Table 1

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

Schoenflies

Orbifold

Order

Comments


Cn

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

Dn

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


Bilateral symmetry can be viewed as a special case of Cnv or Cnh with a rotation of order 1.

Savriama and Klingenberg BMC Evolutionary Biology 2011 11:280   doi:10.1186/1471-2148-11-280

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