We prove certain cases of nonabelian reciprocity between mod p Galois repre
sentations and Hecke eigenclasses in the mod p cohomology of GL(3, Z), usin
g the symmetric square liftings from GL(2) and congruences on Hecke eigenva
lues. We need to develop some modular representation theory for GL(3, Zip).
In some cases we also need to invoke certain computer calculations to fini
sh the proof. (C) 1999 Academic Press.