For which $n$ is the symmetric group $S_n$ a subgroup of the special orthogonal group $SO(3)$?

Question

For which $n$ is the symmetric group $S_n$ a subgroup of the special orthogonal group $SO(3)$? For example, this holds for $n≤4$, however I don't know if it holds for $n=5$ or what happens for larger $n$.

Answer 1

This question can be attacked either as a question about the finite subgroups of $SO(3)$ or as a question about the representation theory of $S_n$. Each of these approaches will prove that $S_n \hookrightarrow SO(3)$ iff $n \le 4$.

Finite subgroups: It's a classical result that the complete list of finite subgroups of $SO(3)$ is the following:

• the cyclic groups $C_n$,
• the dihedral groups $D_n$,
• the tetrahedral group $A_4$,
• the octahedral group $S_4$, or
• the icosahedral group $A_5$.

$S_4$ appears on this list but $S_n$ does not for $n \ge 5$.

Representation theory: The symmetric group $S_n$ is known to have the property that the lowest-dimensional faithful representation has dimension $n-1$ for $n \ge 5$, as reuns points out in the comments.