The full symmetry group of the icosahedron ( including reflections ) is known as the full icosahedral group, and is isomorphic to the product of the rotational symmetry group and the group " C " 2 of size two, which is generated by the reflection through the center of the icosahedron.
22.
For example, the regular quaternionic lines are in a one-to-one correspondence with the finite subgroups of " U " 1 ( "'H "') : the binary cyclic groups, binary dihedral groups, binary tetrahedral group, binary octahedral group, and binary icosahedral group.
23.
The binary icosahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism \ operatorname { Spin } ( 3 ) \ cong \ operatorname { Sp } ( 1 ) where Sp ( 1 ) is the multiplicative group of unit quaternions . ( For a description of this homomorphism see the article on quaternions and spatial rotations .)
24.
Therefore, it is also called a " dihedron " ( Greek : solid with two faces ), which explains the name " dihedral group " ( in analogy to " tetrahedral ", " octahedral " and " icosahedral group ", referring to the proper symmetry groups of a regular tetrahedron, octahedron, and icosahedron respectively ).
25.
The binary tetrahedral, octahedral and icosahedral groups are central extensions of the rotational symmetry groups of the platonic solids; these rotational symmetry groups are A _ 4, S _ 4 and A _ 5 respectively . 2T and 2 I can also be described as SL ( 2, \ mathbb { F } _ 3 ) and SL ( 2, \ mathbb { F } _ 5 ).
26.
The symmetry group of the compound is the ( rotational ) icosahedral group " I " of order 60, while the stabilizer of a single chosen tetrahedron is the ( rotational ) tetrahedral group " T " of order 12, and the orbit space " I " / " T " ( of order 60 / 12 = 5 ) is naturally identified with the 5 tetrahedra the coset " gT " corresponds to which tetrahedron " g " sends the chosen tetrahedron to.
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