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We report the first experimental results of the magnetoresistance, Hall effect, and quantum Shubnikov-de Haas oscillations on single crystals of ZrTe, which was recently predicted to be a new type of topological semimetal hosting both triply degenerate crossing points and Weyl fermion state. The analysis of Hall effect and quantum oscillations indicate that ZrTe is a multiband system with low carrier density, high carrier mobility, small cross-sectional area of Fermi surface, and light cyclotron effective mass, as observed in many topological semimetals. Meanwhile, the angular dependence of the magnetoresistance and the quantum-oscillation frequencies further suggest that ZrTe possesses a three-dimensional Fermi surface that is rather complex. Our results provide a new platform to realize exotic quantum phenomena related to the new three-component fermions distinct from Dirac and Weyl fermions.