Here are five cat tiles that complete the 17 plane groups. See things 155268, 158268, and 159025 - Two Cat Tiles, More Cat Tiles, and More Cat Tiles 02. They are based on plane groups p4m, p3, p3m1, p31m and p6. The patterns came from things 58639 and 60958 - Cat Coasters 03 and 04. OpenSCAD programs like the one used in thing 151488 - Child's Puzzle 5 - were used to cut a repeating unit from a large plate.

Ten stl files are provided:

tile_p4m.stl and 4_tiles_p4m.stl
tile_p3.stl and 3_tiles_p3.stl
tile_p3m1.stl and 3_tiles_p3m1.stl
tile_p31m.stl and 3_tiles_p31m.stl
tile_p6.stl and 6_tiles_p3.stl

Multiple copies of the tiles can be printed and assembled as an aid to the study of the plane groups or to make interesting patterns.

All the stl files, including the ones with multiple copies, can be printed directly on the build platform.

I believe that these simple examples of the plane groups display the correct symmetry. One can check their symmetry elements against those shown for the wallpaper patterns - see for example David Joyce's wallpaper groups site:

http://www.clarku.edu/~djoyce/wallpaper/