  Let $M$ be a convex polytope with non-empty interior in a finite dimensional
vector space, such that each vertex of $M$ is contained in exactly $n$ edges of
$M$. The Lie group of all smooth diffeomorphisms of $M$ contains the Lie
subgroup of all diffeomorphisms which restrict to the identity on the boundary
of the polytope. We give the quotient of the diffeomorphism group mod subgroup
a Lie group structure. This construction turns the canonical quotient
homomorphism into a smooth submersion, and we then obtain related results. We
also study controllability on $M$ and show that the identity component of the
diffeomorphism group is generated by the exponential image.
