A frieze group contains either a translation or a glide reflection. It may optionally contain a reflection along the long axis of the strip and a reflection along the narrow axis of the strip. It may also contain a 180° rotation. (It must be able to be generated by either a single translation or a single glide reflection, along with rotations and reflections.)
There are seven distinct subgroups (up to scaling) in the discrete frieze group generated by a translation, reflection (along the same axis) and a 180° rotation. Each of these subgroups is the symmetry group of one of the frieze patterns. The seven different patterns can be described as follows:
Basic descriptions
In other words, a frieze group may be generated from any set of translations, as long as the result is a symmetry group of a strip, and there aren't infinitely many translations within a finite amount of "translation space"--the translation space expands in two directions; translations are only allowed to occur in one dimension, in a straight line as opposed to a plane. The trivial group (C1) is also excluded.
There are exactly seven different frieze groups, up to translation.