  We study Kitaev-Heisenberg-$\Gamma$ Hamiltonian on finite size clusters and
investigate various zero and finite temperature properties in detail and
explore the key differences due to relative sign and magnitude of Kitaev($K$),
Heisenberg $(J)$) and $\Gamma$ interaction in the presence of magnetic field.
Nearest-neighbor spin-spin correlation function shows step like character when
$J$ is positive only irrespective of sign of $K$. Similarly the low energy
spectrum under magnetic field depends on the sign of $K$. Dynamics of Majorana
fermions depend greatly on external magnetic field which can even stabilize the
gauge fields to Kitaev limit in the presence of $J$. Majorana dynamics also
shows that $K-J-\Gamma$ model to be a testbed to manipulate quantum speed
limit. For small magnetic field, magnetization shows a dome like structure for
an intermediate temperature range for positive $K$. However a positive $J$ can
cause this for negative $K$ also. The zero temperature magnetisation and
susceptibility shows large dependence on external magnetic field and relative
sign and magnitude of $K$ and $J$. Either a negative $K$ or $J$ is shown to
give qualitatively similar susceptibility for different magnetic field only.
For the specific heat a negative $K$ or $J$ favors mostly single peak structure
while a two peak structure is favored for positive $K$ or $J$, though the
magnitude of magnetic field can causes re-entrant behavior also. We briefly
discuss effect of $\Gamma$ and compare with recent experiments.
