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.
