On the structure of Specht modules in the principal block of FΣ3pFΣ3p

Let F be a field of characteristic p   at least 5. We study the Loewy structures of Specht modules in the principal block of FΣ3pFΣ3p. We show that a Specht module in the block has Loewy length at most 4 and composition length at most 14. Furthermore, we classify which Specht modules have Loewy length 1, 2, 3, or 4, produce a Specht module having 14 composition factors, describe the second radical layer and the socle of the reducible Specht modules, and prove that if a Specht module corresponds to a partition that is p-regular and p-restricted then the head of the Specht module does not extend the socle.