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 (x, y and z), rotations about those three coordinates (Rx, Ry and Rz), 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.
Read more about this topic: List Of Character Tables For Chemically Important 3D Point Groups
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