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^{1}), but perhaps it is easier to find it in higher dimension (?).

^{1}*Davies, Rhys*, **The expanding zoo of Calabi-Yau threefolds**, Adv. High Energy Phys. 2011, Article ID 901898, 18 p. (2011). ZBL1234.81110, MR2821564.