A RANK-3 SIMPLE-GROUP OF ORDER 2(14)3(3)5(3)7 . 13 . 29 .1.

AbstractAfter a preliminary discussion of rank 3 permutation groups, attention is focused on the case where the stabilizer of a point is the Ree group F = 2F4(2) and the subdegrees are 1755 and 2304. If there is a rank 3 permutation group with F as point stabilizer and the above subdegrees then it is shown to be a simple group G of order 21433537 · 13 · 29 and the conjugacy classes of elements of G are determined. It is shown that such a group G has a proper covering group Ĝ with center of order two and also that G has no outer automorphisms

A RANK-3 SIMPLE GROUP-G OF ORDER 2(14)3(3)5(3)7 . 13 . 29 .2. CHARACTERS OF G AND G

AbstractProper coverings of finite (especially simple) groups are discussed in general, with emphasis on determining the conjugacy classes of elements of the covering group. This is used to determine the conjugacy classes of elements of the 2-fold proper covering group Ĝ of the group G of the title. During this determination it is proved that Ĝ must have a pair of complex conjugate characters of degree 28 with values in Z[i], the ring of Gaussian integers. A brief general discussion is given of Schur modules (the irreducible components of the tensor powers of the natural module for the general linear group) and their characters. This is then applied to the above pair of 28-dimensional characters of Ĝ to obtain many irreducible characters of Ĝ and G. The remaining irreducible characters of Ĝ and G are then obtained from certain induced characters

More About Divisible Design Graphs

Abstract: Divisible design graphs (DDG for short) have been recently defined by Kharaghani, Meulenberg and the second author as a generalization of (v, k, λ)-graphs. In this paper we give some new constructions of DDGs, most of them using Hadamard matrices and (v, k, λ)-graphs. For three parameter sets we give a nonexistence proof. Furthermore, we find conditions for a DDG to be walk-regular. It follows that most of the known examples are walk-regular, but some are not. In case walk-regularity of a DDG is forced by the parameters, necessary conditions for walk-regularity lead to new nonexistence results for DDGs. We examine all feasible parameter sets for DDGs on at most 27 vertices, establish existence in all but one cases, and decide on existence of a walk-regular DDG in all cases.