We report on the results of an analysis of Chandra, XMM-Newton and new GMRT data of the X-ray bright compact group of galaxies HCG 62, which is one of the few groups known to possess clear, small X-ray cavities in the inner regions. This is part of an ongoing X-ray/low-frequency radio study of 18 groups, initially chosen for the availability of good-quality X-ray data and evidence for AGN/hot gas interaction. At higher frequency (1.4 GHz), the HCG 62 cavity system shows minimal if any radio emission, but the new GMRT observations at 235 MHz and 610 MHz clearly detect extended low-frequency emission from radio lobes corresponding to the cavities. By means of the synergy of X-ray and low-frequency radio observations, we compare and discuss the morphology, luminosity and pressure of the gas and of the radio source. We find that the radio source is radiatively inefficient, with a ratio of radio luminosity to mechanical cavity power of $sim 10^{-4}$, and that the radio pressure of the lobes is about one order of magnitude lower than the X-ray pressure of the surrounding thermal gas. Thanks to the high spatial resolution of the Chandra surface brightness and temperature profiles, we also identify a shock front located at 36 kpc to the south-west of the group center, close to the southern radio lobe, with a Mach number $sim 1.5$ and a total power which is about one order of magnitude higher than the cavity power. Such a shock may have heated the gas in the southern region, as indicated by the temperature map. The shock may also explain the arc-like region of enriched gas seen in the iron abundance map, as this may be produced by a non-Maxwellian electron distribution near its front.
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