# Symmetry group (InstanceClass, 3)

### From Hi.gher. Space

**WIP**

## Three dimensions

The following table contains all the symmetry groups used by CRF polyhedra. The order of each group is given after an asterisk:

n = | 2 | 3 | 4 | 5 |
---|---|---|---|---|

special
| C_{s} *2Plane | |||

C_{n} *nChiral axial | C_{5} *5Chiral rhodoaxial | |||

C_{nv} *2nAxial | C_{2v} *4Biaxial (biplane) | C_{3v} *6Pyroaxial | C_{4v} *8Stauroaxial | C_{5v} *10Rhodoaxial |

D_{n} *2nChiral antiprismatic | D_{3} *6Chiral pyroantiprismatic | D_{4} *8Chiral stauroantiprismatic | D_{5} *10Chiral rhodoantiprismatic | |

D_{nd} *4nAntiprismatic | D_{2d} *8Biantiprismatic | D_{3d} *12Pyroantiprismatic | D_{4d} *16Stauroantiprismatic | D_{5d} *20Rhodoantiprismatic |

D_{nh} *4nPrismatic | D_{2h} *8Biprismatic (triplane, brick) | D_{3h} *12Pyroprismatic | D_{4h} *16Stauroprismatic | D_{5h} *20Rhodoprismatic |

Chiral polyhedral | O *24 Chiral staurohedral | I *60 Chiral rhodohedral | ||

Polyhedral | T_{d} *24Pyrohedral | O_{h} *48Staurohedral | I_{h} *120Rhodohedral |

There are five more 3D symmetry groups, which are not seen in CRF polyhedra:

- C
_{1}*1 (asymmetry) - C
_{nh}*2*n*(cyclic prismatic symmetry) - S
_{n}**n*(for even*n*only) (S_{2}being point symmetry) - T *12 (chiral pyrohedral symmetry)
- T
_{h}*24 (pyritohedral symmetry).