## Quick Reference

A system for categorizing symmetries of molecules. *C _{n}* groups contain only an

*n*-fold rotation axis.

*C*groups, in addition to the

_{nv}*n*-fold rotation axis, have a mirror plane that contains the axis of rotation (and mirror planes associated with the existence of the

*n*-fold axis).

*C*groups, in addition to the

_{nh}*n*-fold rotation axis, have a mirror plane perpendicular to the axis.

*S*groups have an

_{n}*n*-fold rotation-reflection axis.

*D*groups have an

_{n}*n*-fold rotation axis and a two-fold axis perpendicular to the

*n*-fold axis (and two-fold axes associated with the existence of the

*n*-fold axis).

*D*groups have all the symmetry operations of

_{nh}*D*and also a mirror plane perpendicular to the

_{n}*n*-fold axis.

*D*groups contain all the symmetry operations of

_{nd}*D*and also mirror planes that contain the

_{n}*n*-fold axis and bisect the angles between the two-fold axes. In the Schoenflies notation

*C*stands for ‘cyclic’,

*S*stands for ‘spiegel’ (mirror), and

*D*stands for ‘dihedral’. The subscripts

*h*,

*v*, and

*d*stand for horizontal, vertical, and diagonal respectively, where these words refer to the position of the mirror planes with respect to the

*n*-fold axis (considered to be vertical). In addition to the noncubic groups referred to so far, there are cubic groups, which have several rotation axes with the same value of

*n*. These are the tetrahedral groups

*T*,

*T*, and

_{h}*T*, the octahedral groups

_{d}*O*and

*O*and the icosahedral group I. The Schoenflies system is commonly used for isolated molecules, while the Hermann-Mauguin system is commonly used in crystallography.

_{h},*Subjects:*
Chemistry — Physics.