In this article, we study symmetric designs D with parameters (v,k,λ) admitting a flag- transitive and point-primitive automorphism group G whose socle X is a finite simple exceptional group of Lie type. In conclusion, D is the symmetric design with parameters ((q6 − 1)/(q − 1), q5 , q4 (q − 1)) associated to the generalised hexagon H (q) or its dual and X = G2(q) with point-stabiliser ˆ[q5] : GL2(q), or D is the orthogonal design with parameters (351, 126, 45) or (378, 117, 36) respectively for ε = − or ε = +, and X = G 2 (3) with point-stabiliser S Lε3 (3) : 2. Our analysis depends heavily on detailed information about actions of finite exceptional almost simple groups of Lie type on the cosets of their large maximal subgroups. In particular, properties derived in the paper about large subgroups and the subdegrees of such actions may be of independent interest.
