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In three dimensions, the hyperoctahedral group is known as O × S 2 where O ≅ S 4 is the octahedral group, and S 2 is a symmetric group (here a cyclic group) of order 2. Geometric figures in three dimensions with this symmetry group are said to have octahedral symmetry , named after the regular octahedron , or 3- orthoplex .
O h, *432, [4,3], or m3m of order 48 – achiral octahedral symmetry or full octahedral symmetry. This group has the same rotation axes as O, but with mirror planes, comprising both the mirror planes of T d and T h. This group is isomorphic to S 4.C 2, and is the full symmetry group of the cube and octahedron. It is the hyperoctahedral group ...
There are 3 uniform colorings of the octahedron, named by the triangular face colors going around each vertex: 1212, 1112, 1111. The octahedron's symmetry group is O h, of order 48, the three dimensional hyperoctahedral group.
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere.It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices.
A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahedron, and a 4-dimensional cross-polytope is a 16-cell. Its facets are simplexes of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.
In two dimensions, the dihedral groups, which are the symmetry groups of regular polygons, form the series I 2 (p), for p ≥ 3. In three dimensions, the symmetry group of the regular dodecahedron and its dual, the regular icosahedron, is H 3, known as the full icosahedral group.
Since the symmetric group S 2 of degree 2 is isomorphic to the hyperoctahedral group is a special case of a generalized symmetric group. [ 6 ] The smallest non-trivial wreath product is Z 2 ≀ Z 2 {\displaystyle \mathbb {Z} _{2}\wr \mathbb {Z} _{2}} , which is the two-dimensional case of the above hyperoctahedral group.
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