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Tetrahedral symmetry
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Selected <a href="/facts/Point_groups_in_three_dimensions/Uc4MRIyl">point groups in three dimensions</a><table><tbody><tr><td><a href="/facts/List_of_spherical_symmetry_groups/3X3YcHoQ">Involutional symmetry</a>Cs, (*)[ ] = </td><td><a href="/facts/Cyclic_symmetry_in_three_dimensions/wQUCezbQ">Cyclic symmetry</a>Cnv, (*nn)[n] = </td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF">Dihedral symmetry</a>Dnh, (*n22)[n,2] = </td></tr><tr><th colspan="4"><a href="/facts/Polyhedral_group/ynwsQLXE">Polyhedral group</a>, [n,3], (*n32)</th></tr><tr><td>Tetrahedral symmetryTd, (*332)[3,3] = </td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm">Octahedral symmetry</a>Oh, (*432)[4,3] = </td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23">Icosahedral symmetry</a>Ih, (*532)[5,3] = </td></tr></tbody></table> <p>A regular <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedron</a> has 12 rotational (or <a href="/facts/Orientation-preserving/lGXs42Nw">orientation-preserving</a>) symmetries, and a <a href="/facts/Symmetry_order/tmtq9goq">symmetry order</a> of 24 including transformations that combine a reflection and a rotation. </p><p>The group of all (not necessarily orientation preserving) symmetries is isomorphic to the group S4, the <a href="/facts/Symmetric_group/RUksK5l5">symmetric group</a> of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientation-preserving symmetries forms a group referred to as the <a href="/facts/Alternating_group/yethKwyo">alternating subgroup</a> A4 of S4. </p>