Symmetric Group S5 and Alternating Group A5

An old familiar question comes into my mind: "How to prove A5 is simple?" But I forgot the answer.

I remember there is a proof in Michael Artin's book, by considering some "concrete" rotation of some polyhedron. But I thought that proof is not nice.

Tried to google the proof on web. Found the good website Groupprops. And then found the page for S5 and A5.

Yes, there is a proof by considering the conjugacy classes of A5. And by induction, one can prove that is simple for .

A question: is there an intuitive way to see that in A5, the conjugacy class containing (1 2 3 4 5) and the conjugacy class containing (1 3 5 2 4) are different?