A6-extensions of Q and the modp cohomology of GL3(Z)

We present six examples of 3-dimensional mod p Galois representations of projective type A6 for which we were able to obtain computational evidence for the generalization of Serre's Conjecture proposed by Ash, Doud, Pollack, and Sinnott (Duke Math. J. 112 (2002) 521). Five of these examples arise from A6 extensions of Q; one of them arises from a 3.A6 extension of Q. We also propose a further refinement of the conjecture corresponding to the “peu vs. très ramifiée” distinction of Serre.