 Frieze group

A discrete subgroup of Euclidean isometries is a frieze group when its translation subgroup is infinite cyclic and has finite index. Translations along a line alone give $\mathbb{Z}$; adding perpendicular mirrors gives the infinite dihedral group, while adding a mirror along the translation direction gives $\mathbb{Z}\times C_2$. These groups describe patterns periodic in one planar direction.