Constructions of symmetric (176,50,14) designs by subgroups of the
sporadic simple group of Higman and Sims
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In 1967, G. Higman discovered a symmetric (176,50,14) design D that
admits the simple group HS of order 44.352.000 (which - at this time -
had already been discoverd by D.G.Higman and C.C.Sims) as its full automorphism
group. Until 1994, D was the only known symmetric (176,50,14) design.
One possibility to get new designs with these parameters is the
following:
(1) take a subgroup G of HS
(2) determine the stabilizers of the points and blocks of D in G
(3) find all symmetric (176,50,14) designs, on which
G acts with stabilizers isomorphic to those occuring
when G acts on D
While step (2) is a group theoretical routine work, step (3) combines
group theoretical and combinatorial methods. The possibilities of
constructing a design that permits the requested action are given
by action of the block-stabilizers on the point-stabilizers in G.
Application of this method led to two new designs having automorphism
groups of orders 960 and 11.520, respectively. G was chosen to be
isomorphic to an extension of an elementary abelian group of order
16 by the alternating group of degree 5; there are - up to
isomorphism - exactly two such extensions.
These techniques may always be used, when there is a design with
a sufficiently large automorphism group, e.g., the symplectic designs.
A really interesting application would be the search for a non-desarguian
projective plane of prime order with subgroups of PGammaL(n,p).