  We present a detailed study of the ESO 3060170 galaxy group combining
Chandra, XMM and optical observations. The system is found to be a fossil
galaxy group. The group X-ray emission is composed of a central dense cool core
(10 kpc in radius) and an isothermal medium beyond the central 10 kpc. The
region between 10 and 50 kpc (the cooling radius) has the same temperature as
the gas from 50 kpc to 400 kpc although the gas cooling time between 10 and 50
kpc (2 - 6 Gyr) is shorter than the Hubble time. Thus, the ESO 3060170 group
does not have a group-sized cooling core. We suggest that the group cooling
core may have been heated by a central AGN outburst in the past and the small
dense cool core is the truncated relic of a previous cooling core. The Chandra
observations also reveal a variety of X-ray features in the central region,
including a ``finger'', an edge-like feature and a small ``tail'', all aligned
along a north-south axis, as are the galaxy light and group galaxy
distribution. The proposed AGN outburst may cause gas ``sloshing'' around the
center and produce these asymmetric features. The observed flat temperature
profile to 1/3 R_vir is not consistent with the predicted temperature profile
in recent numerical simulations. We compare the entropy profile of the ESO
3060170 group with those of three other groups and find a flatter relation than
that predicted by simulations involving only shock heating, S $\propto$ r$^{~
0.85}$. This is direct evidence for the importance of non-gravitational
processes in group centers. We derive the mass profiles within 1/3 R_vir and
find the ESO 3060170 group is the most massive fossil group known (1 - 2 X
10$^{14}$ M$_{\odot}$). The M/L ratio of the system, ~ 150 at 0.3 R_vir, is
normal.
