For every dimension d ≥ ≥ 2 there exists a SIC-POVM whose elements are the orbit of a positive rank-one operator E 0 under the Weyl–Heisenberg group H d . What is more, E 0 commutes with an element T of the Jacobi group J d = H d ⋊ ⋊ S L ( 2 , Z d ) . The action of T on H d modulo the center has order three.