The set of permutations generated by cyclic shift is studied using a number system coding for these permutations. The system allows to find the rank of a permutation given how it has been generated, and to determine a permutation given its rank. It defines a code describing structural and symmetry properties of the set of permutations ordered according to generation by cyclic shift. The code is associated with an Hamiltonian cycle in a regular weighted digraph. This Hamiltonian cycle is conjectured to be of minimal weight, leading to a combinatorial Gray code listing the set of permutations.