Symmetry operations include the improper rotation, inversion operation, mirror plane, and rotation. Together, these operations create 32 crystal classes corresponding to the 32 point groups.
The inversion operation takes
 |
and is denoted 
.
When used in conjunction with a rotation, it becomes
an improper rotation. An improper
rotation by 
is denoted 
(or 
).
For periodic crystals, the crystallography
restriction allows only the improper rotations 
, 
, 
, 
, and 
.
The mirror plane symmetry operation takes
 |
etc., which is equivalent to 
. Invariance under reflection can be denoted 
or 
. The rotation symmetry
operation for 
is denoted 
(or 
).
For periodic crystals, crystallography
restriction allows only 1, 2, 3, 4, and 6.
Symmetry operations can be indicated with symbols such as 
, 
, 
, 
, 
, and 
.
1. 
indicates rotation about an 
-fold symmetry axis.
2. 
indicates improper rotation about an 
-fold symmetry axis.
3. 
(or 
)
indicates invariance under translation.
4. 
indicates a center of symmetry under inversion.
5. 
indicates invariance under 
vertical reflections.
6. 
indicates invariance under 
horizontal reflections.
