We present optical VLT spectroscopy of 16 dwarf elliptical galaxies (or dEs) comparable in mass to NGC 205, and belonging to the Fornax cluster and to nearby groups of galaxies. Using ULySS and STECKMAP, we derive radial profiles of the SSP-equivalent ages, metallicities and star-formation histories. The old stellar population of the dEs, which dominates their mass, is likely coeval with that of massive ellipticals or bulges, but the star formation efficiency is lower. Important intermediate age (1-5 Gyr) populations, and frequently tails of star formation until recent times are detected. These histories are reminiscent of their lower mass dSph counterparts of the Local Group. Most galaxies (10/16) show significant metallicity gradients, with metallicity declining by 0.5 dex over one half-light radius on average. These gradients are already present in the old population. The flattened (or discy), rotating objects (6/16) have flat metallicity profiles. This may be consistent with a distinct origin for these galaxies or it may be due to their geometry. The central SSP-equivalent age varies between 1 and 6 Gyr, with the age slowly increasing with radius in the vast majority of objects. The group and cluster galaxies have similar radial gradients and star-formation histories. The strong and old metallicity gradients place important constraints on the possible formation scenarios of dEs. Numerical simulations of the formation of spherical low-mass galaxies reproduce these gradients, but they require a longer time for them to build up. A gentle depletion of the gas, by ram-pressure stripping or starvation, could drive the gas-rich, star-forming progenitors to the present dEs.
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