We show that given any finite group G of cardinality k + 1 k + 1 , there is a Riemannian sphere S k − 1 {S^{k - 1}} (imbeddable isometrically as a hypersurface in R k {{\mathbf {R}}^k} ) such that its full isometry group is isomorphic to G. We also show the existence of a finite metric space of cardinality k ( k + 1 ) k(k + 1) whose full isometry group is isomorphic to G.