	C₆	C₃	C₂	C₃²	C₆⁵	θ = e^{2πi /6}
A	1	1	1	1	1	1	R_z, z	x² + y², z²
B	1	−1	1	−1	1	−1
E₁	1 1	θ θ^C	−θ^C −θ	−1 −1	−θ −θ^C	θ^C −θ	(R_x, R_y), (x, y)	(xz, yz)
E₂	1 1	−θ^C −θ	−θ −θ^C	1 1	−θ^C −θ	−θ −θ^C		(x² − y², xy)

C₈

Z₈

8

	E	C₈	C₄	C₈³	C₂	C₈⁵	C₄³	C₈⁷	θ = e^{2πi /8}
A	1	1	1	1	1	1	1	1	R_z, z	x² + y², z²
B	1	−1	1	−1	1	−1	1	−1
E₁	1 1	θ θ^C	i −i	−θ^C −θ	−1 −1	−θ −θ^C	−i i	θ^C θ	(R_x, R_y), (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	−θ −θ^C	i −i	θ^C θ	−1 −1	θ θ^C	−i i	−θ^C −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

4

	E	C₂	i	σ_h
A_g	1	1	1	1	R_z	x², y², z², xy
B_g	1	−1	1	−1	R_x, R_y	xz, yz
A_u	1	1	−1	−1	z
B_u	1	−1	−1	1	x, y

C_3h

Z₆

6

	E	C₃	C₃²	σ_h	S₃	S₃⁵	θ = e^{2πi /3}
A'	1	1	1	1	1	1	R_z	x² + y², z²
E'	1 1	θ θ^C	θ^C θ	1 1	θ θ^C	θ^C θ	(x, y)	(x² − y², xy)
A''	1	1	1	−1	−1	−1	z
E''	1 1	θ θ^C	θ^C θ	−1 −1	−θ −θ^C	−θ^C −θ	(R_x, R_y)	(xz, yz)

C_4h

Z₂ × Z₄

8

	E	C₄	C₂	C₄³	i	S₄³	σ_h	S₄
A_g	1	1	1	1	1	1	1	1	R_z	x² + y², z²
B_g	1	−1	1	−1	1	−1	1	−1		x² − y², xy
E_g	1 1	i −i	−1 −1	−i i	1 1	i −i	−1 −1	−i i	(R_x, R_y)	(xz, yz)
A_u	1	1	1	1	−1	−1	−1	−1	z
B_u	1	−1	1	−1	−1	1	−1	1
E_u	1 1	i −i	−1 −1	−i i	−1 −1	−i i	1 1	i −i	(x, y)

C_5h

Z₁₀

10

	E	C₅	C₅²	C₅³	C₅⁴	σ_h	S₅	S₅⁷	S₅³	S₅⁹	θ = e^{2πi /5}
A'	1	1	1	1	1	1	1	1	1	1	R_z	x² + y², z²
E₁'	1 1	θ θ^C	θ² (θ²)^C	(θ²)^C θ²	θ^C θ	1 1	θ θ^C	θ² (θ²)^C	(θ²)^C θ²	θ^C θ	(x, y)
E₂'	1 1	θ² (θ²)^C	θ^C θ	θ θ^C	(θ²)^C θ²	1 1	θ² (θ²)^C	θ^C θ	θ θ^C	(θ²)^C θ²		(x² - y², xy)
A''	1	1	1	1	1	−1	−1	−1	−1	−1	z
E₁''	1 1	θ θ^C	θ² (θ²)^C	(θ²)^C θ²	θ^C θ	−1 −1	−θ -θ^C	−θ² −(θ²)^C	−(θ²)^C −θ²	−θ^C −θ	(R_x, R_y)	(xz, yz)
E₂''	1 1	θ² (θ²)^C	θ^C θ	θ θ^C	(θ²)^C θ²	−1 −1	−θ² −(θ²)^C	−θ^C −θ	−θ −θ^C	−(θ²)^C −θ²

C_6h

Z₂ × Z₆

12

	E	C₆	C₃	C₂	C₃²	C₆⁵	i	S₃⁵	S₆⁵	σ_h	S₆	S₃	θ = e^{2πi /6}
A_g	1	1	1	1	1	1	1	1	1	1	1	1	R_z	x² + y², z²
B_g	1	−1	1	−1	1	−1	1	−1	1	−1	1	−1
E_1g	1 1	θ θ^C	−θ^C −θ	−1 −1	−θ −θ^C	θ^C θ	1 1	θ θ^C	−θ^C −θ	−1 −1	−θ −θ^C	θ^C θ	(R_x, R_y)	(xz, yz)
E_2g	1 1	−θ^C −θ	−θ −θ^C	1 1	−θ^C −θ	−θ −θ^C	1 1	−θ^C −θ	−θ −θ^C	1 1	−θ^C −θ	−θ −θ^C		(x² − y², xy)
A_u	1	1	1	1	1	1	−1	−1	−1	−1	−1	−1	z
B_u	1	−1	1	−1	1	−1	−1	1	−1	1	−1	1
E_1u	1 1	θ θ^C	−θ^C −θ	−1 −1	−θ −θ^C	θ^C θ	−1 −1	−θ −θ^C	θ^C θ	1 1	θ θ^C	−θ^C −θ	(x, y)
E_2u	1 1	−θ^C −θ	−θ −θ^C	1 1	−θ^C −θ	−θ −θ^C	−1 −1	θ^C θ	θ θ^C	−1 −1	θ^C θ	θ θ^C

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)

4

	E	C₂	σ_v	σ_v'
A₁	1	1	1	1	z	x² , y², z²
A₂	1	1	−1	−1	R_z	xy
B₁	1	−1	1	−1	R_y, x	xz
B₂	1	−1	−1	1	R_x, y	yz

C_3v

D₃

6

	E	2 C₃	3 σ_v
A₁	1	1	1	z	x² + y², z²
A₂	1	1	−1	R_z
E	2	−1	0	(R_x, R_y), (x, y)	(x² − y², xy), (xz, yz)

C_4v

D₄

8

	E	2 C₄	C₂	2 σ_v	2 σ_d
A₁	1	1	1	1	1	z	x² + y², z²
A₂	1	1	1	−1	−1	R_z
B₁	1	−1	1	1	−1		x² − y²
B₂	1	−1	1	−1	1		xy
E	2	0	−2	0	0	(R_x, R_y), (x, y)	(xz, yz)

C_5v

D₅

10

	E	2 C₅	2 C₅²	5 σ_v	θ = 2π/5
A₁	1	1	1	1	z	x² + y², z²
A₂	1	1	1	−1	R_z
E₁	2	2 cos(θ)	2 cos(2θ)	0	(R_x, R_y), (x, y)	(xz, yz)
E₂	2	2 cos(2θ)	2 cos(θ)	0		(x² − y², xy)

C_6v

D₆

12

	E	2 C₆	2 C₃	C₂	3 σ_v	3 σ_d
A₁	1	1	1	1	1	1	z	x² + y², z²
A₂	1	1	1	1	−1	−1	R_z
B₁	1	−1	1	−1	1	−1
B₂	1	−1	1	−1	−1	1
E₁	2	1	−1	−2	0	0	(R_x, R_y), (x, y)	(xz, yz)
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_i group in the nonaxial groups section. S_n groups with an odd value of n are identical to C_nh groups of same n and are therefore not considered here (in particular, S₁ is identical to C_s).

The S₈ table reflects the 2007 discovery of errors in older references.[4] Specifically, (R_x, R_y) transform not as E₁ but rather as E₃.

Point
Group

Canonical
group

Order

Character Table

S₄

Z₄

4

	E	S₄	C₂	S₄³
A	1	1	1	1	R_z,	x² + y², z²
B	1	−1	1	−1	z	x² − y², xy
E	1 1	i −i	−1 −1	−i i	(R_x, R_y), (x, y)	(xz, yz)

S₆

Z₆

6

	E	S₆	C₃	i	C₃²	S₆⁵	θ = e^{2πi /6}
A_g	1	1	1	1	1	1	R_z	x² + y², z²
E_g	1 1	θ^C θ	θ θ^C	1 1	θ^C θ	θ θ^C	(R_x, R_y)	(x² − y², xy), (xz, yz)
A_u	1	−1	1	−1	1	−1	z
E_u	1 1	−θ^C −θ	θ θ^C	−1 −1	θ^C θ	−θ −θ^C	(x, y)

S₈

Z₈

8

	E	S₈	C₄	S₈³	i	S₈⁵	C₄²	S₈⁷	θ = e^{2πi /8}
A	1	1	1	1	1	1	1	1	R_z	x² + y², z²
B	1	−1	1	−1	−1	−1	1	−1	z
E₁	1 1	θ θ^C	i −i	−θ^C −θ	−1 −1	−θ −θ^C	−i i	θ^C θ	(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	−θ^C −θ	−i i	θ θ^C	−1 −1	θ^C θ	i −i	−θ −θ^C	(R_x, R_y)	(xz, yz)

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)

4

	E	C₂(z)	C₂(x)	C₂(y)
A	1	1	1	1		x², y², z²
B₁	1	1	−1	−1	R_z, z	xy
B₂	1	−1	−1	1	R_y, y	xz
B₃	1	−1	1	−1	R_x, x	yz

D₃

6

	E	2 C₃	3 C'₂
A₁	1	1	1		x² + y², z²
A₂	1	1	−1	R_z, z
E	2	−1	0	(R_x, R_y), (x, y)	(x² − y², xy), (xz, yz)

D₄

8

	E	2 C₄	C₂	2 C₂'	2 C₂''
A₁	1	1	1	1	1		x² + y², z²
A₂	1	1	1	−1	−1	R_z, z
B₁	1	−1	1	1	−1		x² − y²
B₂	1	−1	1	−1	1		xy
E	2	0	−2	0	0	(R_x, R_y), (x, y)	(xz, yz)

D₅

10

	E	2 C₅	2 C₅²	5 C₂	θ=2π/5
A₁	1	1	1	1		x² + y², z²
A₂	1	1	1	−1	R_z, z
E₁	2	2 cos(θ)	2 cos(2θ)	0	(R_x, R_y), (x, y)	(xz, yz)
E₂	2	2 cos(2θ)	2 cos(θ)	0		(x² − y², xy)

D₆

12

	E	2 C₆	2 C₃	C₂	3 C₂'	3 C₂''
A₁	1	1	1	1	1	1		x² + y², z²
A₂	1	1	1	1	−1	−1	R_z, z
B₁	1	−1	1	−1	1	−1
B₂	1	−1	1	−1	−1	1
E₁	2	1	−1	−2	0	0	(R_x, R_y), (x, y)	(xz, yz)
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.[4] 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)

8

	E	C₂	C₂(x)	C₂(y)	i	σ(xy)	σ(xz)	σ(yz)
A_g	1	1	1	1	1	1	1	1		x², y², z²
B_1g	1	1	−1	−1	1	1	−1	−1	R_z	xy
B_2g	1	−1	−1	1	1	−1	1	−1	R_y	xz
B_3g	1	−1	1	−1	1	−1	−1	1	R_x	yz
A_u	1	1	1	1	−1	−1	−1	−1
B_1u	1	1	−1	−1	−1	−1	1	1	z
B_2u	1	−1	−1	1	−1	1	−1	1	y
B_3u	1	−1	1	−1	−1	1	1	−1	x

D_3h

D₆

12

	E	2 C₃	3 C₂	σ_h	2 S₃	3 σ_v
A₁'	1	1	1	1	1	1		x² + y², z²
A₂'	1	1	−1	1	1	−1	R_z
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	(R_x, R_y)	(xz, yz)

D_4h

Z₂×D₄

16

	E	2 C₄	C₂	2 C₂'	2 C₂''	i	2 S₄	σ_h	2 σ_v	2 σ_d
A_1g	1	1	1	1	1	1	1	1	1	1		x² + y², z²
A_2g	1	1	1	−1	−1	1	1	1	−1	−1	R_z
B_1g	1	−1	1	1	−1	1	−1	1	1	−1		x² − y²
B_2g	1	−1	1	−1	1	1	−1	1	−1	1		xy
E_g	2	0	−2	0	0	2	0	−2	0	0	(R_x, R_y)	(xz, yz)
A_1u	1	1	1	1	1	−1	−1	−1	−1	−1
A_2u	1	1	1	−1	−1	−1	−1	−1	1	1	z
B_1u	1	−1	1	1	−1	−1	1	−1	−1	1
B_2u	1	−1	1	−1	1	−1	1	−1	1	−1
E_u	2	0	−2	0	0	−2	0	2	0	0	(x, y)

D_5h

D₁₀

20

	E	2 C₅	2 C₅²	5 C₂	σ_h	2 S₅	2 S₅³	5 σ_v	θ=2π/5
A₁'	1	1	1	1	1	1	1	1		x² + y², z²
A₂'	1	1	1	−1	1	1	1	−1	R_z
E₁'	2	2 cos(θ)	2 cos(2θ)	0	2	2 cos(θ)	2 cos(2θ)	0	(x, y)
E₂'	2	2 cos(2θ)	2 cos(θ)	0	2	2 cos(2θ)	2 cos(θ)	0		(x² − y², xy)
A₁''	1	1	1	1	−1	−1	−1	−1
A₂''	1	1	1	−1	−1	−1	−1	1	z
E₁''	2	2 cos(θ)	2 cos(2θ)	0	−2	−2 cos(θ)	−2 cos(2θ)	0	(R_x, R_y)	(xz, yz)
E₂''	2	2 cos(2θ)	2 cos(θ)	0	−2	−2 cos(2θ)	−2 cos(θ)	0

D_6h

Z₂×D₆

24

	E	2 C₆	2 C₃	C₂	3 C₂'	3 C₂''	i	2 S₃	2 S₆	σ_h	3 σ_d	3 σ_v
A_1g	1	1	1	1	1	1	1	1	1	1	1	1		x² + y², z²
A_2g	1	1	1	1	−1	−1	1	1	1	1	−1	−1	R_z
B_1g	1	−1	1	−1	1	−1	1	−1	1	−1	1	−1
B_2g	1	−1	1	−1	−1	1	1	−1	1	−1	−1	1
E_1g	2	1	−1	−2	0	0	2	1	−1	−2	0	0	(R_x, R_y)	(xz, yz)
E_2g	2	−1	−1	2	0	0	2	−1	−1	2	0	0		(x² − y², xy)
A_1u	1	1	1	1	1	1	−1	−1	−1	−1	−1	−1
A_2u	1	1	1	1	−1	−1	−1	−1	−1	−1	1	1	z
B_1u	1	−1	1	−1	1	−1	−1	1	−1	1	−1	1
B_2u	1	−1	1	−1	−1	1	−1	1	−1	1	1	−1
E_1u	2	1	−1	−2	0	0	−2	−1	1	2	0	0	(x, y)
E_2u	2	−1	−1	2	0	0	−2	1	1	−2	0	0

D_8h

Z₂×D₈

32

	E	2 C₈	2 C₈³	2 C₄	C₂	4 C₂'	4 C₂''	i	2 S₈³	2 S₈	2 S₄	σ_h	4 σ_d	4 σ_v	θ=2^1/2
A_1g	1	1	1	1	1	1	1	1	1	1	1	1	1	1		x² + y², z²
A_2g	1	1	1	1	1	−1	−1	1	1	1	1	1	−1	−1	R_z
B_1g	1	−1	−1	1	1	1	−1	1	−1	−1	1	1	1	−1
B_2g	1	−1	−1	1	1	−1	1	1	−1	−1	1	1	−1	1
E_1g	2	θ	−θ	0	−2	0	0	2	θ	−θ	0	−2	0	0	(R_x, R_y)	(xz, yz)
E_2g	2	0	0	−2	2	0	0	2	0	0	−2	2	0	0		(x² − y², xy)
E_3g	2	−θ	θ	0	−2	0	0	2	−θ	θ	0	−2	0	0
A_1u	1	1	1	1	1	1	1	−1	−1	−1	−1	−1	−1	−1
A_2u	1	1	1	1	1	−1	−1	−1	−1	−1	−1	−1	1	1	z
B_1u	1	−1	−1	1	1	1	−1	−1	1	1	−1	−1	−1	1
B_2u	1	−1	−1	1	1	−1	1	−1	1	1	−1	−1	1	−1
E_1u	2	θ	−θ	0	−2	0	0	−2	−θ	θ	0	2	0	0	(x, y)
E_2u	2	0	0	−2	2	0	0	−2	0	0	2	−2	0	0
E_3u	2	−θ	θ	0	−2	0	0	−2	θ	−θ	0	2	0	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

8

	E	2 S₄	C₂	2 C₂'	2 σ_d
A₁	1	1	1	1	1		x², y², z²
A₂	1	1	1	−1	−1	R_z
B₁	1	−1	1	1	−1		x² − y²
B₂	1	−1	1	−1	1	z	xy
E	2	0	−2	0	0	(R_x, R_y), (x, y)	(xz, yz)

D_3d

D₆

12

	E	2 C₃	3 C₂	i	2 S₆	3 σ_d
A_1g	1	1	1	1	1	1		x² + y², z²
A_2g	1	1	−1	1	1	−1	R_z
E_g	2	−1	0	2	−1	0	(R_x, R_y)	(x² − y², xy), (xz, yz)
A_1u	1	1	1	−1	−1	−1
A_2u	1	1	−1	−1	−1	1	z
E_u	2	−1	0	−2	1	0	(x, y)

D_4d

D₈

16

	E	2 S₈	2 C₄	2 S₈³	C₂	4 C₂'	4 σ_d	θ=2^1/2
A₁	1	1	1	1	1	1	1		x² + y², z²
A₂	1	1	1	1	1	−1	−1	R_z
B₁	1	−1	1	−1	1	1	−1
B₂	1	−1	1	−1	1	−1	1	z
E₁	2	θ	0	−θ	−2	0	0	(x, y)
E₂	2	0	−2	0	2	0	0		(x² − y², xy)
E₃	2	−θ	0	θ	−2	0	0	(R_x, R_y)	(xz, yz)

D_5d

D₁₀

20

	E	2 C₅	2 C₅²	5 C₂	i	2 S₁₀	2 S₁₀³	5 σ_d	θ=2π/5
A_1g	1	1	1	1	1	1	1	1		x² + y², z²
A_2g	1	1	1	−1	1	1	1	−1	R_z
E_1g	2	2 cos(θ)	2 cos(2θ)	0	2	2 cos(2θ)	2 cos(θ)	0	(R_x, R_y)	(xz, yz)
E_2g	2	2 cos(2θ)	2 cos(θ)	0	2	2 cos(θ)	2 cos(2θ)	0		(x² − y², xy)
A_1u	1	1	1	1	−1	−1	−1	−1
A_2u	1	1	1	−1	−1	−1	−1	1	z
E_1u	2	2 cos(θ)	2 cos(2θ)	0	−2	−2 cos(2θ)	−2 cos(θ)	0	(x, y)
E_2u	2	2 cos(2θ)	2 cos(θ)	0	−2	−2 cos(θ)	−2 cos(2θ)	0

D_6d

D₁₂

24

	E	2 S₁₂	2 C₆	2 S₄	2 C₃	2 S₁₂⁵	C₂	6 C₂'	6 σ_d	θ=3^1/2
A₁	1	1	1	1	1	1	1	1	1		x² + y², z²
A₂	1	1	1	1	1	1	1	−1	−1	R_z
B₁	1	−1	1	−1	1	−1	1	1	−1
B₂	1	−1	1	−1	1	−1	1	−1	1	z
E₁	2	θ	1	0	−1	−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	0
E₄	2	−1	−1	2	−1	−1	2	0	0
E₅	2	−θ	1	0	−1	θ	−2	0	0	(R_x, R_y)	(xz, yz)

Polyhedral 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

12

	E	4 C₃	4 C₃²	3 C₂	θ=e^{2π i/3}
A	1	1	1	1		x² + y² + z²
E	1 1	θ θ^C	θ^C θ	1 1		(2 z² − x² − y², x² − y²)
T	3	0	0	−1	(R_x, R_y, R_z), (x, y, z)	(xy, xz, yz)

T_d

S₄

24

	E	8 C₃	3 C₂	6 S₄	6 σ_d
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	−1	(R_x, R_y, R_z)
T₂	3	0	−1	−1	1	(x, y, z)	(xy, xz, yz)

T_h

Z₂×A₄

24

	E	4 C₃	4 C₃²	3 C₂	i	4 S₆	4 S₆⁵	3 σ_h	θ=e^{2π i/3}
A_g	1	1	1	1	1	1	1	1		x² + y² + z²
A_u	1	1	1	1	−1	−1	−1	−1
E_g	1 1	θ θ^C	θ^C θ	1 1	1 1	θ θ^C	θ^C θ	1 1		(2 z² − x² − y², x² − y²)
E_u	1 1	θ θ^C	θ^C θ	1 1	−1 −1	−θ −θ^C	−θ^C −θ	−1 −1
T_g	3	0	0	−1	3	0	0	−1	(R_x, R_y, R_z)	(xy, xz, yz)
T_u	3	0	0	−1	−3	0	0	1	(x, y, z)

O

S₄

24

	E	6 C₄	3 C₂ (C₄²)	8 C₃	6 C'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	−1	(R_x, R_y, R_z), (x, y, z)
T₂	3	−1	−1	0	1		(xy, xz, yz)

O_h

Z₂×S₄

48

	E	8 C₃	6 C₂	6 C₄	3 C₂ (C₄²)	i	6 S₄	8 S₆	3 σ_h	6 σ_d
A_1g	1	1	1	1	1	1	1	1	1	1		x² + y² + z²
A_2g	1	1	−1	−1	1	1	−1	1	1	−1
E_g	2	−1	0	0	2	2	0	−1	2	0		(2 z² − x² − y², x² − y²)
T_1g	3	0	−1	1	−1	3	1	0	−1	−1	(R_x, R_y, R_z)
T_2g	3	0	1	−1	−1	3	−1	0	−1	1		(xy, xz, yz)
A_1u	1	1	1	1	1	−1	−1	−1	−1	−1
A_2u	1	1	−1	−1	1	−1	1	−1	−1	1
E_u	2	−1	0	0	2	−2	0	1	−2	0
T_1u	3	0	−1	1	−1	−3	−1	0	1	1	(x, y, z)
T_2u	3	0	1	−1	−1	−3	1	0	1	−1

Icosahedral groups

These polyhedral groups are characterized by having a C₅ proper rotation axis.

Point
Group

Canonical
group

Order

Character Table

I

A₅

60

	E	12 C₅	12 C₅²	20 C₃	15 C₂	θ=π/5
A	1	1	1	1	1		x² + y² + z²
T₁	3	2 cos(θ)	2 cos(3θ)	0	−1	(R_x, R_y, R_z), (x, y, z)
T₂	3	2 cos(3θ)	2 cos(θ)	0	−1
G	4	−1	−1	1	0
H	5	0	0	−1	1		(2 z² − x² − y², x² − y², xy, xz, yz)

I_h

Z₂×A₅

120

	E	12 C₅	12 C₅²	20 C₃	15 C₂	i	12 S₁₀	12 S₁₀³	20 S₆	15 σ	θ=π/5
A_g	1	1	1	1	1	1	1	1	1	1		x² + y² + z²
T_1g	3	2 cos(θ)	2 cos(3θ)	0	−1	3	2 cos(3θ)	2 cos(θ)	0	−1	(R_x, R_y, R_z)
T_2g	3	2 cos(3θ)	2 cos(θ)	0	−1	3	2 cos(θ)	2 cos(3θ)	0	−1
G_g	4	−1	−1	1	0	4	−1	−1	1	0
H_g	5	0	0	−1	1	5	0	0	−1	1		(2 z² − x² − y², x² − y², xy, xz, yz)
A_u	1	1	1	1	1	−1	−1	−1	−1	−1
T_1u	3	2 cos(θ)	2 cos(3θ)	0	−1	−3	−2 cos(3θ)	−2 cos(θ)	0	1	(x, y, z)
T_2u	3	2 cos(3θ)	2 cos(θ)	0	−1	−3	−2 cos(θ)	−2 cos(3θ)	0	1
G_u	4	−1	−1	1	0	−4	1	1	−1	0
H_u	5	0	0	−1	1	−5	0	0	1	−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

	E	2 C_∞^Φ	...	∞ σ_v
A₁=Σ⁺	1	1	...	1	z	x² + y², z²
A₂=Σ⁻	1	1	...	−1	R_z
E₁=Π	2	2 cos(Φ)	...	0	(x, y), (R_x, R_y)	(xz, yz)
E₂=Δ	2	2 cos(2Φ)	...	0		(x² - y², xy)
E₃=Φ	2	2 cos(3Φ)	...	0
...	...	...	...	...

D_∞h

	E	2 C_∞^Φ	...	∞ σ_v	i	2 S_∞^Φ	...	∞ C₂
Σ_g⁺	1	1	...	1	1	1	...	1		x² + y², z²
Σ_g⁻	1	1	...	−1	1	1	...	−1	R_z
Π_g	2	2 cos(Φ)	...	0	2	−2 cos(Φ)	..	0	(R_x, R_y)	(xz, yz)
Δ_g	2	2 cos(2Φ)	...	0	2	2 cos(2Φ)	..	0		(x² − y², xy)
...	...	...	...	...	...	...	...	...
Σ_u⁺	1	1	...	1	−1	−1	...	−1	z
Σ_u⁻	1	1	...	−1	−1	−1	...	1
Π_u	2	2 cos(Φ)	...	0	−2	2 cos(Φ)	..	0	(x, y)
Δ_u	2	2 cos(2Φ)	...	0	−2	−2 cos(2Φ)	..	0
...	...	...	...	...	...	...	...	...

Notes

Drago, Russell S. (1977). Physical Methods in Chemistry. W.B. Saunders Company. ISBN 0-7216-3184-3.
Cotton, F. Albert (1990). Chemical Applications of Group Theory. John Wiley & Sons: New York. ISBN 0-471-51094-7.
Gelessus, Achim (2007-07-12). "Character tables for chemically important point groups". Jacobs University, Bremin; Computational Laboratory for Analysis, Modeling, and Visualization. Retrieved 2007-07-12.
Shirts, Randall B. (2007). "Correcting Two Long-Standing Errors in Point Group Symmetry Character Tables". Journal of Chemical Education. American Chemical Society. 84 (1882): 1882. Bibcode:2007JChEd..84.1882S. doi:10.1021/ed084p1882. Retrieved 2007-10-16.
Vanovschi, Vitalii. "POINT GROUP SYMMETRY CHARACTER TABLES". WebQC.Org. Retrieved 2008-10-29.
Mulliken, Robert S. (1933-02-15). "Electronic Structures of Polyatomic Molecules and Valence. IV. Electronic States, Quantum Theory of the Double Bond". Physical Review. American Physical Society (APS). 43 (4): 279–302. Bibcode:1933PhRv...43..279M. doi:10.1103/physrev.43.279. ISSN 0031-899X.
Salthouse, J.A.; Ware, M.J. (1972). Point group character tables and related data. Cambridge: Cambridge University Press. pp. 88 + v. ISBN 0-521-08139-4.

External links

Character tables for many more point groups (includes symmetry transformations of Cartesian products up to sixth order)

List of character tables for chemically important 3D point groups

Notation

Character tables

Nonaxial symmetries

Cyclic symmetries

Cyclic groups (C_n)

Reflection groups (C_nh)

Pyramidal groups (C_nv)

Improper rotation groups (S_n)

Dihedral symmetries

Dihedral groups (D_n)

Prismatic groups (D_nh)

Antiprismatic groups (D_nd)

Polyhedral symmetries

Cubic groups

Icosahedral groups

Linear (cylindrical) groups

See also

Notes

External links

Further reading

List of character tables for chemically important 3D point groups

Notation

Character tables

Nonaxial symmetries

Cyclic symmetries

Cyclic groups (Cn)

Reflection groups (Cnh)

Pyramidal groups (Cnv)

Improper rotation groups (Sn)

Dihedral symmetries

Dihedral groups (Dn)

Prismatic groups (Dnh)

Antiprismatic groups (Dnd)

Polyhedral symmetries

Cubic groups

Icosahedral groups

Linear (cylindrical) groups

See also

Notes

External links

Further reading

Cyclic groups (C_n)

Reflection groups (C_nh)

Pyramidal groups (C_nv)

Improper rotation groups (S_n)

Dihedral groups (D_n)

Prismatic groups (D_nh)

Antiprismatic groups (D_nd)