A planar crystallographic group or wallpaper group is a group of isometries of the plane which has a subgroup of translations that is isomorphic to ℤ2. (Compare frieze groups.)
The symmetry group of the pattern in the figure is one such wallpaper group. It contains translations and glide reflections but has no non-trivial rotational symmetry. In all there are 17 wallpaper groups (up to isomorphism).
In three dimensions there are 230 crystallographic groups and in four dimensions there are 4783.