The paper describes a construction of abstract polytopes from Cayley graphs of symmetric groups. Given any connected graph G with p vertices and q edges, we associate with G a Cayley graph G(G) of the symmetric group S p and then construct a vertex-transitive simple polytope of rank q, the graphicahedron, whose 1-skeleton (edge graph) is G(G). The graphicahedron of a graph G is a generalization of the well-known permutahedron; the latter is obtained when the graph is a path. We also discuss symmetry properties of the graphicahedron and determine its structure when G is small.