Classification of Automorphism Groups of Smooth Cubic Threefolds and Fourfolds

We classify the automorphism groups of smooth cubic threefolds and fourfolds. We also show that there are $156$ (respectively, $40$) connected families of smooth cubic fourfolds (respectively, threefolds) with specified automorphism group action. For each family, the group action and defining equations are also given. The classification combines both representation theory (based on GAP) and lattice theory (based on SageMath).