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.
It takes (x,y,z)\rightarrow(-x,-y,-z) and is denoted i. When used in conjunction with a rotation, it becomes an improper rotation. An improper rotation by 360\degree/n degrees is denoted \bar{n} (or S_n). For periodic crystals, the crystallography restriction allows only the improper rotations \bar{1}, \bar{2}, \bar{3}, \bar{4}, and \bar{6}.
It takes (x,y,z)\rightarrow(x,y,-z),(x,-y,z),(-x,y,z), etc., which is equivalent to \bar{2}. Invariance under reflection can be denoted n{\sigma}_v or n\sigma_h. The rotation symmetry operation for 360\degree/n is denoted n (or C_n). For periodic crystals, crystallography restriction allows only 1, 2, 3, 4, and 6.
Symmetry operations can be indicated with symbols such as C_n, S_n, E, i, n\sigma_v, and n\sigma_h.
C_n indicates rotation about an n-fold symmetry axis.
S_n indicates improper rotation about an n-fold symmetry axis.
E (or I) indicates invariance under translation.
i indicates a center of symmetry under inversion.
n\sigma_v indicates invariance under n vertical reflections.
n\sigma_h indicates invariance under n horizontal reflections.