Summary: We develop the theory of discrete time Lagrangian mechanics on Lie groups, originated in the work of A. P. Veselov and J. Moser [Commun. Math. Phys. 139, No. 2, 217-243 (1991; Zbl 0754.58017)], and the theory of Lagrangian reduction in the discrete time setting. The results thus obtained are applied to the investigation of an integrable time discretization of a famous integrable system of classical mechanics – the Lagrange top. We recall the derivation of the Euler-Poinsot equations of motion both in the frame moving with the body and in the rest frame (the latter ones being less widely known). We find then a discrete time Lagrange function turning into the known continuous time Lagrangian in the continuous limit, and elaborate on both descriptions of the resulting discrete time system, namely in the body frame and in the rest frame. This system naturally inherits Poisson properties of the continuous time system, the integrals of motion being deformed. The discrete time Lax representations are found, and Kirchhoff’s kinetic analogy between elastic curves and motions of the Lagrange top is generalised to the discrete context.