Partitions of AG(4,3)AG(4,3) into maximal caps

In a geometry, a maximal cap is a collection of points of largest size no three of which are collinear. In AG(4,3)AG(4,3), maximal caps contain 20 points; the 81 points of AG(4,3)AG(4,3) can be partitioned into 4 mutually disjoint maximal caps together with a single point PP, where every pair of points that makes a line with PP lies entirely inside one of those caps. The caps in a partition can be paired up so that both pairs are either in exactly one partition or they are both in two different partitions. This difference determines the two equivalence classes of partitions of AG(4,3)AG(4,3) under the action by affine transformations.