We prove certain cases of nonabelian reciprocity between mod p Galois representations and Hecke eigenclasses in the mod p cohomology of GL(3,Z), using the symmetric square liftings from GL(2) and congruences on Hecke eigenvalues. We need to develop some modular representation theory for GL(3,Z/p). In some cases we also need to invoke certain computer calculations to finish the proof.