3D
- Space
- Only the trivial isometry group C1 leaves the whole space fixed.
- Plane
- Cs with respect to a plane leaves that plane fixed.
- Line
- Isometry groups leaving a line fixed are isometries which in every plane perpendicular to that line have common 2D point groups in two dimensions with respect to the point of intersection of the line and the planes.
- Cn ( n > 1 ) and Cnv ( n > 1 )
- cylindrical symmetry without reflection symmetry in a plane perpendicular to the axis
- cases in which the symmetry group is an infinite subset of that of cylindrical symmetry
- Point
- All other point groups in three dimensions
- No fixed points
- The isometry group contains translations or a screw operation.
Read more about this topic: Fixed Points Of Isometry Groups In Euclidean Space