PSL ( 2, 5 ) E " " A " 5, which can also be thought of as the binary icosahedral group covering the icosahedral group.
12.
The fundamental group of the Poincar?sphere is isomorphic to the binary icosahedral group, as the Poincar?sphere is the quotient of a 3-sphere by the binary icosahedral group.
13.
The fundamental group of the Poincar?sphere is isomorphic to the binary icosahedral group, as the Poincar?sphere is the quotient of a 3-sphere by the binary icosahedral group.
14.
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.
15.
It should not be confused with the full icosahedral group, which is a different group of order 120, and is rather a subgroup of the orthogonal group O ( 3 ).
16.
For example the binary icosahedral group covers the icosahedral group, an alternating group of degree 5, and the binary tetrahedral group covers the tetrahedral group, an alternating group of degree 4.
17.
For example the binary icosahedral group covers the icosahedral group, an alternating group of degree 5, and the binary tetrahedral group covers the tetrahedral group, an alternating group of degree 4.
18.
The alternating group " A " 5 is perfect but not superperfect : it has a non-trivial central extension, the binary icosahedral group ( which is in fact its UCE, and is superperfect ).
19.
Further, the binary icosahedral group is superperfect, meaning abstractly that its first two group homology groups vanish : H _ 1 ( 2I; \ mathbf { Z } ) \ cong H _ 2 ( 2I; \ mathbf { Z } ) \ cong 0.
20.
:Yes-the symmetry group of the snub cube is " O ", and the symmetry group of the snub dodecahedron is " I "-see octahedral group and icosahedral group at MathWorld .-- Gandalf61 16 : 06, Nov 22, 2003 ( UTC)
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