  A group-theoretical framework to describe vacancy ordering and magnetism in
the Fe$_{1-x}$S system is developed. This framework is used to determine the
sequence of crystal structures consistent with the observed magnetic structures
of troilite (FeS), and to determine the crystallographic nature of the
low-temperature Besnus transition in Fe$_{0.875}$S. We conclude that the Besnus
transition is a magnetically driven transition characterised by the rotation of
the moments out of the ac-plane, accompanied by small atomic displacements that
lower the symmetry to triclinic at low temperatures. Based on our phase
diagram, we predict related magnetically driven phase transitions at low
temperatures in all the commensurate superstructures of pyrrhotite. The exact
nature of the transition is determined by the symmetry of the vacancy ordered
state Based on this we predict spin-flop transitions in 3C and 5C pyrrhotite
and a transition akin to the Besnus transition in 6C pyrrhotite. Furthermore,
we clarify that 3C and 4C pyrrhotite carry a ferrimagnetic moment whereas 5C
and 6C are antiferromagnetic.
