Geodesic
The classification of flag-transitive Steiner 4-designs
Journal of Algebraic Combinatorics, Tome 26 (2007) no. 2, pp. 183-207.

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Summary: Among the properties of homogeneity of incidence structures flag-transitivity obviously is a particularly important and natural one. Consequently, in the last decades flag-transitive Steiner $t$-designs (i.e. flag-transitive $t-( v, k,1)$ designs) have been investigated, whereas only by the use of the classification of the finite simple groups has it been possible in recent years to essentially characterize all flag-transitive Steiner 2-designs. However, despite the finite simple group classification, for Steiner $t$-designs with parameters $t > 2$ such characterizations have remained challenging open problems for about 40 years (cf. [11, p. 147] and [12 p. 273], but presumably dating back to around 1965). The object of the present paper is to give a complete classification of all flag-transitive Steiner 4-designs. Our result relies on the classification of the finite doubly transitive permutation groups and is a continuation of the author's work [20, 21] on the classification of all flag-transitive Steiner 3-designs.
Classification : Primary, 51E10, cdot, Secondary, 05B05, cdot, 20B25
Keywords: keywords Steiner designs, flag-transitive group of automorphisms, 2-transitive permutation group 
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Huber, Michael. The classification of flag-transitive Steiner 4-designs. Journal of Algebraic Combinatorics, Tome 26 (2007) no. 2, pp. 183-207. http://geodesic.mathdoc.fr/item/JAC_2007__26_2_a3/