Only four infinite families of 2-designs with gcd(r,lambda)=1 admit flag-transitive almost simple automorphism groups with exceptional Lie type socle: Suzuki designs, Ree designs, and Ree unitals.
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Flag-transitive block designs and finite exceptional simple groups of Lie type
Only four infinite families of 2-designs with gcd(r,lambda)=1 admit flag-transitive almost simple automorphism groups with exceptional Lie type socle: Suzuki designs, Ree designs, and Ree unitals.