Schoenflies notation
The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is one of two conventions commonly used to describe point groups. This notation is used in spectroscopy. The other convention is the Hermann–Mauguin notation, also known as the International notation. A point group in the Schoenflies convention is completely adequate to describe the symmetry of a molecule; this is sufficient for spectroscopy. The Hermann–Mauguin notation is able to describe the space group of a crystal lattice, while the Schoenflies notation isn't. Thus the Hermann–Mauguin notation is used in crystallography.
Symmetry elements are denoted by i for centers of inversion, C for proper rotation axes, σ for mirror planes, and S for improper rotation axes (rotation-reflection axes). C and S are usually followed by a subscript number (abstractly denoted n) denoting the order of rotation possible.
By convention, the axis of proper rotation of greatest order is defined as the principal axis. All other symmetry elements are described in relation to it. A vertical mirror plane (containing the principal axis) is denoted σv; a horizontal mirror plane (perpendicular to the principal axis) is denoted σh.
In three dimensions, there are an infinite number of point groups, but all of them can be classified by several families.
All groups that do not contain several higher-order axes (order 3 or more) can be arranged in a table, as shown below; symbols marked in red should not be used.
In crystallography, due to the crystallographic restriction theorem, n is restricted to the values of 1, 2, 3, 4, or 6. The noncrystallographic groups are shown with grayed backgrounds. D4d and D6d are also forbidden because they contain improper rotations with n = 8 and 12 respectively. The 27 point groups in the table plus T, Td, Th, O and Oh constitute 32 crystallographic point groups.
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