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:
The number after p is the highest order of rotation, e.g. if it is 6 then there is a rotation which is 1/6 of a revolution.
m is for a (mirror) reflection perpendicular to the "x-axis"
g means a glide reflection but no reflection perpendicular to the "x-axis"
The "x-axis" is actually the left vertical edge of a cell
1 means no symmetry axis perpendicular to the "x-axis"
The last symbol is for a symmetry axis at an angle to the "x-axis"
If we were to use the above guidelines to name the groups rather than the standard abbreviations, here are the changes:
p2 |
pm |
pmm |
pg |
pgg |
pmg |
cm |
cmm |
p4m |
p4g |
p6m |
p211 |
p1m1 |
p2mm |
p1g1 |
p2gg |
p2mg |
c1m1 |
c2mm |
p4mm |
p4gm |
p6mm |
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