  Galaxy groups form the environment of the majority of galaxies in the local
Universe, and many host an extended hot intra-group medium whose radiative
cooling appears to fuel, and be stabilised by, feedback from AGN in
group-central galaxies. Unfortunately studies of the physical properties of
groups and the influence of AGN on their member galaxies and gaseous haloes
have been limited by a lack of reliable representative samples of groups in the
local Universe. To address this problem, we have assembled the Complete
Local-Volume Groups Sample (CLoGS), an optically-selected
statistically-complete sample of 53 groups within 80 Mpc, which we aim to
observe in both low-frequency radio and X-ray wavebands. We here describe
results from the first half of the sample, for which X-ray and radio
observations are complete. Roughly 55% of the groups have group-scale X-ray
halos, of which ~65% have cool cores a similar fraction to that found in galaxy
clusters. While 25 of the 26 group central galaxies host radio AGN, among the
X-ray bright groups only the cool core systems are found to support central jet
sources.
