We explicitly realize the group 17T7, an extension of SL(2,16) by C_2, as a Galois group over the rationals. The group arises from the field of definition of the 2-torsion on an abelian fourfold with real multiplication defined over a real quadratic field. We find such fourfolds using Hilbert modular forms, first numerically, then certifying the result with exact methods.