We investigate the kinematic properties of a large (N=998) sample of COSMOS spectroscopic galaxy members distributed among 79 groups. We identify the Brightest Group Galaxies (BGGs) and cross-match our data with the VLA-COSMOS Deep survey at 1.4 GHz, classifying our parent sample into radio/non-radio BGGs and radio/non-radio satellites. The radio luminosity distribution spans from $L_Rsim2times10^{21}$ W Hz$^{-1}$ to $L_Rsim3times$10$^{25}$ W Hz$^{-1}$. A phase-space analysis, performed by comparing the velocity ratio (line-of-sight velocity divided by the group velocity dispersion) with the galaxy-group centre offset, reveals that BGGs (radio and non-radio) are mostly ($sim$80%) ancient infallers. Furthermore, the strongest ($L_R>10^{23}$ W Hz$^{-1}$) radio galaxies are always found within 0.2$R_{rm vir}$ from the group centre. Comparing our samples with HORIZON-AGN, we find that the velocities and offsets of simulated galaxies are more similar to radio BGGs than to non-radio BGGs, albeit statistical tests still highlight significant differences between simulated and real objects. We find that radio BGGs are more likely to be hosted in high-mass groups. Finally, we observe correlations between the powers of BGG radio galaxies and the X-ray temperatures, $T_{rm x}$, and X-ray luminosities, $L_{rm x}$, of the host groups. This supports the existence of a link between the intragroup medium and the central radio source. The occurrence of powerful radio galaxies at group centres can be explained by Chaotic Cold Accretion, as the AGN can feed from both the galactic and intragroup condensation, leading to the observed positive $L_{rm R}-T_{rm x}$ correlation.
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