I know that $$|G| = |{\rm Orb}_G(i)|\times |{\rm stab}_G(i)|.$$
A regular soccer ball has $20$ regular hexagons and $12$ regular pentagons. If I consider $12$ pentagons, the orbit of any of them has $12$ elements since a pentagon can be moved to any of the $12$ places by rotations and it remains in its place by five different rotations, ${\rm Stab}(S)$ has $5$ elements.
So, number of symmetries of a regular soccer ball is $12\times 5 = 60$.
But if I work this out using the $20$ hexagons ($20$ elements in orbit and $6$ in stabilizer), result comes out to be $20\times 6 = 120$.
What am I doing wrong?