The Complete Local Volume Groups Sample -- III. Characteristics of group central radio galaxies in the Local Universe

Using new 610 MHz and 235 MHz observations from the Giant Metrewave Radio Telescope (GMRT) in combination with archival GMRT and Very Large Array (VLA) survey data we present the radio properties of the dominant early-type galaxies in the low$-$richness sub-sample of the Complete Local-volume Groups Sample (CLoGS; 27 galaxy groups) and provide results for the radio properties of the full CLoGS sample for the first time. We find a high radio detection rate in the dominant galaxies of the low-richness sub-sample of 82% (22/27); for the full CLoGS sample, the detection rate is 87% (46/53). The group-dominant galaxies exhibit a wide range of radio power, 10$^{20}$ $-$ 10$^{25}$ W Hz$^{-1}$ in the 235 and 610 MHz bands, with the majority (53%) presenting point-like radio emission, 19% hosting currently active radio jets, 6% having remnant jets, 9% being diffuse and 13% having no detected radio emission. The mean spectral index of the detected radio sources in the 235$-$610 MHz frequency range is found to be $alpha_{235}^{610}sim$0.68, and $alpha_{235}^{1400}sim$0.59 in the 235$-$1400 MHz one. In agreement with earlier studies, we find that the fraction of ultra-steep spectrum sources ($alpha>$1.3) is $sim$4%, mostly dependent on the detection limit at 235 MHz. The majority of point-like systems are found to reside in dynamically young groups, whereas jet systems show no preference between spiral-rich and spiral-poor group environments. The mechanical power of the jet sources in the low$-$richness sample groups is estimated to be $sim$10$^{42}$ $-$ 10$^{44}$ erg s$^{-1}$ with their black hole masses ranging between 2$times$10$^{8}$ $-$ 5$times$10$^{9}$ M$_{odot}$. We confirm previous findings that, while radio jet sources tend to be associated with more massive black holes, black hole mass is not the decisive factor in determining jet activity or power.