# Pyritohedral symmetry

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Pyritohedral symmetry | |
---|---|

Rank | 3 |

Space | Spherical |

Order | 24 |

Elements | |

Axes | 3 × K_{3}, 4 × (G_{2}+×A_{1})/2 |

**Pyritohedral symmetry**, also known as **pyritic symmetry** and notated as **B _{3}/2**, is a 3D spherical symmetry group. It is the symmetry group of the pyritohedral icosahedron and its dual pyritohedron. It is similar to cubic symmetry, but the square axes turn into rectangular axes and the triangular axes turn into chiral triangular ones. Unlike full cubic or tetrahedral symmetries, it is a subgroup of dodecahedral symmetry.

### Subgroups[edit | edit source]

- Chiral tetrahedral symmetry (maximal)
- Protriangular antiprismatic symmetry (maximal)
- Chiral triangular prismatic symmetry
- Digonal prismatic symmetry (maximal)
- Chiral digonal prismatic symmetry
- Rectangular pyramidal symmetry
- Prodigonal prismatic symmetry
- Chiral digonal pyramidal symmetry
- Inversion symmetry
- Reflection symmetry
- Identity symmetry

### Convex polytopes with B_{3}/2 symmetry[edit | edit source]

- Cube (regular)/Octahedron (regular)
- Cuboctahedron (isogonal)/Rhombic dodecahedron (isotopic)
- Truncated cube (isogonal)/Triakis octahedron (isotopic)
- Pyritohedral icosahedron (isogonal)/Pyritohedron (isotopic)
- Pyritosnub cube (isogonal)/Tetragonal icositetrahedron (isotopic)