>>14 >5次交代群は表現とかわかるの？
下記が参考になるだろう
(参考)（”A5 < SO3(R)”の図解があるので 参考になるよ）
https://en.wikipedia.org/wiki/Alternating_group Alternating group
A5 is the smallest non-abelian simple group, having order 60, and thus the smallest non-solvable group.
Generators and relations
For n ≥ 3, An is generated by 3-cycles, since 3-cycles can be obtained by combining pairs of transpositions. This generating set is often used to prove that An is simple for n ≥ 5.
Exceptional isomorphisms
There are some exceptional isomorphisms between some of the small alternating groups and small groups of Lie type, particularly projective special linear groups. These are:
A5 is isomorphic to PSL2(4), PSL2(5), and the symmetry group of chiral icosahedral symmetry. (See[1] for an indirect isomorphism of PSL2(F5) → A5 using a classification of simple groups of order 60, and here for a direct proof).
Example A5 as a subgroup of 3-space rotations
A5 is the group of isometries of a dodecahedron in 3-space, so there is a representation A5 → SO3(R).
In this picture the vertices of the polyhedra represent the elements of the group, with the center of the sphere representing the identity element. Each vertex represents a rotation about the axis pointing from the center to that vertex, by an angle equal to the distance from the origin, in radians. Vertices in the same polyhedron are in the same conjugacy class. Since the conjugacy class equation for A5 is 1 + 12 + 12 + 15 + 20 = 60, we obtain four distinct (nontrivial) polyhedra.
The vertices of each polyhedron are in bijective correspondence with the elements of its conjugacy class, with the exception of the conjugacy class of (2,2)-cycles, which is represented by an icosidodecahedron on the outer surface, with its antipodal vertices identified with each other. The reason for this redundancy is that the corresponding rotations are by π radians, and so can be represented by a vector of length π in either of two directions. Thus the class of (2,2)-cycles contains 15 elements, while the icosidodecahedron has 30 vertices.
The two conjugacy classes of twelve 5-cycles in A5 are represented by two icosahedra, of radii 2π/5 and 4π/5, respectively. The nontrivial outer automorphism in Out(A5) ≃ Z2 interchanges these two classes and the corresponding icosahedra.
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