The icosahedron has $20$ triangular faces, $30$ edges and $12$ vertices. The rotational
symmetry group is isomorphic to the alternating permutation group $A_5$ (of order $60$).
To see that there are $20$ faces, we can divide the icosahedron as shown. The
top and bottom caps have $5$ faces each, and the band around the middle has
$10$ faces.
To see that there are $30$ edges, note that we have $20$ faces with $3$
edges each, which apparently gives $20\times 3=60$ edges. However, each edge is
shared between two adjacent faces, so there are really only $60/2=30$ edges.
To see that there are $12$ vertices, note that we have $20$ faces with $3$
vertices each, which apparently gives $20\times 3=60$ vertices. However,
each vertex is shared between $5$ adjacent faces, so there are really only
$60/5=12$ vertices.
Octahedron
The octahedron has $8$ triangular faces, $12$ edges and $6$ vertices. The rotational
symmetry group is isomorphic to the permutation group $S_4$ (of order $24$).
Tetrahedron
The tetrahedron has $4$ triangular faces, $6$ edges and $4$ vertices. The rotational
symmetry group is isomorphic to the alternating permutation group $A_4$ (of order $12$).
Cube
The cube has $6$ square faces, $12$ edges and $8$ vertices. The rotational
symmetry group is isomorphic to the permutation group $S_4$ (of order $24$).
Dodecahedron
The dodecahedron has $12$ pentagonal faces, $30$ edges and $20$ vertices. The rotational
symmetry group is isomorphic to the alternating permutation group $A_5$ (of order $60$).