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Co₃Sn₂S₂ is a newly discovered magnetic Weyl semimetal with a kagome lattice of cobalt ions and has triggered intense interest for rich fantastic phenomena. It was proposed that it may exhibit a coexistence of ferromagnetic order and antiferromagnetic order below T_C≈ 175 K, followed by a pure ferromagnetic order below T_A≈ 135 K. We employed half-polarized neutron technology and confirmed the ferromagnetic order along the c axis below T_C≈ 175 K. Inelastic neutron scattering revealed highly anisotropic magnon dispersions and linewidths below T_C, and paramagnetic excitations above T_C. The spin-wave spectra in the ferromagnetic order is dominated by ferromagnetic third-neighbor “across-hexagon” J_d coupling with a weak frustrated next-nearest-neighbor bond.  Our density functional theory calculations reveal that both the symmetry-allowed 120° antiferromagnetic orders support Weyl points in the intermediate temperature region T_AT_C, with distinct numbers and the locations of Weyl points. Our study unveils the unusual magnetic coupling and the interplay between various magnetic orders and electronic band topology in Co₃Sn₂S₂.