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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>Co</m:mi> <m:mn>3</m:mn> </m:msub> </m:math> \mathrm{Co}_{3} , 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">G</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>Co</m:mi> <m:mn>3</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathcal{G}(\mathrm{Co}_{3}) . The 2-local geometries for the other Conway groups <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>Co</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> \mathrm{Co}_{1} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>Co</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> \mathrm{Co}_{2} 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, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">G</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>Co</m:mi> <m:mn>3</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathcal{G}(\mathrm{Co}_{3}) seems to be studied less. There is another geometry associated with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>Co</m:mi> <m:mn>3</m:mn> </m:msub> </m:math> \mathrm{Co}_{3} (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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">G</m:mi> <m:mn>276</m:mn> </m:msub> </m:math> \mathcal{G}_{276} correspond to cliques of size 1, 2, 3, and 6, respectively, in a double cover of the complete graph on 276 vertices. The group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>Co</m:mi> <m:mn>3</m:mn> </m:msub> </m:math> \mathrm{Co}_{3} acts doubly transitively on the vertex set of the complete graph and flag-transitively on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">G</m:mi> <m:mn>276</m:mn> </m:msub> </m:math> \mathcal{G}_{276} . This double cover is naturally associated with the well-known 2-graph of <jats:alternat