A property of the Birkhoff polytope

The Birkhoff polytope $B_n$ is the convex hull of all $n\times n$ permutation matrices in ${\mathbb{R}}^{n\times n}$. We compute the combinatorial symmetry group of the Birkhoff polytope. A representation polytope is the convex hull of some finite matrix group $G\le \mathrm{GL}(d,\mathbb{R})$. We show that the group of permutation matrices is essentially the only finite matrix group which yields a representation polytope with the same face lattice as the Birkhoff polytope.