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 ($sim3.4sigma$) 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.
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