Given a $r$-graph $G$ with edge chromatic number $r$, there exist a natural construction of an abstract $r$- Polytope $P_G$, called the Colorful Polytope, such that the 1- skeleton of such polytope is the graph $G$. In particular when the graph is a Cayley graph of the symmetric group $S_p$, $P_G$ is a generalization of the Permuthahedron called the Graphicahedron. In this talk we will discuss the construction, and explore some combinatorial symmetry properties of such polytope, analyze transitivity properties of their automorphism groups, discuss some interesting cases which are intimately related to the geometry of the infinite euclidean Coxeter group. Furthermore we will observe an interesting relation of this polytopes with PL-manifolds.