So...that's definitely a possible fundamental group of a 3-manifold.  But if you consider all the other conditions of the fibering conjecture, it's not allowed.  H_1 infinite means there's a non-separating surface; irrreduciblity means it can't be a sphere.  Compressing the surface gives a pi_1-injective surface, so the 3-manifold group contains a surface subgroup, which can't be the case if it's the integers (since the sphere is excluded).

Actually, the conclusion that the manifold fibers implies that the group contains a surface subgroup.  The surface can't be a sphere (because of irreducibility), so we have a more complicated surface whose fundamental group injects into the fundamental group of the 3-manifold.