Is there a known example of a Calabi-Yau manifold (say,a Kähler compact manifold with $c_1$ torsion) with finite simple (non cyclic) fundamental group, for instance $\mathfrak{A}_5$? I am pretty sure such an example is not known in dimension 3 (see [this paper](http://downloads.hindawi.com/journals/ahep/2011/901898.pdf)), but perhaps it is easier to find it in higher dimension (?).