 # 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 2 /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 2k /n radians. Both n and k are integers. Sn An improper rotation (clockwise) through an angle of 2 /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). A mirror plane (from the German word for mirror - Spiegel). h Horizontal reflection plane - passing through the origin and perpendicular to the axis with the `highest' symmetry. v Vertical reflection plane - passing through the origin and the axis with the `highest' symmetry. d 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 v.

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).