  For $N\ge 4$ we present a series of *-homomorphisms
$\varphi_n:C(S_N^+)\rightarrow B_n$ where $S_N^+$ is the quantum permutation
group. They are not necessarily representations of the quantum group $S_N^+$
but they yield good operator algebraic models of quantum permutation matrices.
The C*-algebras $B_n$ allow the construction of an inverse limit $B_{\infty}$
which defines a compact matrix quantum group $S_N\subsetneq G\subseteq S_N^+$.
We know $G=S_N^+$ for $N=4,5$ from Banica's work, but we have to leave open the
case $N\ge 6$.
