Hausdorff density

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 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 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: 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	$E$ $i$ $A_g$ $1$ $1$ $R_x$, $R_y$, $R_z$ $x^2$, $y^2$, $z^2$, $xy$, $xz$, $yz$ $A_u$ $1$ $-1$ $x$, $y$, $z$
$C_s$	$Z_2$	$2$	$E$ ${\sigma }_h$ $A^{\prime }$ $1$ $1$ $x$, $y$, $R_z$ $x^2$, $y^2$, $z^2$, $xy$ $A^{″}$ $1$ $-1$ $z$, $R_x$, $R_y$ $yz$, $xz$

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 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 (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
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 (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
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 (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	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 (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 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 (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
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 (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
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

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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     ... ... ... ... ... ... ... ... ...

See also

• Linear combination of atomic orbitals molecular orbital method
• Raman spectroscopy
• Vibrational spectroscopy (molecular vibration)
• List of small groups
• Cubic harmonics

Notes

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External links

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

Further reading

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