# 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: Zn: cyclic group of order n, Dn: dihedral group isomorphic to the symmetry group of an n–sided regular polygon, Sn: symmetric group on n letters, and An: 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 σh, 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 two rightmost columns indicate which irreducible representations describe the symmetry transformations of the three Cartesian coordinates (xy and z), rotations about those three coordinates (RxRy and Rz), and functions of the quadratic terms of the coordinates(x2y2z2xyxz, and yz).

The symbol i used in the body of the table denotes the imaginary unit: i 2 = −1. Used in a column heading, it denotes the operation of inversion. A superscripted uppercase "C" denotes complex conjugation.

## 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 (Cn)

The cyclic groups are denoted by Cn. These groups are characterized by an n-fold proper rotation axis Cn. 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 Rz, z x2, y2, z2, xy B 1 −1 Rx, Ry, x, y xz, yz
C3 Z3 3
 E C3 C32 θ = e2πi /3 A 1 1 1 Rz, z x2 + y2 E 1 1 θ  θC θC θ (Rx, Ry), (x, y) (x2 - y2, xy), (xz, yz)
C4 Z4 4
 E C4 C2 C43 A 1 1 1 1 Rz, z x2 + y2, z2 B 1 −1 1 −1 x2 − y2, xy E 1 1 i −i −1 −1 −i i (Rx, Ry), (x, y) (xz, yz)
C5 Z5 5
 E C5 C52 C53 C54 θ = e2πi /5 A 1 1 1 1 1 Rz, z x2 + y2, z2 E1 1 1 θ  θC θ2 (θ2)C (θ2)C θ2 θC θ (Rx, Ry), (x, y) (xz, yz) E2 1 1 θ2 (θ2)C θC θ θ  θC (θ2)C θ2 (x2 - y2, xy)
C6 Z6 6
 E C6 C3 C2 C32 C65 θ = e2πi /6 A 1 1 1 1 1 1 Rz, 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 (x2 − y2, xy)
C8 Z8 8
 E C8 C4 C83 C2 C85 C43 C87 θ = e2πi /8 A 1 1 1 1 1 1 1 1 Rz, 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 (x2 − y2, xy) E3 1 1 −θ  −θC i −i θC θ −1 −1 θ  θC −i i −θC −θ

#### Reflection groups (Cnh)

The reflection groups are denoted by Cnh. These groups are characterized by i) an n-fold proper rotation axis Cn; ii) a mirror plane σh normal to Cn. The C1h group is the same as the Cs group in the nonaxial groups section.

Point
Group
Canonical
group
Order Character Table
C2h Z2 × Z2 4
 E C2 i σh Ag 1 1 1 1 Rz x2, y2, z2, xy Bg 1 −1 1 −1 Rx, Ry xz, yz Au 1 1 −1 −1 z Bu 1 −1 −1 1 x, y
C3h Z6 6
 E C3 C32 σh S3 S35 θ = e2πi /3 A' 1 1 1 1 1 1 Rz x2 + y2, z2 E' 1 1 θ  θC θC θ 1 1 θ  θC θC θ (x, y) (x2 − 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) (xz, yz)
C4h Z2 × Z4 8
 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)
C5h Z10 10
 E C5 C52 C53 C54 σh S5 S57 S53 S59 θ = e2πi /5 A' 1 1 1 1 1 1 1 1 1 1 Rz 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 (x2 - y2, xy) A'' 1 1 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) (xz, yz) 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 12
 E C6 C3 C2 C32 C65 i S35 S65 σh S6 S3 θ = e2πi /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 θ  θC −θC −θ −1 −1 −θ  −θC θC θ 1 1 θ  θC −θC −θ −1 −1 −θ  −θC θC θ (Rx, Ry) (xz, yz) 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 (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 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 (Cnv)

The pyramidal groups are denoted by Cnv. These groups are characterized by i) an n-fold proper rotation axis Cn; ii) n mirror planes σv which contain Cn. The C1v group is the same as the Cs group in the nonaxial groups section.

Point
Group
Canonical
group
Order Character Table
C2v Z2 × Z2
(=D2)
4
 E C2 σv σv' A1 1 1 1 1 z x2, y2, z2 A2 1 1 −1 −1 Rz xy B1 1 −1 1 −1 Ry, x xz B2 1 −1 −1 1 Rx, y yz
C3v D3 6
 E 2 C3 3 σv A1 1 1 1 z x2 + y2, z2 A2 1 1 −1 Rz E 2 −1 0 (Rx, Ry), (x, y) (x2 − y2, xy), (xz, yz)
C4v D4 8
 E 2 C4 C2 2 σv 2 σd 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)
C5v D5 10
 E 2 C5 2 C52 5 σv θ = 2π/5 A1 1 1 1 1 z x2 + y2, z2 A2 1 1 1 −1 Rz E1 2 2 cos(θ) 2 cos(2θ) 0 (Rx, Ry), (x, y) (xz, yz) E2 2 2 cos(2θ) 2 cos(θ) 0 (x2 − y2, xy)
C6v D6 12
 E 2 C6 2 C3 C2 3 σv 3 σd 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)

#### Improper rotation groups (Sn)

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

The S8 table reflects the 2007 discovery of errors in older references.[4] Specifically, (Rx, Ry) 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 Rz, x2 + y2, z2 B 1 −1 1 −1 z x2 − y2, xy E 1 1 i −i −1 −1 −i i (Rx, Ry), (x, y) (xz, yz)
S6 Z6 6
 E S6 C3 i C32 S65 θ = e2πi /6 Ag 1 1 1 1 1 1 Rz 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 8
 E S8 C4 S83 i S85 C42 S87 θ = e2πi /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 θ  θC i −i −θC −θ −1 −1 −θ  −θC −i i θC 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 −θC −θ −i i θ  θC −1 −1 θC θ i −i −θ −θC (Rx, Ry) (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 (Dn)

The dihedral groups are denoted by Dn. These groups are characterized by i) an n-fold proper rotation axis Cn; ii) n 2-fold proper rotation axes C2 normal to Cn. 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 Rz, z xy B2 1 −1 −1 1 Ry, y xz B3 1 −1 1 −1 Rx, x yz
D3 D3 6
 E 2 C3 3 C2 A1 1 1 1 x2 + y2, z2 A2 1 1 −1 Rz, z E 2 −1 0 (Rx, Ry), (x, y) (x2 − y2, xy), (xz, yz)
D4 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 Rz, z 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)
D5 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 Rz, z E1 2 2 cos(θ) 2 cos(2θ) 0 (Rx, Ry), (x, y) (xz, yz) E2 2 2 cos(2θ) 2 cos(θ) 0 (x2 − y2, xy)
D6 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 Rz, z 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 groups (Dnh)

The prismatic groups are denoted by Dnh. These groups are characterized by i) an n-fold proper rotation axis Cn; ii) n 2-fold proper rotation axes C2 normal to Cn; iii) a mirror plane σh normal to Cn and containing the C2s. The D1h group is the same as the C2v group in the pyramidal groups section.

The D8h 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
D2h Z2×Z2×Z2
(=Z2×D2)
8
 E C2 C2 (x) C2 (y) 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
D3h D6 12
 E 2 C3 3 C2 σh 2 S3 3 σv 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)
D4h Z2×D4 16
 E 2 C4 C2 2 C2' 2 C2'' i 2 S4 σh 2 σv 2 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)
D5h D10 20
 E 2 C5 2 C52 5 C2 σh 2 S5 2 S53 5 σv θ=2π/5 A1' 1 1 1 1 1 1 1 1 x2 + y2, z2 A2' 1 1 1 −1 1 1 1 −1 Rz 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 (Rx, Ry) (xz, yz) E2'' 2 2 cos(2θ) 2 cos(θ) 0 −2 −2 cos(2θ) −2 cos(θ) 0
D6h Z2×D6 24
 E 2 C6 2 C3 C2 3 C2' 3 C2'' i 2 S3 2 S6 σh 3 σd 3 σv 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
D8h Z2×D8 32
 E 2 C8 2 C83 2 C4 C2 4 C2' 4 C2'' i 2 S83 2 S8 2 S4 σh 4 σd 4 σv θ=21/2 A1g 1 1 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 −1 −1 Rz B1g 1 −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 1 E1g 2 θ −θ 0 −2 0 0 2 θ −θ 0 −2 0 0 (Rx, Ry) (xz, yz) E2g 2 0 0 −2 2 0 0 2 0 0 −2 2 0 0 (x2 − y2, xy) E3g 2 −θ θ 0 −2 0 0 2 −θ θ 0 −2 0 0 A1u 1 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 1 z B1u 1 −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 1 −1 E1u 2 θ −θ 0 −2 0 0 −2 −θ θ 0 2 0 0 (x, y) E2u 2 0 0 −2 2 0 0 −2 0 0 2 −2 0 0 E3u 2 −θ θ 0 −2 0 0 −2 θ −θ 0 2 0 0

#### Antiprismatic groups (Dnd)

The antiprismatic groups are denoted by Dnd. These groups are characterized by i) an n-fold proper rotation axis Cn; ii) n 2-fold proper rotation axes C2 normal to Cn; iii) n mirror planes σd which contain Cn. The D1d group is the same as the C2h group in the reflection groups section.

Point
Group
Canonical
group
Order Character Table
D2d D4 8
 E 2 S4 C2 2 C2' 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 xy E 2 0 −2 0 0 (Rx, Ry), (x, y) (xz, yz)
D3d D6 12
 E 2 C3 3 C2 i 2 S6 3 σd 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 D8 16
 E 2 S8 2 C4 2 S83 C2 4 C2' 4 σd θ=21/2 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 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 (Rx, Ry) (xz, yz)
D5d D10 20
 E 2 C5 2 C52 5 C2 i 2 S10 2 S103 5 σd θ=2π/5 A1g 1 1 1 1 1 1 1 1 x2 + y2, z2 A2g 1 1 1 −1 1 1 1 −1 Rz E1g 2 2 cos(θ) 2 cos(2θ) 0 2 2 cos(2θ) 2 cos(θ) 0 (Rx, Ry) (xz, yz) E2g 2 2 cos(2θ) 2 cos(θ) 0 2 2 cos(θ) 2 cos(2θ) 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 2 cos(θ) 2 cos(2θ) 0 −2 −2 cos(2θ) −2 cos(θ) 0 (x, y) E2u 2 2 cos(2θ) 2 cos(θ) 0 −2 −2 cos(θ) −2 cos(2θ) 0
D6d D12 24
 E 2 S12 2 C6 2 S4 2 C3 2 S125 C2 6 C2' 6 σd θ=31/2 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 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 (Rx, Ry) (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=e2π i/3 A 1 1 1 1 x2 + y2 + z2 E 1 1 θ  θC θC θ 1 1 (2 z2 − x2 − y2, x2 − y2) T 3 0 0 −1 (Rx, Ry, Rz), (x, y, z) (xy, xz, yz)
Td S4 24
 E 8 C3 3 C2 6 S4 6 σd 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 (Rx, Ry, Rz) T2 3 0 −1 −1 1 (x, y, z) (xy, xz, yz)
Th Z2×A4 24
 E 4 C3 4 C32 3 C2 i 4 S6 4 S65 3 σh θ=e2π i/3 Ag 1 1 1 1 1 1 1 1 x2 + y2 + z2 Au 1 1 1 1 −1 −1 −1 −1 Eg 1 1 θ  θC θC θ 1 1 1 1 θ  θC θC θ 1 1 (2 z2 − x2 − y2, x2 − y2) Eu 1 1 θ  θC θC θ 1 1 −1 −1 −θ  −θC −θC −θ −1 −1 Tg 3 0 0 −1 3 0 0 −1 (Rx, Ry, Rz) (xy, xz, yz) Tu 3 0 0 −1 −3 0 0 1 (x, y, z)
O S4 24
 E 6 C4 3 C2  (C42) 8 C3 6 C2 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 (Rx, Ry, Rz), (x, y, z) T2 3 −1 −1 0 1 (xy, xz, yz)
Oh Z2×S4 48
 E 8 C3 6 C2 6 C4 3 C2  (C42) i 6 S4 8 S6 3 σh 6 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 Eg 2 −1 0 0 2 2 0 −1 2 0 (2 z2 − 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 (xy, xz, yz) 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

#### 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 (Rx, Ry, Rz), (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)
Ih Z2×A5 120
 E 12 C5 12 C52 20 C3 15 C2 i 12 S10 12 S103 20 S6 15 σ 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 −1 3 2 cos(3θ) 2 cos(θ) 0 −1 (Rx, Ry, Rz) T2g 3 2 cos(3θ) 2 cos(θ) 0 −1 3 2 cos(θ) 2 cos(3θ) 0 −1 Gg 4 −1 −1 1 0 4 −1 −1 1 0 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 −1 −1 T1u 3 2 cos(θ) 2 cos(3θ) 0 −1 −3 −2 cos(3θ) −2 cos(θ) 0 1 (x, y, z) T2u 3 2 cos(3θ) 2 cos(θ) 0 −1 −3 −2 cos(θ) −2 cos(3θ) 0 1 Gu 4 −1 −1 1 0 −4 1 1 −1 0 Hu 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 ... ∞ σv A1=Σ+ 1 1 ... 1 z x2 + y2, z2 A2=Σ− 1 1 ... −1 Rz E1=Π 2 2 cos(Φ) ... 0 (x, y), (Rx, Ry) (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 ... ∞ σv i 2 S∞Φ ... ∞ C2 Σg+ 1 1 ... 1 1 1 ... 1 x2 + y2, z2 Σg− 1 1 ... −1 1 1 ... −1 Rz Πg 2 2 cos(Φ) ... 0 2 −2 cos(Φ) .. 0 (Rx, Ry) (xz, yz) Δg 2 2 cos(2Φ) ... 0 2 2 cos(2Φ) .. 0 (x2 − y2, xy) ... ... ... ... ... ... ... ... ... Σu+ 1 1 ... 1 −1 −1 ... −1 z Σu− 1 1 ... −1 −1 −1 ... 1 Πu 2 2 cos(Φ) ... 0 −2 2 cos(Φ) .. 0 (x, y) Δu 2 2 cos(2Φ) ... 0 −2 −2 cos(2Φ) .. 0 ... ... ... ... ... ... ... ... ...