We present magnetoresistance studies of the quasi-two-dimensional organic conductor $kappa$-(BETS)$_2$Mn[N(CN)$_2$]$_3$, where BETS stands for bis-(ethylene-dithio)-tetra-selena-fulvalene. Under a moderate pressure of 1.4,kbar, required for stabilizing the metallic ground state, Shubnikov - de Haas oscillations, associated with a classical and a magnetic-breakdown cyclotron orbits on the cylindrical Fermi surface, have been found at fields above 10,T. The effective cyclotron masses evaluated from the temperature dependence of the oscillation amplitudes reveal strong renormalization due to many-body interactions. The analysis of the relative strength of the oscillations corresponding to the different orbits and its dependence on magnetic field suggests an enhanced role of electron-electron interactions on flat parts of the Fermi surface.
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