The 24-Cell and Calabi-Yau Threefolds with Hodge Numbers (1,1)
  Calabi-Yau threefolds with h^11(X)=h^21(X)=1 are constructed as free quotients of a hypersurface in the ambient toric variety defined by the 24-cell. Their fundamental groups are SL(2,3), a semidirect product of Z_3 and Z_8, and Z_3 x Q_8. 
Volker Braun
hep-th