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	Full automorphism group	Orientation-preserving
$p1$	None	Trivial	Yes	No	Yes
$pg$	Glide reflection	Two-element subgroup generated by reflection	No	No	No
$pm$	Reflection	Two-element subgroup generated by reflection	Yes	No	No
$cm$	Reflection and glide reflection	Klein four-group generated by two reflections	No	No	No
$p2$	Rotation by $\pi$	Two-element subgroup generated by $\pi$-rotation	Yes	Yes	Yes
$pmg$	Reflection and rotation by $\pi$	Klein four-group generated by two reflections	No	No	No
$cmm$	Reflection, rotation by $\pi$ with rotation center off reflection axis	Klein four-group generated by two reflections	No	No	No
$pmm$	Two orthogonal reflections	Klein four-group generated by two reflections	Yes	Yes	No
$pgg$	Rotation by $\pi$, glide reflection	Klein four-group generated by two reflections	No	No	No
$p4$	Rotation by $\pi /2$	Four-element subgroup generated by $\pi /2$-rotation	Yes	No	Yes
$p4g$	Rotation by $\pi /2$, Reflection	Dihedral group of order eight	No	No	No
$p3$	Rotation by $2\pi /3$	Three-element subgroup generated by rotation	Yes	No	Yes
$p3m1$	Rotation by $2\pi /3$, Reflection	Dihedral group of order six	Yes	No	No
$p31m$	Rotation by $2\pi /3$, Reflection, with rotation center off reflection axis	Dihedral group of order six	No	No	No
$p6$	Rotation by $\pi /3$	Six-element subgroup generated by rotation	Yes	No
$p6m$	Rotation by $\pi /3$, Reflection	Dihedral group:D12	Yes	Yes	No

Facts used

1. Crystallographic restriction: This is the main lemma used in the classification, and it states that if a lattice possesses a nontrivial rotational symmetry, then it is spanned by two shortest vectors of equal length at an angle of $\pi /2,\pi /3,2\pi /3$. Note that the case of $\pi /3$ and $2\pi /3$ gives equivalent lattices.