Glasnik Matematicki, Vol. 41, No.2 (2006), 189-194.

BLOCK DESIGNS AND STRONGLY REGULAR GRAPHS CONSTRUCTED FROM THE GROUP U(3,4)

Dean Crnković and Vedrana Mikulić

Abstract.   We show a construction of the projective plane PG(2,16) and the Hermitian unital S(2,5,65) from the unitary group U(3,4) Further, we construct two block designs, a 2-(65,15,21) design and a 2-(65,26,250) design, and two strongly regular graphs with parameters (208,75,30,25) and (416,100,36,20). These incidence structures are defined on the elements of the conjugacy classes of the maximal subgroups of U(3,4). The group U(3,4) acts transitively as an automorphism group of the so constructed designs and strongly regular graphs. The strongly regular graph with parameters (416,100,36,20) has the full automorphism group of order 503193600, isomorphic to G(2,4) : Z₂. Since the Janko group J₂ is a subgroup of G(2,4), J₂ acts as an automorphism group of the constructed SRG(416,100,36,20).

2000 Mathematics Subject Classification.   05B05, 05E20, 05E30.

Key words and phrases.   Unitary group, block design, projective plane, Steiner system, strongly regular graph.

DOI: 10.3336/gm.41.2.01

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