Finite exceptional groups of Lie type and symmetric designs

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 finite simple exceptional of Lie type. conclusion, the design q 6 − 1 / 5 4 associated to generalised hexagon H or its dual = 2 point-stabiliser ˆ [ ] : L orthogonal 351 126 45 378 117 36 respectively for ϵ + 3 S . Our analysis depends heavily on detailed information about actions almost groups type cosets their large maximal subgroups. particular, properties derived in paper subgroups subdegrees such may be independent interest.