We study the breaking of time-reversal invariance (TRI) by the application of a magnetic field in the quantum kicked rotor (QKR), using Izrailev's finite-dimensional model. There is a continuous crossover from TRI to time-reversal noninvariance (TRNI) in the spectral and eigenvector fluctuations of the QKR. We show that the properties of this TRI to TRNI transition depend on α2/N , where α is the chaos parameter of the QKR and N is the dimensionality of the evolution operator matrix. For α2/N ≳N , the transition coincides with that in random matrix theory. For α2/N <N , the transition shows a marked deviation from random matrix theory. Further, the speed of this transition as a function of the magnetic field is significantly enhanced as α2/N decreases.