# 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.

## 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, 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: i$^2$ = −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(x$^2$, y$^2$, z$^2$, xy, xz, and yz).
A further column is included in some tables, such as those of Salthouse and Ware 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 C$_1$ group is covered in the nonaxial groups section.
| Point Group | Canonical Group | Order | Character Table |
| C$_2$ | Z$_2$ | 2 | EC$_2$  A11R$_z$, z x$^2$, y$^2$, z$^2$, xy B1−1R$_x$, R$_y$, x, y xz, yz |
| C$_3$ | Z$_3$ | 3 | EC$_3$ C$_3$$^2$ θ = e$^{2πi/3}$ A111R$_z$, z x$^2$ + y$^2$ E1 1 θ θ$^C$ θ$^C$ θ  (R$_x$, R$_y$), (x, y) (x$^2$ - y$^2$, xy), (xz, yz) |
| C$_4$ | Z$_4$ | 4 | EC$_4$  C$_2$ C$_4$$^3$   A1111R$_z$, z x$^2$ + y$^2$, z$^2$ B1−11−1  x$^2$ − y$^2$, xy E1 1i −i−1 −1−i i (R$_x$, R$_y$), (x, y)(xz, yz) |
| C$_5$ | Z$_5$ | 5 | E  C$_5$ C$_5$$^2$ C$_5$$^3$C$_5$$^4$ θ = e$^{2πi/5}$ A11111R$_z$, z x$^2$ + y$^2$, z$^2$ E$_1$ 1 1 θ θ$^C$ θ$^2$ (θ$^2$)$^C$ (θ$^2$)$^C$ θ$^2$ θ$^C$ θ  (R$_x$, R$_y$), (x, y)(xz, yz) E$_2$ 1 1 θ$^2$ (θ$^2$)$^C$ θ$^C$ θ  θ θ$^C$ (θ$^2$)$^C$ θ$^2$  (x$^2$ - y$^2$, xy) |
| C$_6$ | Z$_6$ | 6 | E  C$_6$ C$_3$  C$_2$ C$_3$$^2$ C$_6$$^5$ θ = e$^{2πi/6}$ A111111R$_z$, z x$^2$ + y$^2$, z$^2$ B1−11−11−1  E$_1$ 1 1 θ θ$^C$ −θ$^C$ −θ  −1 −1 −θ −θ$^C$ θ$^C$ −θ  (R$_x$, R$_y$), (x, y) (xz, yz) E$_2$ 1 1 −θ$^C$ −θ  −θ −θ$^C$ 1 1 −θ$^C$ −θ  −θ −θ$^C$  (x$^2$ − y$^2$, xy) |
| C$_8$ | Z$_8$ | 8 | E  C$_8$ C$_4$  C$_8$$^3$C$_2$  C$_8$$^5$C$_4$$^3$ C$_8$$^7$ θ = e$^{2πi/8}$ A11111111R$_z$, z x$^2$ + y$^2$, z$^2$ B1−11−11−11−1  E$_1$ 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) E$_2$ 1 1 i −i −1 −1 −i i 1 1 i −i −1 −1 −i i  (x$^2$ − y$^2$, xy) E$_3$ 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$_{2h}$ | Z$_2$ × Z$_2$ | 4 | EC$_2$  iσ$_h$  A$_g$1111R$_z$ x$^2$, y$^2$, z$^2$, xy B$_g$1−11−1R$_x$, R$_y$ xz, yz A$_u$11−1−1z  B$_u$1−1−11x, y |
| C$_{3h}$ | Z$_6$ | 6 | EC$_3$  C$_3$$^2$σ$_h$  S$_3$ S$_3$$^5$ θ = e$^{2πi/3}$ A'111111R$_z$ x$^2$ + y$^2$, z$^2$ E'1 1 θ θ$^C$ θ$^C$ θ  1 1 θ θ$^C$ θ$^C$ θ  (x, y)(x$^2$ − y$^2$, xy) A''111−1−1−1z  E''1 1 θ θ$^C$ θ$^C$ θ  −1 −1 −θ −θ$^C$ −θ$^C$ −θ  (R$_x$, R$_y$)(xz, yz) |
| C$_{4h}$ | Z$_2$ × Z$_4$ | 8 | EC$_4$  C$_2$ C$_4$$^3$ iS$_4$$^3$σ$_h$  S$_4$  A$_g$11111111 R$_z$x$^2$ + y$^2$, z$^2$ B$_g$1−11−11−11−1  x$^2$ − y$^2$, xy E$_g$1 1i −i−1 −1 −i i1 1i −i −1 −1−i i (R$_x$, R$_y$)(xz, yz) A$_u$1111−1−1−1−1z  B$_u$1−11−1−11−11  E$_u$1 1i −i−1 −1 −i i−1 −1−i i 1 1i −i (x, y) |
| C$_{5h}$ | Z$_{10}$ | 10 | E  C$_5$ C$_5$$^2$ C$_5$$^3$C$_5$$^4$ σ$_h$ S$_5$  S$_5$$^7$S$_5$$^3$ S$_5$$^9$ θ = e$^{2πi/5}$ A'1111111111R$_z$ x$^2$ + y$^2$, z$^2$ E$_1$' 1 1 θ θ$^C$ θ$^2$ (θ$^2$)$^C$ (θ$^2$)$^C$ θ$^2$ θ$^C$ θ  1 1 θ θ$^C$ θ$^2$ (θ$^2$)$^C$ (θ$^2$)$^C$ θ$^2$ θ$^C$ θ  (x, y)  E$_2$' 1 1 θ$^2$ (θ$^2$)$^C$ θ$^C$ θ  θ θ$^C$ (θ$^2$)$^C$ θ$^2$ 1 1 θ$^2$ (θ$^2$)$^C$ θ$^C$ θ  θ θ$^C$ (θ$^2$)$^C$ θ$^2$  (x$^2$ - y$^2$, xy) A''11111 −1−1−1−1−1 z  E$_1$'' 1 1 θ θ$^C$ θ$^2$ (θ$^2$)$^C$ (θ$^2$)$^C$ θ$^2$ θ$^C$ θ  −1 −1 −θ -θ$^C$ −θ$^2$ −(θ$^2$)$^C$ −(θ$^2$)$^C$ −θ$^2$ −θ$^C$ −θ  (R$_x$, R$_y$)(xz, yz) E$_2$'' 1 1 θ$^2$ (θ$^2$)$^C$ θ$^C$ θ  θ θ$^C$ (θ$^2$)$^C$ θ$^2$ −1 −1 −θ$^2$ −(θ$^2$)$^C$ −θ$^C$ −θ  −θ −θ$^C$ −(θ$^2$)$^C$ −θ$^2$ |
| C$_{6h}$ | Z$_2$ × Z$_6$ | 12 | E  C$_6$ C$_3$  C$_2$ C$_3$$^2$ C$_6$$^5$iS$_3$$^5$ S$_6$$^5$σ$_h$  S$_6$ S$_3$  θ = e$^{2πi/6}$ A$_g$111111111111 R$_z$x$^2$ + y$^2$, z$^2$ B$_g$1−11−11−1 1−11−11−1   E$_{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$^C$  (x$^2$ − y$^2$, xy) A$_u$111111 −1−1−1−1−1−1 z  B$_u$1−11−11−1 −11−11−11   E$_{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$_{2v}$ | Z$_2$ × Z$_2$ (=D$_2$) | 4 | EC$_2$  σ$_v$  σ$_v$'    A$_1$1111z x$^2$ , y$^2$, z$^2$ A$_2$11−1−1R$_z$xy B$_1$1−11−1R$_y$, xxz B$_2$1−1−11R$_x$, yyz |
| C$_{3v}$ | D$_3$ | 6 | E2 C$_3$  3 σ$_v$    A$_1$111z x$^2$ + y$^2$, z$^2$ A$_2$11−1R$_z$  E2−10(R$_x$, R$_y$), (x, y) (x$^2$ − y$^2$, xy), (xz, yz) |
| C$_{4v}$ | D$_4$ | 8 | E2 C$_4$  C$_2$ 2 σ$_v$  2 σ$_d$  A$_1$11111 zx$^2$ + y$^2$, z$^2$ A$_2$111−1−1R$_z$  B$_1$1−111−1  x$^2$ − y$^2$ B$_2$1−11−11 xy E20−200 (R$_x$, R$_y$), (x, y)(xz, yz) |
| C$_{5v}$ | D$_5$ | 10 | E  2 C$_5$ 2 C$_5$$^2$ 5 σ$_v$ θ = 2π/5 A$_1$1111z x$^2$ + y$^2$, z$^2$ A$_2$111−1R$_z$  E$_1$22 cos(θ)2 cos(2θ)0 (R$_x$, R$_y$), (x, y)(xz, yz) E$_2$22 cos(2θ)2 cos(θ)0  (x$^2$ − y$^2$, xy) |
| C$_{6v}$ | D$_6$ | 12 | E  2 C$_6$ 2 C$_3$  C$_2$ 3 σ$_v$  3 σ$_d$  A$_1$111111 zx$^2$ + y$^2$, z$^2$ A$_2$1111−1−1R$_z$  B$_1$1−11−11−1  B$_2$1−11−1−11  E$_1$21−1−200 (R$_x$, R$_y$), (x, y)(xz, yz) E$_2$2−1−1200  (x$^2$ − y$^2$, 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 S$_2$ 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, S$_1$ is identical to C$_s$).
The S$_8$ table reflects the 2007 discovery of errors in older references. Specifically, (R$_x$, R$_y$) transform not as E$_1$ but rather as E$_3$.
| Point Group | Canonical group | Order | Character Table |
| S$_4$ | Z$_4$ | 4 | ES$_4$  C$_2$ S$_4$$^3$   A1111R$_z$,  x$^2$ + y$^2$, z$^2$ B1−11−1z x$^2$ − y$^2$, xy E1 1i −i−1 −1 −i i (R$_x$, R$_y$), (x, y)(xz, yz) |
| S$_6$ | Z$_6$ | 6 | E  S$_6$ C$_3$  iC$_3$$^2$S$_6$$^5$ θ = e$^{2πi/6}$ A$_g$111111R$_z$ x$^2$ + y$^2$, z$^2$ E$_g$ 1 1 θ$^C$ θ  θ θ$^C$ 1 1 θ$^C$ θ  θ θ$^C$ (R$_x$, R$_y$) (x$^2$ − y$^2$, xy), (xz, yz) A$_u$1−11−11−1z  E$_u$ 1 1 −θ$^C$ −θ  θ θ$^C$ −1 −1 θ$^C$ θ  −θ −θ$^C$ (x, y) |
| S$_8$ | Z$_8$ | 8 | E  S$_8$ C$_4$  S$_8$$^3$i S$_8$$^5$C$_4$$^2$ S$_8$$^7$ θ = e$^{2πi/8}$ A11111111R$_z$ x$^2$ + y$^2$, z$^2$ B1−11−1−1−11−1z  E$_1$ 1 1 θ θ$^C$ i −i −θ$^C$ −θ  −1 −1 −θ −θ$^C$ −i i θ$^C$ θ  (x, y)(xz, yz) E$_2$ 1 1i −i−1 −1 −i i1 1i −i −1 −1−i i  (x$^2$ − y$^2$, xy) E$_3$ 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 C$_2$ normal to C$_n$. The D$_1$ group is the same as the C$_2$ group in the cyclic groups section.
| Point Group | Canonical group | Order | Character Table |
| D$_2$ | Z$_2$ × Z$_2$ (=D$_2$) | 4 | EC$_2$ (z) C$_2$ (x) C$_2$ (y)  A1111  x$^2$, y$^2$, z$^2$ B$_1$11−1−1R$_z$, zxy B$_2$1−1−11R$_y$, yxz B$_3$1−11−1R$_x$, xyz |
| D$_3$ | D$_3$ | 6 | E2 C$_3$  3 C'$_2$  A$_1$111  x$^2$ + y$^2$, z$^2$ A$_2$11−1R$_z$, z  E2−10(R$_x$, R$_y$), (x, y) (x$^2$ − y$^2$, xy), (xz, yz) |
| D$_4$ | D$_4$ | 8 | E2 C$_4$  C$_2$ 2 C$_2$'  2 C$_2$''    A$_1$11111  x$^2$ + y$^2$, z$^2$ A$_2$111−1−1R$_z$, z  B$_1$1−111−1  x$^2$ − y$^2$ B$_2$1−11−11 xy E20−200 (R$_x$, R$_y$), (x, y)(xz, yz) |
| D$_5$ | D$_5$ | 10 | E  2 C$_5$ 2 C$_5$$^2$ 5 C$_2$ θ=2π/5 A$_1$1111  x$^2$ + y$^2$, z$^2$ A$_2$111−1R$_z$, z  E$_1$22 cos(θ)2 cos(2θ)0 (R$_x$, R$_y$), (x, y)(xz, yz) E$_2$22 cos(2θ)2 cos(θ)0  (x$^2$ − y$^2$, xy) |
| D$_6$ | D$_6$ | 12 | E  2 C$_6$ 2 C$_3$  C$_2$ 3 C$_2$'  3 C$_2$''    A$_1$111111  x$^2$ + y$^2$, z$^2$ A$_2$1111−1−1 R$_z$, z  B$_1$1−11−11−1  B$_2$1−11−1−11  E$_1$21−1−200 (R$_x$, R$_y$), (x, y)(xz, yz) E$_2$2−1−1200  (x$^2$ − y$^2$, 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 C$_2$ normal to C$_n$; iii) a mirror plane σ$_h$ normal to C$_n$ and containing the C$_2$s. 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 2S$_8$ and 2S$_8$$^3$ were reversed in the older references.
| Point Group | Canonical group | Order | Character Table |
| D$_{2h}$ | Z$_2$×Z$_2$×Z$_2$ (=Z$_2$×D$_2$) | 8 | EC$_2$  C$_2$ (x) C$_2$ (y)i σ(xy)  σ(xz)  σ(yz)  A$_g$11111111  x$^2$, y$^2$, z$^2$ B$_{1g}$11−1−111−1−1 R$_z$xy B$_{2g}$1−1−111−11−1 R$_y$xz B$_{3g}$1−11−11−1−11 R$_x$yz A$_u$1111 −1−1−1−1  B$_{1u}$11−1−1 −1−111z  B$_{2u}$1−1−11 −11−11y  B$_{3u}$1−11−1 −111−1x |
| D$_{3h}$ | D$_6$ | 12 | E2 C$_3$  3 C$_2$ σ$_h$  2 S$_3$ 3 σ$_v$    A$_1$'111111  x$^2$ + y$^2$, z$^2$ A$_2$'11−111−1R$_z$  E'2−102−10(x, y) (x$^2$ − y$^2$, xy) A$_1$''111−1−1−1   A$_2$''11−1−1−11 z  E''2−10−210 (R$_x$, R$_y$)(xz, yz) |
| D$_{4h}$ | Z$_2$×D$_4$ | 16 | E2 C$_4$  C$_2$ 2 C$_2$'  2 C$_2$'' i 2 S$_4$ σ$_h$  2 σ$_v$  2 σ$_d$  A$_{1g}$1111111111  x$^2$ + y$^2$, z$^2$ A$_{2g}$111−1−1 111−1−1 R$_z$  B$_{1g}$1−111−1 1−111−1  x$^2$ − y$^2$ B$_{2g}$1−11−11 1−11−11  xy E$_g$20−20020−200 (R$_x$, R$_y$)(xz, yz) A$_{1u}$11111 −1−1−1−1−1   A$_{2u}$111−1−1 −1−1−111 z  B$_{1u}$1−111−1 −11−1−11   B$_{2u}$1−11−11 −11−11−1   E$_u$20−200−20200 (x, y) |
| D$_{5h}$ | D$_{10}$ | 20 | E  2 C$_5$ 2 C$_5$$^2$ 5 C$_2$  σ$_h$ 2 S$_5$  2 S$_5$$^3$5 σ$_v$  θ=2π/5 A$_1$'11111111  x$^2$ + y$^2$, z$^2$ A$_2$'111−1111−1 R$_z$  E$_1$'22 cos(θ)2 cos(2θ)02 2 cos(θ)2 cos(2θ)0(x, y)  E$_2$'22 cos(2θ)2 cos(θ)02 2 cos(2θ)2 cos(θ)0  (x$^2$ − y$^2$, xy) A$_1$''1111 −1−1−1−1   A$_2$''111−1 −1−1−11 z  E$_1$''22 cos(θ) 2 cos(2θ)0−2−2 cos(θ) −2 cos(2θ)0 (R$_x$, R$_y$)(xz, yz) E$_2$''22 cos(2θ) 2 cos(θ)0−2−2 cos(2θ) −2 cos(θ)0 |
| D$_{6h}$ | Z$_2$×D$_6$ | 24 | E  2 C$_6$ 2 C$_3$  C$_2$ 3 C$_2$'  3 C$_2$'' i 2 S$_3$ 2 S$_6$  σ$_h$ 3 σ$_d$  3 σ$_v$  A$_{1g}$111111111111  x$^2$ + y$^2$, z$^2$ A$_{2g}$1111−1−1 1111−1−1 R$_z$  B$_{1g}$1−11−11−1 1−11−11−1   B$_{2g}$1−11−1−11 1−11−1−11   E$_{1g}$21−1−200 21−1−200 (R$_x$, R$_y$)(xz, yz) E$_{2g}$2−1−1200 2−1−1200  (x$^2$ − y$^2$, xy) A$_{1u}$111111 −1−1−1−1−1−1   A$_{2u}$1111−1−1 −1−1−1−111 z  B$_{1u}$1−11−11−1 −11−11−11   B$_{2u}$1−11−1−11 −11−111−1   E$_{1u}$21−1−200 −2−11200 (x, y)  E$_{2u}$2−1−1200 −211−200 |
| D$_{8h}$ | Z$_2$×D$_8$ | 32 | E  2 C$_8$ 2 C$_8$$^3$ 2 C$_4$ C$_2$  4 C$_2$'  4 C$_2$'' i 2 S$_8$$^3$2 S$_8$  2 S$_4$  σ$_h$  4 σ$_d$  4 σ$_v$  θ=2$^{1/2}$ A$_{1g}$1111111 1111111  x$^2$ + y$^2$, z$^2$ A$_{2g}$11111−1−1 11111−1−1R$_z$  B$_{1g}$1−1−1111−1 1−1−1111−1  B$_{2g}$1−1−111−11 1−1−111−11  E$_{1g}$2θ−θ0−200 2θ−θ0−200 (R$_x$, R$_y$)(xz, yz) E$_{2g}$200−2200 200−2200  (x$^2$ − y$^2$, xy) E$_{3g}$2−θθ0−200 2−θθ0−200   A$_{1u}$1111111 −1−1−1−1−1−1−1  A$_{2u}$11111−1−1 −1−1−1−1−111z  B$_{1u}$1−1−1111−1 −111−1−1−11  B$_{2u}$1−1−111−11 −111−1−11−1   E$_{1u}$2θ−θ0−200 −2−θθ0200 (x, y)  E$_{2u}$200−2200 −2002−200  E$_{3u}$2−θθ0−200 −2θ−θ0200 |

#### 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 C$_2$ 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$_{2d}$ | D$_4$ | 8 | E 2 S$_4$  C$_2$ 2 C$_2$'  2 σ$_d$  A$_1$11111  x$^2$, y$^2$, z$^2$ A$_2$111−1−1R$_z$  B$_1$1−111−1  x$^2$ − y$^2$ B$_2$1−11−11zxy E20−200 (R$_x$, R$_y$), (x, y)(xz, yz) |
| D$_{3d}$ | D$_6$ | 12 | E 2 C$_3$  3 C$_2$ i  2 S$_6$  3 σ$_d$    A$_{1g}$111111  x$^2$ + y$^2$, z$^2$ A$_{2g}$11−111−1 R$_z$  E$_g$2−102−10 (R$_x$, R$_y$) (x$^2$ − y$^2$, xy), (xz, yz) A$_{1u}$111−1−1−1  A$_{2u}$11−1−1−11z  E$_u$2−10−210(x, y) |
| D$_{4d}$ | D$_8$ | 16 | E 2 S$_8$  2 C$_4$ 2 S$_8$$^3$ C$_2$ 4 C$_2$'  4 σ$_d$  θ=2$^{1/2}$ A$_1$1111111  x$^2$ + y$^2$, z$^2$ A$_2$11111−1−1 R$_z$  B$_1$1−11−111−1  B$_2$1−11−11−11z  E$_1$2θ0−θ−200 (x, y)  E$_2$20−20200  (x$^2$ − y$^2$, xy) E$_3$2−θ0θ−200 (R$_x$, R$_y$)(xz, yz) |
| D$_{5d}$ | D$_{10}$ | 20 | E  2 C$_5$ 2 C$_5$$^2$ 5 C$_2$ i  2 S$_{10}$ 2 S$_{10}$$^3$ 5 σ$_d$  θ=2π/5 A$_{1g}$11111111  x$^2$ + y$^2$, z$^2$ A$_{2g}$111−1111−1 R$_z$  E$_{1g}$22 cos(θ)2 cos(2θ)0 22 cos(2θ)2 cos(θ)0 (R$_x$, R$_y$)(xz, yz) E$_{2g}$22 cos(2θ)2 cos(θ)0 22 cos(θ)2 cos(2θ)0  (x$^2$ − y$^2$, xy) A$_{1u}$1111 −1−1−1−1  A$_{2u}$111−1 −1−1−11z  E$_{1u}$22 cos(θ)2 cos(2θ)0 −2−2 cos(2θ)−2 cos(θ)0 (x, y)  E$_{2u}$22 cos(2θ)2 cos(θ)0 −2−2 cos(θ)−2 cos(2θ)0 |
| D$_{6d}$ | D$_{12}$ | 24 | E  2 S$_{12}$ 2 C$_6$  2 S$_4$ 2 C$_3$  2 S$_{12}$$^5$C$_2$  6 C$_2$' 6 σ$_d$  θ=3$^{1/2}$ A$_1$111111111  x$^2$ + y$^2$, z$^2$ A$_2$1111111−1−1 R$_z$  B$_1$1−11−11−111−1   B$_2$1−11−11−11−11 z  E$_1$2θ10−1 −θ−200(x, y)  E$_2$21−1−2−11200  (x$^2$ − y$^2$, xy) E$_3$20−2020−200   E$_4$2−1−12−1−1200   E$_5$2−θ10−1 θ−200 (R$_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 C$_5$ proper rotation axis.
| Point Group | Canonical group | Order | Character Table |
| T | A$_4$ | 12 | E4 C$_3$  4 C$_3$$^2$ 3 C$_2$  θ=e$^{2πi/3}$ A1111  x$^2$ + y$^2$ + z$^2$ E1 1θ θ$^C$ θ$^C$ θ  1 1  (2 z$^2$ − x$^2$ − y$^2$, x$^2$ − y$^2$) T300−1 (R$_x$, R$_y$, R$_z$), (x, y, z) (xy, xz, yz) |
| T$_d$ | S$_4$ | 24 | E8 C$_3$  3 C$_2$ 6 S$_4$  6 σ$_d$    A$_1$11111  x$^2$ + y$^2$ + z$^2$ A$_2$111−1−1  E2−1200  (2 z$^2$ − x$^2$ − y$^2$, x$^2$ − y$^2$) T$_1$30−11−1 (R$_x$, R$_y$, R$_z$)  T$_2$30−1−11 (x, y, z)(xy, xz, yz) |
| T$_h$ | Z$_2$×A$_4$ | 24 | E4 C$_3$  4 C$_3$$^2$ 3 C$_2$ i 4 S$_6$ 4 S$_6$$^5$ 3 σ$_h$  θ=e$^{2πi/3}$ A$_g$11111111  x$^2$ + y$^2$ + z$^2$ A$_u$1111−1−1−1−1   E$_g$1 1 θ θ$^C$ θ$^C$ θ  1 11 1 θ θ$^C$ θ$^C$ θ  1 1   (2 z$^2$ − x$^2$ − y$^2$, x$^2$ − y$^2$) E$_u$1 1 θ θ$^C$ θ$^C$ θ  1 1−1 −1 −θ −θ$^C$ −θ$^C$ −θ  −1 −1   T$_g$300−1300−1 (R$_x$, R$_y$, R$_z$) (xy, xz, yz) T$_u$300−1−3001 (x, y, z) |
| O | S$_4$ | 24 | E  6 C$_4$  3 C$_2$ (C$_4$$^2$) 8 C$_3$ 6 C'$_2$    A$_1$11111  x$^2$ + y$^2$ + z$^2$ A$_2$1−111−1  E202−10  (2 z$^2$ − x$^2$ − y$^2$, x$^2$ − y$^2$) T$_1$31−10−1 (R$_x$, R$_y$, R$_z$), (x, y, z)   T$_2$3−1−101  (xy, xz, yz) |
| O$_h$ | Z$_2$×S$_4$ | 48 | E  8 C$_3$ 6 C$_2$  6 C$_4$  3 C$_2$ (C$_4$$^2$) i6 S$_4$  8 S$_6$ 3 σ$_h$  6 σ$_d$    A$_{1g}$1111111111  x$^2$ + y$^2$ + z$^2$ A$_{2g}$11−1−111−111−1   E$_g$2−100220−120   (2 z$^2$ − x$^2$ − y$^2$, x$^2$ − y$^2$) T$_{1g}$30−11−1310−1−1 (R$_x$, R$_y$, R$_z$)   T$_{2g}$301−1−13−10−11  (xy, xz, yz) A$_{1u}$11111 −1−1−1−1−1   A$_{2u}$11−1−11 −11−1−11   E$_u$2−1002−201−20   T$_{1u}$30−11−1 −3−1011 (x, y, z)  T$_{2u}$301−1−1 −3101−1 |

#### Icosahedral groups

These polyhedral groups are characterized by having a C$_5$ proper rotation axis.
| Point Group | Canonical group | Order | Character Table |
| I | A$_5$ | 60 | E12 C$_5$  12 C$_5$$^2$ 20 C$_3$  15 C$_2$  θ=π/5 A11111  x$^2$ + y$^2$ + z$^2$ T$_1$32 cos(θ)2 cos(3θ)0−1 (R$_x$, R$_y$, R$_z$), (x, y, z)  T$_2$32 cos(3θ)2 cos(θ)0−1   G4−1−110  H500−11  (2 z$^2$ − x$^2$ − y$^2$, x$^2$ − y$^2$, xy, xz, yz) |
| I$_h$ | Z$_2$×A$_5$ | 120 | E12 C$_5$  12 C$_5$$^2$ 20 C$_3$  15 C$_2$ i 12 S$_{10}$  12 S$_{10}$$^3$ 20 S$_6$  15 σ θ=π/5 A$_g$1111111111  x$^2$ + y$^2$ + z$^2$ T$_{1g}$32 cos(θ)2 cos(3θ)0−1 32 cos(3θ)2 cos(θ)0−1 (R$_x$, R$_y$, R$_z$)  T$_{2g}$32 cos(3θ)2 cos(θ)0−1 32 cos(θ)2 cos(3θ)0−1  G$_g$4−1−1104−1−110   H$_g$500−11500−11  (2 z$^2$ − x$^2$ − y$^2$, x$^2$ − y$^2$, xy, xz, yz) A$_u$11111 −1−1−1−1−1   T$_{1u}$32 cos(θ)2 cos(3θ)0−1 −3−2 cos(3θ)−2 cos(θ)01 (x, y, z)  T$_{2u}$32 cos(3θ)2 cos(θ)0−1 −3−2 cos(θ)−2 cos(3θ)01   G$_u$4−1−110 −411−10   H$_u$500−11−5001−1 |

### 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$_{∞v}$ | E2 C$_∞$$^Φ$ ... ∞ σ$_v$    A$_1$=Σ$^+$11...1z x$^2$ + y$^2$, z$^2$ A$_2$=Σ$^−$11...−1R$_z$   E$_1$=Π22 cos(Φ)...0 (x, y), (R$_x$, R$_y$)(xz, yz) E$_2$=Δ22 cos(2Φ)...0  (x$^2$ - y$^2$, xy) E$_3$=Φ22 cos(3Φ)...0   ............... |
| D$_{∞h}$ | E2 C$_∞$$^Φ$... ∞ σ$_v$ i 2 S$_∞$$^Φ$...∞ C$_2$    Σ$_g$$^+$11...111...1  x$^2$ + y$^2$, z$^2$ Σ$_g$$^−$11... −111...−1 R$_z$  Π$_g$22 cos(Φ)...02−2 cos(Φ)..0 (R$_x$, R$_y$)(xz, yz) Δ$_g$22 cos(2Φ)...022 cos(2Φ)..0  (x$^2$ − y$^2$, xy) ...........................  Σ$_u$$^+$11... 1−1−1...−1 z  Σ$_u$$^−$11... −1−1−1...1   Π$_u$22 cos(Φ)... 0−22 cos(Φ)..0 (x, y)  Δ$_u$22 cos(2Φ)... 0−2−2 cos(2Φ)..0   ........................... |