  We introduce a group naturally acting on aperiodic necklaces of length $n$
with two colours using the 1--1 correspondences between aperiodic necklaces and
irreducible polynomials over the field $\F_2$ 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.
