  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 of the symmetric group S_p and then
construct a vertex-transitive simple polytope of rank q, called the
graphicahedron, whose 1-skeleton (edge graph) is the Cayley graph. 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.
