Deficiency One Parallelisms of PG(5, 2)

A spread in PG(n, q) is a set of lines such that each point is in exactly one line. A parallelism is a partition of the set of lines of PG(n, q) to spreads. A deficiency one parallelism is a parallelism with exactly one missing spread. Most of the results on parallelisms are for PG(3, q). PG(5, 2) is the smallest projective space of a dimension greater than 3. All point-transitive and all cyclic parallelisms of PG(5, 2) are known. Parallelisms with some definite automorphism groups have been constructed too. No transitive deficiency one parallelisms are known. The present work concerns deficiency one parallelisms of PG(5, 2). We present all parallelisms of PG(5, 2) which are invariant under a group of order 30 and show that the full automorphism group of a possible transitive deficiency one parallelism of PG(5, 2) cannot have a normal subgroup of order 15.