The candidate SX Phe star KIC 11754974 shows a remarkably high number of combination frequencies in the Fourier amplitude spectrum: 123 of the 166 frequencies in our multifrequency fit are linear combinations of independent modes. Predictable patterns in frequency spacings are seen in the Fourier transform of the light curve. We present an analysis of 180 d of short-cadence Kepler photometry and of new spectroscopic data for this evolved, late A-type star. We infer from the 1150-d, long-cadence light curve, and in two different ways, that our target is the primary of a 343-d, non-eclipsing binary system. According to both methods, the mass function is similar, f(M) = 0.0207 ± 0.0003 M⊙. The observed pulsations are modelled extensively, using separate, state-of-the-art, time-dependent convection (TDC) and rotating models. The models match the observed temperature and low metallicity, finding a mass of 1.50-1.56 M⊙. The models suggest that the whole star is metal poor, and that the low metallicity is not just a surface abundance peculiarity. This is the best frequency analysis of an SX Phe star, and the only Kepler δ Sct star to be modelled with both TDC and rotating models.