In this paper, we introduce a new generalisation of Johnson graphs. The study of these graphs is linked to the study of intransitive triple factorisations Sym ( Ω ) = ABA of the (fi- nite)symmetricgroup,wherethesubgroupsAandBareintransitivesubgroupsofSym ( Ω ) . Indeed, we give combinatorial arguments to investigate the conditions under which such factorisations exist. We also use combinatorial arguments to study those conditions for which Sym ( Ω ) is a Geometric ABA-group, that is to say, Sym ( Ω ) = ABA, A ̸⊆ B, B ̸⊆ A and AB ∩ BA = A ∪ B
