### The (56, 11, 2) design of Hall, Lane, and Wales

W Jónsson
1973 Journal of combinatorial theory. Series A
In Hall, Lane, and Wales [3] , two classes of symmetric designs are constructed from rank three permutation groups (see [3] and [4] for definitions and notation). As an example of one of these constructions, they exhibit a symmetric design D with the parameters (56, 11, 2) , that is to say, with 56 points, 56 blocks, 11 points per block and any 2 distinct points determine exactly 2 blocks. The rank three representation of PSL(3, 4) of degree 56 is used in the construction. This design can be
more » ... structed with the aid of the Steiner system S" associated with the Mathieu group M" on 22 symbols. The hint for this construction comes from the remark in [3] that the points of D can be looked upon as a set of ovoids in the projective plane PG(2,4) upon which PSL(3, 4) acts. There are 168 ovals in this plane and under the action of PSL(3,4) they are divided into 3 orbits of 56 each. But one of these orbits is just the set of blocks avoiding a particular point in a certain representation of S" (see for example [6] ). Using the representation of D derived from S" it is possible to compute that D has a group of collineations which is isomorphic to a subgroup of index 3 in the group of all collineations and correlations of the geometry PG(2,4). In order to show that this is the full group of collineations, one needs to know that the stabilizer of a point in the representation PSL(3,4) on 56 symbols is PSL(2,9) acting in its natural representation as a group on 10 symbols on one of the orbits and in its representation on the 45 unordered pairs from the 10 symbols on the other orbit. Wales [7] discusses this representation of PSL(3, 4) and the associated graph [4] in detail. THE CONSTRUCTION OF D Let S" = (g, g', I) where B is the point set, @ the block set and I the incidence relation, be the Steiner system associated with A4" (see [5] ). 113