Co$_{1/3}$NbS$_2$ is the only magnetically intercalated layered transition metal dichalcogenide (TMD) suggested to experience the complete suppression of magnetic order under pressure. From elastic neutron scattering we report the direct evidence for the reduction of the antiferromagnetic ordering temperature under pressure, up to complete suppression of magnetic order around 1.7 GPa. The static and ac magnetic susceptibility measurements reveal strong frustration in the magnetic subsystem, and spin canting responsible for the appearance of ferromagnetic (FM) component in dominantly antiferromagnetic (AF) ordered state. The electric transport in directions perpendicular and parallel to layers is explored for the first time in magnetically intercalated TMDs, in the wide temperature and pressure ranges. We show that electric transport reacts differently to magnetic ordering in directions along and perpendicular to layers, with the in-plane conductivity increasing, and the out-of-plane conductivity decreasing in the ordered state. At pressures above 3 GPa, we identify the appearance of the Kondo scattering regime. We use ab-initio calculations to explore the electronic structure in magnetically ordered state, the nature of magnetic interactions, and the mechanism responsible for the changes observed under pressure. The mechanisms of suppression of magnetic order under pressure are scrutinized in the light of these experimental and theoretical findings. We conclude that magnetic couplings beyond nearest-neighbors determine the nature of magnetic ordering. The suppression of ordering under pressure is ascribed to the pressure-induced shift in balance between super-exchange and Ruderman-Kittel-Kasuya-Yosida (RKKY) magnetic couplings, leading to amplified magnetic frustration.
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