  The quantum permutation group of the set $X_n=\{1,..., n\}$ corresponds to
the Hopf algebra $A_{aut}(X_n)$. This is an algebra constructed with generators
and relations, known to be isomorphic to $\cc (S_n)$ for $n\leq 3$, and to be
infinite dimensional for $n\geq 4$. In this paper we find an explicit
representation of the algebra $A_{aut}(X_n)$, related to Clifford algebras. For
$n=4$ the representation is faithful in the discrete quantum group sense.
