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), Z_3 × Z_8 , and Z_3 × Q_8