  The kinematic morphology-density relation of galaxies is normally attributed
to a changing distribution of galaxy stellar masses with the local environment.
However, earlier studies were largely focused on slow rotators; the dynamical
properties of the overall population in relation to environment have received
less attention. We use the SAMI Galaxy Survey to investigate the dynamical
properties of $\sim$1800 early and late-type galaxies with
$\log(M_*/M_{\odot})>9.5$ as a function of mean environmental overdensity
($\Sigma_{5}$) and their rank within a group or cluster. By classifying
galaxies into fast and slow rotators, at fixed stellar mass above
$\log(M_*/M_{\odot})>10.5$, we detect a higher fraction ($\sim3.4\sigma$) of
slow rotators for group and cluster centrals and satellites as compared to
isolated-central galaxies. Focusing on the fast-rotator population, we also
detect a significant correlation between galaxy kinematics and their stellar
mass as well as the environment they are in. Specifically, by using
inclination-corrected or intrinsic $\lambda_{R_e}$ values, we find that, at
fixed mass, satellite galaxies on average have the lowest
$\lambda_{\,R_e,intr}$, isolated-central galaxies have the highest
$\lambda_{\,R_e,intr}$, and group and cluster centrals lie in between.
Similarly, galaxies in high-density environments have lower mean
$\lambda_{\,R_e,intr}$ values as compared to galaxies at low environmental
density. However, at fixed $\Sigma_{5}$, the mean $\lambda_{\,R_e,intr}$
differences for low and high-mass galaxies are of similar magnitude as when
varying $\Sigma_{5}$ {($\Delta \lambda_{\,R_e,intr} \sim 0.05$. Our results
demonstrate that after stellar mass, environment plays a significant role in
the creation of slow rotators, while for fast rotators we also detect an
independent, albeit smaller, impact of mass and environment on their kinematic
properties.
