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A symmetry operation is a transformation or manipulation of an object that leaves it apparently unchanged. The set of symmetry operations that can be done to an object (or lattice of objects) defines its symmetry and so its point group.

The different types of symmetry operation are:

  • Rotation - This is a rotation about an axis of a particular order of 1/order of a complete rotation
  • Reflection - This is a transformation of the object through a plane
  • Identity - This is an operation that does nothing it is necessary for completeness
  • Inversion - This is the transformation of the object through a centre of symmetry
  • Improper rotation (or rotoinversion) - This is a combination of a rotation and an inversion
I have not given abbreviations for the operations as these are different depending upon how the symmetry is being used (crystallography, geology, etc.) and when and where you learnt your point groups.