A plane crystallographic group is a discrete group of Euclidean plane isometries with compact quotient. Its translation subgroup is a rank-two lattice, and the quotient by translations is its finite point group. This separates the periodic translational geometry from the finite rotational and reflection symmetries.
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A wallpaper group is a classification of two-dimensional repeating patterns, which can be used to describe the symmetry and structure of various kinds of tiling and decorative designs. In mathematical terms, a wallpaper group is one of the 17 different groups that describe the possible patterns that can be formed on a plane, where each pattern can be generated by translations, rotations, reflections, and glide reflections.
17 of them.