Identifying the 17 Plane Symmetry Groups

Type Lattice Smallest
Rotation
Reflections Glide-
Reflections
Generating Region Notes
p1 parallelogram none no no 1 unit the only symmetries are translations
p2 parallelogram 1/2 revolution no no 1/2 unit 4 types of 180 degree rotations.
pm rectangular none yes no 1/2 unit 2 types of parallel reflection axes
pmm rectangular 1/2 revolution yes no 1/4 unit 2 types of reflection axes horizontally and vertically
pg rectangular none no yes 1/2 unit  
pgg rectangular 1/2 revolution no yes 1/4 unit  
pmg rectangular 1/2 revolution yes yes 1/4 unit reflection axes are parallel to each other
cm rhombic none yes yes 1/2 unit  
cmm rhombic 1/2 revolution yes yes 1/4 unit  reflection axes are perpendicular to each other
p4 square 1/4 revolution no no 1/4 unit  
p4m square 1/4 revolution yes yes 1/8 unit 4-fold rotational centers are on reflection axes
p4g square 1/4 revolution yes yes 1/8 unit 4-fold rotational centers are not on reflection axes
p3 hexagonal 1/3 revolution no no 1/3 unit  
p3m1 hexagonal 1/3 revolution yes yes 1/6 unit centers of rotation are on reflection axes
p31m hexagonal 1/3 revolution yes yes 1/6 unit not all  centers of rotation are on reflection axes
p6 hexagonal 1/6 revolution no no 1/6 unit  
p6m hexagonal 1/6 revolution yes yes 1/12 unit  

Click on the name of the group in the table for a pattern which has that group as its group of symmetries.

Crystallographic notation for the symmetry groups
How to interpret the symbols in the notation:

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