The tetrahedral group
is the point group of symmetries of the tetrahedron
including the inversion operation. It is one of the 12 non-Abelian groups of order
24. The tetrahedral group has conjugacy classes 1, , , , and (Cotton 1990, pp. 47 and 434). Its multiplication
table is illustrated above. The tetrahedral group is implemented in the Wolfram
Language as `FiniteGroupData`[`"Tetrahedral"`,
`"PermutationGroupRepresentation"`] and as a point
group as `FiniteGroupData`[`"CrystallographicPointGroup"`,
`"Td"`, `"PermutationGroupRepresentation"`].

has a pure rotational subgroup of
order 12 denoted
(Cotton 1990, pp. 50 and 433). It is isomorphic to the alternating
group
and has conjugacy classes 1, , , and . It has 10 subgroups: one of length 1, three of length
2, 4 of length 3, one of length 4, and one of length 12. Of these, only the trivial
subgroup, subgroup of order 4, and complete subgroup are normal. The pure rotational
tetrahedral subgroup
is implemented in the Wolfram Language
as a point group as `FiniteGroupData`[`"CrystallographicPointGroup"`,
`"T"`, `"PermutationGroupRepresentation"`].

The cycle graph of is illustrated above. The numbers of elements such that for , 2, ..., 12, are 1, 4, 9, 4, 1, 12, 1, 4, 9, 4, 1, 12.

Platonic and Archimedean solids that can be generated by group are illustrated above, with the corresponding basis vector summarized in the following table, where is the golden ratio.

solid | basis vector |

cuboctahedron | |

icosahedron | |

octahedron | |

tetrahedron | |

truncated tetrahedron |

There is also a point group known as . It has conjugacy classes 1, , , , , , , and (Cotton 1990, pp. 50 and 434). The group is implemented in the Wolfram
Language as a point group as `FiniteGroupData`[`"CrystallographicPointGroup"`,
`"Th"`, `"PermutationGroupRepresentation"`].