This is part of a set of 17 things representing all wallpaper groups, designed for learning about symmetry.

"A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles and tiles as well as wallpaper."

https://en.wikipedia.org/wiki/Wallpaper_group

This thing includes tiles for creating patterns with the appropriate symmetry.
The tiles have an embossed swirl pattern that is helpful for observing rotation, translation, reflections and glide planes.
Straight edges indicate mirror symmetry.

Files:
p4g-cutout.stl - the asymmetric unit.
p4g-cutout-swirl.stl - optional insert for the asymmetric unit.
p4g-cutout-m.stl - mirror image asymmetric unit.
p4g-cutout-swirl-m.stl - optional insert for the mirror image asymmetric unit.

The figure shows the symmetry elements for this group.

OpenSCAD files are available upon request.

Wallpaper group p4g has eight tiles per unit cell.

The group p4g has two centres of rotation of order four (90°), which are each other's mirror image, but it has reflections in only two directions, which are perpendicular. There are rotations of order two (180°) whose centres are located at the intersections of reflection axes. It has glide reflections axes parallel to the reflection axes, in between them, and also at an angle of 45° with these.

Orbifold signature: 4*2
Coxeter notation: [4+,4]
Lattice: square
Point group: D4

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