On geometries of the Conway group Co 3 and their McLaughlin group subgeometries

Abstract Recently, the author published a book A. A. Ivanov, Ever-Evolving Groups—an Introduction to Modern Finite Group Theory, Algebr. Appl. 32, Springer, Cham, 2025 where he summarised the recent progress in the geometric theory of sporadic groups and outlined some geometries which require further investigation. Among them was a geometry for the smallest Conway sporadic simple group Co 3 Co₃, with diagram originally introduced in M. A. Ronan and G. Stroth, Minimal parabolic geometries for the sporadic groups, European J. Combin. 5 (1984), 1, 59–91 (cf. M. A. Ronan, Coverings of certain finite geometries, Finite Geometries and Designs, London Math. Soc. Lecture Note Ser. 49, Cambridge University, Cambridge (1981), 316–331, Table 1 and F. Buekenhout, Diagram geometries for sporadic groups, Finite Groups—Coming of Age (Montreal 1982), Contemp. Math. 45, American Mathematical Society, Providence (1985), geometry (23), p. 14), which we denote by G ⁢ (Co 3) G (Co₃). The 2-local geometries for the other Conway groups Co 1 Co₁ and Co 2 Co₂ are the tilde and Petersen geometries which have been intensively studied (cf. A. A. Ivanov and S. V. Shpectorov, The flag-transitive tilde and Petersen-type geometries are all known, Bull. Amer. Math. Soc. (N. S. ) 31 (1994), 2, 173–184). However, G ⁢ (Co 3) G (Co₃) seems to be studied less. There is another geometry associated with Co 3 Co₃ (cf. M. A. Ronan, Coverings of certain finite geometries, Finite Geometries and Designs, London Math. Soc. Lecture Note Ser. 49, Cambridge University, Cambridge (1981), 316–331, Table 1 and F. Buekenhout, Diagram geometries for sporadic groups, Finite Groups—Coming of Age (Montreal 1982), Contemp. Math. 45, American Mathematical Society, Providence (1985), geometry (23), p. 14) with diagram The elements of type 1, 2, 3, and 4 in the geometry G 276 G₂₇₆ correspond to cliques of size 1, 2, 3, and 6, respectively, in a double cover of the complete graph on 276 vertices. The group Co 3 Co₃ acts doubly transitively on the vertex set of the complete graph and flag-transitively on G 276 G₂₇₆. This double cover is naturally associated with the well-known 2-graph of <jats: alternat