A Convex Polytope and an Antimatroid for any Given, Finite Group

For any given finite group G, we construct a convex polytope and an antimatroid whose automorphism groups are both isomorphic to G. The convex polytope is combinatorial in the sense of Naddef and Pulleyblank (1981), in particular it is binary; the diameter of its skeleton is at most 2; any automorphism of the polytope skeleton is the restriction of some isometry. The antimatroid is the antimatroid induced on a set of points in some euclidean space; any of its automorphisms is induced by some isometry.