D4-type subgroups of F4(q)

LAWTHERWe treat the action of the simple group F 4 (q) on the cosets of subgroups D 4 (q), 2 D 4 (q) and 3 D 4 (q) and their extensions by graph automorphisms.We obtain the ranks and decompose the corresponding permutation characters; we show that, even allowing for the application of field automorphisms, the only two primitive multiplicity-free actions arising are those of F 4 (2) on cosets of D 4 (2).S 3 and 3 D 4 (2).3.For these two actions, we calculate the subdegrees; we find that all suborbits are self-paired, but that the action gives rise to no distance-transitive graph.