An improper rotation involves two operation steps: a proper rotation followed by reflection through a plane perpendicular to the rotation axis. The improper rotation is represented by the symbol where is the order. Since the improper rotation is the combination of a proper rotation and a reflection, will always exist whenever and a perpendicular plane exist separately. is usually denoted as , a reflection operation about a mirror plane. is usually denoted as , an inversion operation about an inversion center. When is an even number S_n^n = E, but when is odd S_n^{2n} = E.
