Bilbao Crystallographic Server

Seitz symbols for 3D Point Groups

The Seitz symbols for 3D crystallographic point groups symmetry operations consist of a rotation (or linear) part denoted by R.

The rotation parts R consists of symbols that specify the type and the order of the symmetry operation, and the orientation of the correspondig symmetry element with respect to the basis. The orientation is denoted by the direction of the axis for rotations and rotoinversions, or the direction of the normal to reflection planes.

The nomenclature for the symbol R is:
  • 1 and -1 for the identity and the inversion, respectively
  • m for reflections
  • 2, 3, 4 and 6 are used for rotations
  • -3, -4 and -6 are used for rotoinversions
For rotations and rotoinversions of order higher than 2, a superscript + or - is used to indicate the sense of the rotation. The subscripts of the symbols R denote the characteristic direction of the operation.


Consider the coordinate triplets of the general positions of the 3D Crystallographic point group S4(-4):

(1) x,y,z          (2) -x,-y,z            (3) y,-x,-z                      (4) -y,x,-z

The corresponding geometric interpretations of the symmetry operations are:

(1) 1                (2) 2 0,0,z           (3) -4+ 0,0,z; 0,0,0       (4) -4- 0,0,z; 0,0,0     

In Seitz notation, the symmetry operations are denoted by:

(1) 1                (2) 2001               (3) -4+001                      (4) -4-001