There are some very interesting groups that can be realized in three-space, but I don't want to go there. I will discuss frieze groups, dihedral groups an cyclic groups. I will also mention permutation groups of index three or less. Frieze Groups Frieze groups are closely related to wallpaper groups. They are also discontinuous subgroups of the isometries of the Euclidian plane.
Frieze groups are more restricted than wallpaper groups.
There is only one translation in a frieze group. There are only two possible angles for reflections in a frieze group, parallel to the translation on the center line of the pattern or perpendicular to the translation. Finally, the only rotation allowed is a rotation of order two. The effect of these restrictions is cause the patterns of frieze groups to be strips. That's why some call them strip patterns.