In this paper we show that the symmetric group $S_n$ has a Majorana representation $(S_n, T, W, \phi, \psi)$ only if $n\leq 12$.
The converse follows by a result of Simon Norton, who proved that the Monster group has a subgroup isomorphic to $S_{12}$. Further, we give the irreducible constituents and their dimensions of the ${\mathbb R}[S_n]$-modules generated by the axial vectors for $8\leq n\leq 12$
