I read Lectures on the Ikosahedron and the Solution of Equations of the Fifth Degree, by Felix Klein pp.26-30 If you read this part, you can see the paragraph like below.
We then have the proposition, the deduction of which was the object of our present considerations, viz., that the 60 rotations of the ikosahedral group are given by the following scheme: $$S^{\mu}, S^{\mu}TS^{\nu}, S^{\mu}U, S^{\mu}TS^{\nu}U, (\mu, \nu=0, 1, 2, 3, 4)$$ Here the rotations: $$S^{\mu}, S^{\mu}U$$
I understand that there can be up to 4 combinations of rigid motions of an icosahedron, such as $S^{\mu}TS^{\nu}U$. Is my understanding correct? If not, what are the maximum combinations of alphabets that a rigid motion of an icosahedron can have? And why?
Background to this question
I read this book I solved this way, a living way genius mathematician by Shun'ichi Kimura. And if you read the latter half of the book, you will find the following:
First, perform a reverse rotation of $B$, then perform $A$, and finally perform $B$ (let's call the rotation that combines these three as $C$). However, $C$ makes $g$ invariant. This is because, let's assume that the reverse rotation of $B$ moves $g$ to $r^ig$. Then, the destination of $g$ by $C$ is to first move to $r^ig$ by the reverse rotation of $B$, and since $A$ keeps $g$ invariant, its integer multiple $r^ig$ is also made invariant, and then $r^ig$ returns to $g$ by $B$. Therefore, we know that $C$ moves $g$ to $g$. (This is the critical point of the proof. Lagrange could not think of considering a rotation that reverses $B$, then performs $A$, and then performs $B$.)
When I read this paragraph, I wondered if the author wrote that this was the core of the proof because Lagrange did not think of the three rotations $C$, since three is the maximum number of rotations. Or maybe the maximum number of rotations could be four, five, or more, but he wrote it like this because he had to think of at least three rotations to get there.
