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 


C_{n} 
nn 
n 
Rotational symmetry of order n 
C_{nv} 
*nn 
2n 
Rotational symmetry of order n and reflection symmetry about n planes containing the rotation axis 
C_{nh} 
n* 
2n 
Rotational symmetry of order n and reflection symmetry about a plane perpendicular to the rotation axis 
S_{2n} 
n× 
2n 
Rotational symmetry of order 2n in which oddnumbered 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 
D_{nd} 
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 
D_{nh} 
*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 
T_{d} 
*332 
24 
Complete tetrahedral symmetry, including reflection 
T_{h} 
3*2 
24 
Pyritohedral symmetry 
O 
432 
24 
Octahedral symmetry, rotations only (also applies to cube) 
O_{h} 
*432 
48 
Complete octahedral symmetry, icluding reflection (also applies to cube) 
I 
532 
60 
Icosahedral symmetry, rotations only 
I_{h} 
*532 
120 
Complete icosahedral symmetry, including reflection 


Bilateral symmetry can be viewed as a special case of C_{nv }or C_{nh }with a rotation of order 1. 

Savriama and Klingenberg BMC Evolutionary Biology 2011 11:280 doi:10.1186/1471214811280 