A non-trivial 'Euler characteristic' is exhibited on a class of closed three dimensional manifolds. This is done by taking a class of Seifert fibre spaces, these being in 1- 1 correspondence with their fundamental groups - in-this case infinite cyclic central extensions of Fuchsian groups. By observation of-their finite index subgroups, or initially their presentations courtesy of the Reidemeister - Schreier approach, one can obtain a group -theoretic Euler characteristic which is in general non-trivial.