Types of spreads and duality of the parallelisms of PG(3, 5) with automorphisms of order 13

摘要

A spread is a set of lines of PG(n,q) which partition the point set. A parallelism is a partition of the set of all lines by spreads. Empirical data on parallelisms is of interest both from theoretical point of view, and for different applications. Only 51 explicit examples of parallelisms of PG(3,5) have been known. We construct all (321) parallelisms of PG(3,5) with automorphisms of order 13 and classify them by the order of their automorphism group, the number of reguli in their spreads and duality. There are no regular ones among them. There are 19 self-dual parallelisms. We also claim that PG(3,5) has no point-transitive parallelisms.