Let C be a smooth projective curve of genus 3 over the rational numbers, and let J be its Jacobian. For each prime ell, the action of the absolute Galois group on the ell-torsion points of J gives rise to a six-dimensional Galois representation over the finite field with ell elements. If C is generic, then for all but finitely many primes, the image of this mod ell Galois representation is as large as possible (given the restraint of the Weil pairing). In this talk, we examine the following question: given an explicit generic genus 3 curve, can one determine this finite list of primes where the image of the Galois action is smaller than what would be expected?