There are exactly 17 different plane symmetry groups. They are referred to as wallpaper groups, because they make two-dimensional patterns like wallpaper. A plane symmetry group is made up translations, reflections, glide-reflections, and rotations. Translations shift the pattern some distance from the original and leave the pattern appearing unchanged. (Nelson, Newman, Shipley 2012. Page 4)
17 Plane Symmetry Groups