In crystallography, a crystallographic point group or crystal class is a set of symmetry operations that leave a point fixed, like rotations or reflections, which leave the crystal unchanged. It is a symmetry group. It can be shown that there exist only 32 unique crystallographic point groups.

The point group of a crystal, among other things, determines the symmetry of the crystal's optical properties. For instance, one knows whether it is birefringent, or whether it shows the Pockels effect, simply by knowing its point group.

Notation

The point groups are denoted by their component symmetries. There are a few standard notations used by crystallographers, mineralogists, and physicists.

Schoenflies notation

In Schoenflies notation, point groups are denoted by a letter symbol with a subscript. The symbols mean the following:

• The letter O (for octahedron) indicates that the group has the symmetry of an octahedron (or cube), with (O_h) or without (O) improper operations (those that change handedness).
• The letter T (for tetrahedron) indicates that the group has the symmetry of a tetrahedron. T_d includes improper operations, T excludes improper operations, and T_h is T with the addition of an inversion.
• C_n (for cyclic) indicates that the group has an n-fold rotation axis. C_nh is C_n with the addition of a mirror (reflection) plane perpendicular to the axis of rotation. C_nv is C_n with the addition of a mirror plane parallel to the axis of rotation.
• S_n (for Spiegel, German for mirror) denotes a group that contains only an n-fold rotation-reflection axis.
• D_n (for dihedral, or two-sided) indicates that the group has an n-fold rotation axis plus a two-fold axis perpendicular to that axis. D_nh has, in addition, a mirror plane perpendicular to the n-fold axis. D_nv has, in addition to the elements of D_n, mirror planes parallel to the n-fold axis.