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, rotations about those three coordinates, and functions of the quadratic terms of the coordinates. 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 rotation is considered the identity operation. These groups have involutional symmetry: the only nonidentity operation, if any, is its own inverse. In the group, all functions of the Cartesian coordinates and rotations about them transform as the irreducible representation.
Cyclic symmetries
The families of groups with these symmetries have only one rotation axis.
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.
Reflection groups (''C''nh)
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.
Pyramidal groups (''C''nv)
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.
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 Ci group in the nonaxial groups section. Sn groups with an odd value of n are identical to Cnh groups of same n and are therefore not considered here. The S8 table reflects the 2007 discovery of errors in older references. Specifically, transform not as E1 but rather as E3.
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 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.
Prismatic groups (''D''nh)
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. Specifically, symmetry operation column headers 2S8 and 2S83 were reversed in the older references.
Antiprismatic groups (''D''nd)
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.
