Classification of line-transitive point-imprimitive linear spaces with line size at most 12

In this paper we complete a classification of finite linear spaces $${\mathcal{S}}$$ with line size at most 12 admitting a line-transitive point-imprimitive subgroup of automorphisms. The examples are the Desarguesian projective planes of orders 4, 7, 9 and 11, two designs on 91 points with line size 6, and 467 designs on 729 points with line size 8.

[1]  Andries E. Brouwer,et al.  Block designs , 1996 .

[2]  Alice C. Niemeyer,et al.  Finite line-transitive linear spaces: parameters and normal point-partitions , 2003 .

[3]  Cheryl E. Praeger,et al.  Block-transitive, point-imprimitive designs with lambda = 1 , 1993, Discret. Math..

[4]  A. Camina,et al.  Sporadic groups and automorphisms of linear spaces , 2000 .

[5]  Alexander Shen,et al.  Algorithms and Programming , 1996 .

[6]  L. Thiel,et al.  The Non-Existence of Finite Projective Planes of Order 10 , 1989, Canadian Journal of Mathematics.

[7]  Cheryl E. Praeger,et al.  The block-transitive, point-imprimitive 2-(729, 8, 1) designs , 1992, Applicable Algebra in Engineering, Communication and Computing.

[9]  Alice C. Niemeyer,et al.  Linear Spaces with a Line-transitive Point-imprimitive Automorphism Group and Fang-li Parameter Gcd(k,r) at Most Eight * , 2007 .

[10]  Alan R. Camina,et al.  Line-transitive Automorphism Groups of Linear Spaces , 1993, Electron. J. Comb..

[11]  Cheryl E. Praeger,et al.  Inequalities involving the Delandtsheer-Doyen parameters for finite line-transitive linear spaces , 2003, J. Comb. Theory, Ser. A.

[12]  Alice C. Niemeyer,et al.  Finite line-transitive linear spaces: Theory and search strategies , 2009 .

[13]  Alan R. Camina,et al.  Block transitive automorphism groups of 2-(nu, k, 1) block designs , 1989, J. Comb. Theory, Ser. A.

[14]  J. Conway,et al.  Atlas of finite groups : maximal subgroups and ordinary characters for simple groups , 1987 .

[15]  J. Conway,et al.  ATLAS of Finite Groups , 1985 .

[16]  H. Weyl.  Permutation Groups , 2022 .

[17]  C. Praeger,et al.  Line-transitive, point-imprimitive linear spaces: the grid case , 2008 .

[18]  J. Doyen,et al.  Most block-transitivet-designs are point-primitive , 1989 .