Finite exceptional groups of Lie type and symmetric designs

In this article, we study symmetric designs D with parameters (v, k, lambda) 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 ((q(6) - 1)/(q - 1), q(5), q(4)(q - 1)) associated to the generalised hexagon H(q) or its dual and X = G(2)(q) with point-stabiliser circumflex accent [q(5)] : GL(2)(q), or D is the orthogonal design with parameters (351, 126, 45) or (378, 117, 36) respectively for epsilon = - or epsilon = +, and X = G(2)(3) with point-stabiliser SL3 epsilon(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. (c) 2022 Elsevier B.V. All rights reserved.