The formation and stability of surface gap solitons at the interface between a metal and a self-defocusing periodic media are investigated. Such type of surface gap solitons exist only in finite gaps when the lattice depth exceeds a critical value. In the first and second gap, the existence domain of surface gap solitons includes both stable and unstable domains. The critical value of the lattice depth in the second gap is much larger than the critical value of the lattice depth in the first gap. In the first finite gap, if the lattice depth is fixed, the energy flow of surface gap solitons decreases with an increase of the propagation constant, surface gap solitons have short oscillating tails in the lattice region when the propagation constant is increased. For a given propagation constant, the energy flow of surface gap solitons increases with lattice depth increasing, and surface gap solitons can be switched from unstable state to stable state by changing the lattice depth. Unstable surface gap solitons tend to self-bend toward the positive direction of the transverse axis during propagation. The self-deflection angle decreases with an increase of the lattice depth. The trajectory of the main lobe of surface gap soliton is a zigzag curve. In the second gap, surface gap solitons have more tail oscillations, and the unstable domain is close to the upper bound of the existing domain. With the propagation constant increasing, the unstable domain decreases gradually.