University of Exeter

Symmetry Operations and Character Tables

Exeter Symmetry Pages

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

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 2Pi/n radians, where n is an integer. The axis for which n is greatest is termed the principle axis.
Cnk Rotation (clockwise) through an angle of 2kPi/n radians. Both n and k are integers.
Sn An improper rotation (clockwise) through an angle of 2Pi/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) arrow (-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).
sigma A mirror plane (from the German word for mirror - Spiegel).
sigmah Horizontal reflection plane - passing through the origin and perpendicular to the axis with the `highest' symmetry.
sigmav Vertical reflection plane - passing through the origin and the axis with the `highest' symmetry.
sigmad 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 of sigmav.

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).

The term Cubic

There is an important note to bear in mind regarding Cubic point groups. Within this site, this term is used in the crystallographic sense, but there is (unfortunately) an alternative definition of cubic. The expression 'cubic point groups' may 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.