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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 ∼1800 early and late-type galaxies with log (M ⋆/ M ⊙) > 9.5 as a function of mean environmental overdensity (Σ5) and their rank within a group or cluster. By classifying galaxies into fast and slow rotators, at fixed stellar mass above log (M ⋆/ M ⊙) > 10.5, we detect a higher fraction (∼3.4σ) of slow rotators for group and cluster centrals and satellites as compared to isolated-central galaxies. We find similar results when using Σ5 as a tracer for environment. 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 |$\lambda _{R_{\rm {e}}}$| values, we find that, at fixed mass, satellite galaxies on average have the lowest |$\lambda _{\, R_{\rm {e}},\rm {intr}}$|⁠ , isolated-central galaxies have the highest |$\lambda _{\, R_{\rm {e}},\rm {intr}}$|⁠ , and group and cluster centrals lie in between. Similarly, galaxies in high-density environments have lower mean |$\lambda _{\, R_{\rm {e}},\rm {intr}}$| values as compared to galaxies at low environmental density. However, at fixed Σ5, the mean |$\lambda _{\, R_{\rm {e}},\rm {intr}}$| differences for low and high-mass galaxies are of similar magnitude as when varying Σ5 (⁠|$\Delta \lambda _{\, R_{\rm {e}},\rm {intr}} \sim 0.05$|⁠ , with σrandom = 0.025, and σ syst < 0.03). 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. [ABSTRACT FROM AUTHOR]