  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
$L_R\sim2\times10^{21}$ W Hz$^{-1}$ to $L_R\sim3\times$10$^{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.
