In Schoenflies notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following:
*C (for cyclic) indicates that the group has an n-fold rotation axis. C is C with the addition of a mirror (reflection) plane perpendicular to the axis of rotation. C is C with the addition of n mirror planes parallel to the axis of rotation.
*S (for Spiegel, German for mirror) denotes a group with only a 2n-fold rotation-reflection axis.
*D (for dihedral, or two-sided) indicates that the group has an n-fold rotation axis plus n twofold axes perpendicular to that axis. D has, in addition, a mirror plane perpendicular to the n-fold axis. D has, in addition to the elements of D, mirror planes parallel to the n-fold axis.
*The letter T (for tetrahedron) indicates that the group has the symmetry of a tetrahedron. T includes improper rotation operations, T excludes improper rotation operations, and T is T with the addition of an inversion.
*The letter O (for octahedron) indicates that the group has the symmetry of an octahedron (or cube), with (O) or without (O) improper operations (those that change handedness).
Due to the crystallographic restriction theorem, n = 1, 2, 3, 4, or 6 in 2- or 3-dimensional space.
D and D are actually forbidden because they contain improper rotations with n=8 and 12 respectively. The 27 point groups in the table plus T, T, T, O and O constitute 32 crystallographic point groups.