  We use an updated version of the halo-based galaxy group catalog of Yang et
al., and take the surface brightness of the galaxy group ($\mu_{\rm lim}$)
based on projected positions and luminosities of galaxy members as a
compactness proxy to divide groups into sub-systems with different compactness.
By comparing various properties, including galaxy conditional luminosity
function, stellar population, active galactic nuclei (AGN) activity, and X-ray
luminosity of the intra-cluster medium of carefully controlled high (HC) and
low compactness (LC) group samples, we find that the group compactness plays an
essential role in characterizing the detailed physical properties of the group
themselves and their group members, especially for low mass groups with $M_h
\lesssim 10^{13.5}h^{-1}M_{\odot}$. We find that the low-mass HC groups have a
systematically lower magnitude gap $\Delta m_{12}$ and X-ray luminosity than
their LC counterparts, indicating that the HC groups are probably in the early
stage of group merging. On the other hand, a higher fraction of passive
galaxies is found in the HC group, which however is a result of systematically
smaller halo-centric distance distribution of their satellite population. After
controlling of both $M_h$ and halo-centric distance, we do not find any
differences for both the quenching faction and AGN activity of the member
galaxies between the HC and LC groups. Therefore, we conclude that the halo
quenching effect, which result in the halo-centric dependence of galaxy
population, is a faster process compared to the dynamical relaxed time-scale of
galaxy groups.
