For flag-transitive point-primitive 2-designs with λ prime, exceptional simple socles give exactly the Suzuki-Tits ovoid design and a new G2(q) coset-geometry family, while sporadic socles give only three known designs.
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On flag-transitive automorphism groups of $2$-designs with $\lambda$ prime
For flag-transitive point-primitive 2-designs with λ prime, exceptional simple socles give exactly the Suzuki-Tits ovoid design and a new G2(q) coset-geometry family, while sporadic socles give only three known designs.