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The reflection symmetries have 6, 9, and 15 mirrors respectively. The octahedral symmetry, [4,3] can be seen as the union of 6 tetrahedral symmetry [3,3] mirrors, and 3 mirrors of dihedral symmetry Dih 2, [2,2]. Pyritohedral symmetry is another doubling of tetrahedral symmetry. The conjugacy classes of full tetrahedral symmetry, T d ≅ S 4 ...
D nh is the symmetry group for a "regular" n-gonal prism and also for a "regular" n-gonal bipyramid. D nd is the symmetry group for a "regular" n-gonal antiprism, and also for a "regular" n-gonal trapezohedron. D n is the symmetry group of a partially rotated ("twisted") prism. The groups D 2 and D 2h are noteworthy in that there is no special ...
The order of the symmetry group is the number of symmetries of the polyhedron. One often distinguishes between the full symmetry group, which includes reflections, and the proper symmetry group, which includes only rotations. The symmetry groups of the Platonic solids are a special class of three-dimensional point groups known as polyhedral ...
Conway's notation allows the order of the group to be computed as a product of elements with chiral polyhedral group orders: (T=12, O=24, I=60). In Conway's notation, a (±) prefix implies central inversion, and a suffix (.2) implies mirror symmetry. Similarly Du Val's notation has an asterisk (*) superscript for mirror symmetry.
For example, a cube is face-transitive, while a truncated cube has two symmetry orbits of faces. The same abstract structure may support more or less symmetric geometric polyhedra. But where a polyhedral name is given, such as icosidodecahedron, the most symmetrical geometry is often implied. [citation needed]
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags.A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive.
There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry. This article lists the groups by Schoenflies notation , Coxeter notation , [ 1 ] orbifold notation , [ 2 ] and order.
The full symmetry group of the icosahedron (including reflections) ... Every Platonic graph, including the icosahedral graph, is a polyhedral graph.