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Table 1 Enumeration of all finite symmetry groups in 3D space, with the Schoenflies and orbifold notations and the order of each group [47]

From: Beyond bilateral symmetry: geometric morphometric methods for any type of symmetry

Schoenflies Orbifold Order Comments
C n nn n Rotational symmetry of order n
Cnv *nn 2n Rotational symmetry of order n and reflection symmetry about n planes containing the rotation axis
Cnh n* 2n Rotational symmetry of order n and reflection symmetry about a plane perpendicular to the rotation axis
S2n n× 2n Rotational symmetry of order 2n in which odd-numbered elements are reflected about a plane perpendicular to the rotation axis
D n 22n 2n Dihedral symmetry: rotational symmetry of order n combined with rotational symmetry of order 2 about axes that are perpendicular to the first rotation axis
Dnd 2*n 2n Antiprismatic symmetry: Rotation symmetry of order n and reflection symmetry about n planes containing the rotation axis, as well as rotation symmetry of order 2 about a perpendicular axis in each of the resulting sectors
Dnh *22n 4n Prismatic symmetry: rotational symmetry of order n and reflection symmetries about planes n passing through the rotation axis as well as the plane perpendicular to it.
T 332 12 Tetrahedral symmetry, rotations only
Td *332 24 Complete tetrahedral symmetry, including reflection
Th 3*2 24 Pyritohedral symmetry
O 432 24 Octahedral symmetry, rotations only (also applies to cube)
Oh *432 48 Complete octahedral symmetry, icluding reflection (also applies to cube)
I 532 60 Icosahedral symmetry, rotations only
Ih *532 120 Complete icosahedral symmetry, including reflection
  1. Bilateral symmetry can be viewed as a special case of C nv or C nh with a rotation of order 1.