Classification of wallpaper groups

Statement

This article completely classifies all the space groups in two dimensions, also called the wallpaper groups. The classification up to affine space type is the same as the classification up to crystallographic space type, and there are a total of seventeen types.

The seventeen types are as follows:

IUC name for group	Description of generators other than translations	Projection to linear group (rosette group)	Split
$p1$	None	Trivial	Yes
$pg$	Glide reflection	Two-element subgroup generated by reflection	No
$pm$	Reflection	Two-element subgroup generated by reflection	Yes
$cm$	Reflection and glide reflection	Klein four-group generated by two reflections	No
$pmg$	Reflection and rotation by $\pi$	Klein four-group generated by two reflections	No
$cmm$	Reflection, rotation by $\pi$ with rotation center off reflection axis	Klein four-group generated by two reflections	No
$pmm$	Two orthogonal reflections	Klein four-group generated by two reflections	Yes
$pgg$	Rotation by $\pi$, glide reflection	Klein four-group generated by two reflections	No
$p2$	Rotation by $\pi$	Two-element subgroup generated by $\pi$-rotation	Yes
$p4$	Rotation by $\pi /2$	Four-element subgroup generated by $\pi /2$-rotation	Yes
$p4g$	Rotation by $\pi /2$, Reflection	Dihedral group of order eight	No
$p3$	Rotation by $2\pi /3$	Three-element subgroup generated by rotation	Yes
$p3m1$	Rotation by $2\pi /3$, Reflection	Dihedral group of order six	Yes
$p31m$	Rotation by $2\pi /3$, Reflection, with rotation center off reflection axis	Dihedral group of order six	No
$p6m$	Rotation by $\pi /3$, Reflection	Dihedral group:D12	Yes
$p6$	Rotation by $\pi /3$	Six-element subgroup generated by rotation	Yes