On parallelisms of PG(5, 2) invariant under a cyclic subgroup of order 21

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. The construction of spreads and parallelisms is motivated by their various relations and applications. Most of the presently known explicit constructions of parallelisms are for n = 3. PG(5, 2) is the smallest projective space with n = 5. All point-transitive and all cyclic parallelisms of PG(5, 2) are known. In the present work we establish that up to conjugacy there is only one cyclic subgroup (of the automorphism group of the projective space) of order 21 which can be the automorphism group of a parallelism. We construct all parallelisms invariant under this subgroup that have the greatest possible number of spreads fixed under the assumed subgroup. We compute the automorphism group orders and invariants of the spreads of all the 2138 parallelisms of PG(5, 2) which we obtain.