 # Symmetry Operations and Character Tables All the character tables are laid out in the same way, and some pre-knowledge of group theory is assumed. In brief:

• The top row and first column consist of the symmetry operations and irreducible representations respectively.
• The table elements are the characters.
• The final two columns show the first and second order combinations of Cartesian coordinates.
• Infinitesimal rotations are listed as Ix, Iy, and Iz.

The notation for the symmetry operations is as follows:

 E The identity transformation (E coming from the German Einheit, meaning unity). Cn Rotation (clockwise) through an angle of 2π/n radians, where n is an integer. The axis for which n is greatest is termed the principal axis. Cnk Rotation (clockwise) through an angle of 2kπ/n radians. Both n and k are integers. Sn An improper rotation (clockwise) through an angle of 2π/n radians. Improper rotations are regular rotations followed by a reflection in the plane perpendicular to the axis of rotation. Also known as alternating axis of symmetry and rotation-reflection axis. i The inversion operator (the same as S2). In Cartesian coordinates, (x, y, z)→(−x, −y, −z). Irreducible representations that are even under this symmetry operation are usually denoted with the subscript g for gerade (german=even), and those that are odd are denoted with the subscript u for ungerade (german=odd). σ A mirror plane (from the German word for mirror - Spiegel). σh Horizontal reflection plane - passing through the origin and perpendicular to the axis with the ‘highest’ symmetry. σv Vertical reflection plane - passing through the origin and the axis with the ‘highest’ symmetry. σd Diagonal or dihedral reflection in a plane through the origin and the axis with the ‘highest’ symmetry, but also bisecting the angle between the twofold axes perpendicular to the symmetry axis. This is actually a special case ofσv.

It often occurs for a point group that there are inequivalent operations of the same type. For example, there are three C2 operations in the D2d point group, two of which are inequivalent to the third. In such cases the different operations may be distinguished with a ‘prime’ or by indicating some Cartesian reference (such as the x, y, and z related C2 operations in D2).

A special note on the term cubic: Within this site, cubic is used in the crystallographic sense, but there is (unfortunately) an alternative definition - the expression ’cubic point groups’ may be used to refer to all point groups with indistinguishable cartesian axes. This means that for example all three moments of inertia are identical. All cubic point groups can be derived from the Platonic solids.

A special note on non-cyrstallographic groups: Of course there are an infinte set of non-cyrstallographic point groups, as can be seen in the idea that one can have any n for Cn group. On this site we focus on some of the more important non-crystallographic point groups and other examples.

 Revised: © University of Newcastle upon Tyne, UK