English: A picture of our friend A5! (Imagined as embedded in SO(3) )

In this picture the vertices of the polyhedra represent elements of the group. The center of the sphere is the identity element, and vertices in the same polyhedron are in the same conjugacy class.The action of the vertex-elements should be interpreted as a rotation about the axis pointing from the center to that vertex by an angle given by the distance from the origin. Legend: Grey Sphere of radius Pi - our picture of SO(3) (really a picture of a sphere in its Lie algebra so(3) ) Yellow icosidodecahedron (https://en.wikipedia.org/wiki/Icosidodecahedron) of radius Pi representing the conjugacy class of 2-2-cycles (the class is order 15, and an icosododecahedron has 30 vertices, but in SO(3) the antipodes of the sphere of radius Pi are identified, so antipodal pairs are identified, giving 15 points.) Purple/blue and red icosahedra of radius 4 Pi/5 and 2 Pi/5 respectively. Each icosahedron represents half of the split 5-cycles (https://groupprops.subwiki.org/wiki/Splitting_criterion_for_conjugacy_classes_in_the_alternating_group), each order 12, corresponding to the 12 vertices. (Only the edge sphere has its antipodes identified) Green dodecahedron of radius 2 Pi/3 representing the class of 3-cycles, which is order 20 and as always corresponds to the 20 vertices. And there we have it! 1+12+12+15+20=60 elements in A5.

I got most of this information from here. https://groupprops.subwiki.org/wiki/Element_structure_of_alternating_group:A5#Conjugacy_class_structure