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"].
