Groups acting on necklaces and sandpile groups

We introduce a group naturally acting on aperiodic necklaces of length $n$ with two colours using the 1–1 correspondences between such necklaces and irreducible polynomials of degree $n$ over the field $\mathbb F_2$ of two elements. We notice that this group is isomorphic to the quotient group of non-degenerate circulant matrices of size $n$ over that field modulo a natural cyclic subgroup. Our groups turn out to be isomorphic to the sandpile groups for a special sequence of directed graphs.