Tetrahedral symmetry
Involutional symmetry C_{s}, [1], (*) 
Cyclic symmetry C_{nv}, [n], (*nn) 
Dihedral symmetry D_{nh}, [n,2], (*n22) 

Polyhedral group, [n,3], (*n32)  

Tetrahedral symmetry T_{d}, [3,3], (*332) 
Octahedral symmetry O_{h}, [4,3], (*432) 
Icosahedral symmetry I_{h}, [5,3], (*532) 
A regular tetrahedron has 12 rotational (or orientationpreserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.
The group of all symmetries is isomorphic to the group S_{4}, symmetric group, as a permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientationpreserving symmetries forms a group referred to as the alternating subgroup A_{4} of S_{4}.
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Chiral and full (or achiral) tetrahedral symmetry and pyritohedral symmetry are discrete point symmetries (or equivalently, symmetries on the sphere). They are among the crystallographic point groups of the cubic crystal system.
Chiral tetrahedral symmetry
The tetrahedral rotation group T with fundamental domain; for the triakis tetrahedron, see below, the latter is one full face 
A tetrahedron can be placed in 12 distinct positions by rotation alone. These are illustrated above in the cycle graph format, along with the 180° edge (blue arrows) and 120° vertex (reddish arrows) rotations that permute the tetrahedron through those positions. 
In the tetrakis hexahedron one full face is a fundamental domain; other solids with the same symmetry can be obtained by adjusting the orientation of the faces, e.g. flattening selected subsets of faces to combine each subset into one face, or replacing each face by multiple faces, or a curved surface. 
T, 332, [3,3]^{+}, or 23, of order 12  chiral or rotational tetrahedral symmetry. There are three orthogonal 2fold rotation axes, like chiral dihedral symmetry D_{2} or 222, with in addition four 3fold axes, centered between the three orthogonal directions. This group is isomorphic to A_{4}, the alternating group on 4 elements; in fact it is the group of even permutations of the four 3fold axes: e, (123), (132), (124), (142), (134), (143), (234), (243), (12)(34), (13)(24), (14)(23).
The conjugacy classes of T are:
 identity
 4 × rotation by 120° clockwise (seen from a vertex): (234), (143), (412), (321)
 4 × rotation by 120° anticlockwise (ditto)
 3 × rotation by 180°
The rotations by 180°, together with the identity, form a normal subgroup of type Dih_{2}, with quotient group of type Z_{3}. The three elements of the latter are the identity, "clockwise rotation", and "anticlockwise rotation", corresponding to permutations of the three orthogonal 2fold axes, preserving orientation.
A_{4} is the smallest group demonstrating that the converse of Lagrange's theorem is not true in general: given a finite group G and a divisor d of G, there does not necessarily exist a subgroup of G with order d: the group G = A_{4} has no subgroup of order 6. Although it is a property for the abstract group in general, it is clear from the isometry group of chiral tetrahedral symmetry: because of the chirality the subgroup would have to be C_{6} or D_{3}, but neither applies.
Subgroups of chiral tetrahedral symmetry
Schoenflies notation 
Coxeter notation 
Orbifold notation 
Hermann Mauguin notation 
Order  Index  Relational hierarchy 

T  [3,3]^{+}  332  23  12  1  
D_{2}  [2,2]^{+}  222  222  4  3  
C_{3}  [3]^{+}  33  3  3  4  
C_{2}  [2]^{+}  22  2  2  6  
C_{1}  [ ]^{+}  11  1  1  12 
Achiral tetrahedral symmetry
T_{d}, *332, [3,3] or 43m, of order 24  achiral or full tetrahedral symmetry, also known as the (2,3,3) triangle group. This group has the same rotation axes as T, but with six mirror planes, each through two 3fold axes. The 2fold axes are now S_{4} (4) axes. T_{d} and O are isomorphic as abstract groups: they both correspond to S_{4}, the symmetric group on 4 objects. T_{d} is the union of T and the set obtained by combining each element of O \ T with inversion. See also the isometries of the regular tetrahedron.
The conjugacy classes of T_{d} are:
 identity
 8 × rotation by 120°
 3 × rotation by 180°
 6 × reflection in a plane through two rotation axes
 6 × rotoreflection by 90°
Subgroups of achiral tetrahedral symmetry
Schoenflies notation 
Coxeter notation 
Orbifold notation 
Hermann Mauguin notation 
Order  Index  Subgroup relations 

T_{d}  [3,3]  *332  43m  24  1  
T  [3,3]^{+}  332  23  12  2  
D_{2d}  [2^{+},4]  2*2  42m  8  3  
C_{3v}  [3]  *33  3m  6  4  
D_{2}  [2,2]^{+}  222  222  4  6  
C_{2v}  [2]  *22  mm2  4  6  
S_{4}  [2^{+},4^{+}  2×  4  4  6  
C_{3}  [3]^{+}  33  3  3  8  
C_{2}  [2]^{+}  22  2  2  12  
C_{s}  [ ]  *  2 or m  2  12  
C_{1}  [ ]^{+}  11  1  1  24 
Pyritohedral symmetry
T_{h}, 3*2, [4,3^{+} or m3, of order 24  pyritohedral symmetry. This group has the same rotation axes as T, with mirror planes through two of the orthogonal directions. The 3fold axes are now S_{6} (3) axes, and there is inversion symmetry. T_{h} is isomorphic to T × Z_{2}: every element of T_{h} is either an element of T, or one combined with inversion. Apart from these two normal subgroups, there is also a normal subgroup D_{2h} (that of a cuboid), of type Dih_{2} × Z_{2} = Z_{2} × Z_{2} × Z_{2} . It is the direct product of the normal subgroup of T (see above) with C_{i}. The quotient group is the same as above: of type Z_{3}. The three elements of the latter are the identity, "clockwise rotation", and "anticlockwise rotation", corresponding to permutations of the three orthogonal 2fold axes, preserving orientation.
It is the symmetry of a cube with on each face a line segment dividing the face into two equal rectangles, such that the line segments of adjacent faces do not meet at the edge. The symmetries correspond to the even permutations of the body diagonals and the same combined with inversion. It is also the symmetry of a pyritohedron, which is extremely similar to the cube described, with each rectangle replaced by a pentagon with one symmetry axis and 4 equal sides and 1 different side (the one corresponding to the line segment dividing the cube's face); i.e., the cube's faces bulge out at the dividing line and become narrower there. It is a subgroup of the full icosahedral symmetry group (as isometry group, not just as abstract group), with 4 of the 10 3fold axes.
The conjugacy classes of T_{h} include those of T, with the two classes of 4 combined, and each with inversion:
 identity
 8 × rotation by 120°
 3 × rotation by 180°
 inversion
 8 × rotoreflection by 60°
 3 × reflection in a plane
Subgroups of pyritohedral symmetry
Schoenflies notation 
Coxeter notation 
Orbifold notation 
Hermann Mauguin notation 
Order  Index  Subgroup relations 

T_{h}  [3^{+},4]  3*2  m3  24  1  
T  [3,3]^{+}  332  23  12  2  
D_{2h}  [2,2]  *222  mmm  8  3  
D_{3}  [2,3]^{+}  322  3  6  4  
D_{2}  [2,2]^{+}  222  222  4  6  
C_{2h}  [2,2^{+}  2*  2/m  4  6  
C_{2v}  [2]  *22  mm2  4  6  
C_{3}  [3]^{+}  33  3  3  8  
C_{2}  [2]^{+}  22  2  2  12  
C_{s}  [ ]  *  2 or m  2  12  
S_{2}  [2^{+},2^{+}  ×  1  2  12  
C_{1}  [ ]^{+}  11  1  1  24 
Solids with chiral tetrahedral symmetry
The Icosahedron colored as a snub tetrahedron has chiral symmetry.
Solids with full tetrahedral symmetry
Class  Name  Picture  Faces  Edges  Vertices 

Platonic solid  tetrahedron  4  6  4  
Archimedean solid  truncated tetrahedron  8  18  12  
Catalan solid  triakis tetrahedron  12  18  8  
Nearmiss Johnson solid  Truncated triakis tetrahedron  16  42  28  
Tetrated dodecahedron  28  54  28  
Uniform star polyhedron  Tetrahemihexahedron  7  12  6 
See also
References
 Peter R. Cromwell, Polyhedra (1997), p. 295
 The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, ISBN 9781568812205
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [1]
 N.W. Johnson: Geometries and Transformations, Manuscript, (2011) Chapter 11: Finite symmetry groups
External links
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