Question: compute the order of the group of rotations of soccer ball ? and find it's group of rotations? Further, show that soccer ball can not have $60°$ rotational symmetry about line through the centre of two opposite hexagonal faces.
My attempt: In case of regular icosahedron (a solid with 20 congruent traingles as faces) I just denote each traingle face from 1 through 20 and then used the orbit stabilizer theorem. From which I get $|G|= |{\rm orb}_G(1)| |{\rm stab}_G(1)| =20• 3=60$
But here given that(soccer ball) has 20 faces that regular hexagon and 12 faces that are regular pentagons. So how $|G|$ computed? what to do of that additional 12 pentagonal faces ? How can i use the orbit stabilizer theorem? Please help.....