  From the first two seasons of the Palomar Transient Factory, we identify
three peculiar transients (PTF09dav, PTF10iuv, PTF11bij) with five
distinguishing characteristics: peak luminosity in the gap between novae and
supernovae (M_R = 15.5 to -16.5), rapid photometric evolution (rise-time
~12--15 days), large photospheric velocities (~6000 to 11000 km/s), early
spectroscopic evolution into nebular phase (~1 to 3 months) and peculiar
nebular spectra dominated by Calcium. We also culled the extensive decade-long
Lick Observatory Supernova Search database and identified an additional member
of this group, SN 2007ke. Our choice of photometric and spectroscopic
properties was motivated by SN 2005E (Perets et al. 2010). To our surprise, as
in the case of SN 2005E, all four members of this group are also clearly offset
from the bulk of their host galaxy. Given the well-sampled early and late-time
light curves, we derive ejecta masses in the range of 0.4--0.7 Msun.
Spectroscopically, we find that there may be a diversity in the photospheric
phase, but the commonality is in the unusual nebular spectra. Our extensive
follow-up observations rule out standard thermonuclear and standard
core-collapse explosions for this class of "Calcium-rich gap" transients. If
the progenitor is a white dwarf, we are likely seeing a detonation of the white
dwarf core and perhaps, even shockfront interaction with a previously ejected
nova shell. In the less likely scenario of a massive star progenitor, a very
non-standard channel specific to a low-metallicity environment needs to be
invoked (e.g., ejecta fallback leading to black hole formation). Detection (or
lack thereof) of a faint underlying host (dwarf galaxy, cluster) will provide a
crucial and decisive diagnostic to choose between these alternatives.
