Polyhedral group

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Tetrahedral reflection domains.png
6 mirrors
Octahedral reflection domains.png
3+6 mirrors
Icosahedral reflection domains.png
15 mirrors
Sphere symmetry group td.png
Td, [3,3], *332
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
(Order 24)
Sphere symmetry group oh.png
Oh, [4,3], *432
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
(Order 48)
Sphere symmetry group ih.png
Ih, [5,3], *532
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
(Order 120)
Sphere symmetry group t.png
T, [3,3]+, 332
Sphere symmetry group o.png
O, [4,3]+, 432
Sphere symmetry group i.png
I, [5,3]+, 532
Tetrahedral group
(Order 12)
Octahedral group
(Order 24)
Icosahedral group
(Order 60)

In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids. There are three polyhedral groups:

These symmetries double to 24, 48, 120 respectly for the full reflectional groups. 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 Dih2, [2,2].

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