Construction
The construction of a Hughes plane is based on a nearfield N of order ''p''''2n'' for ''p'' an odd prime whose kernel K has order ''p''''n'' and coincides with the center of N.Properties
A Hughes plane H: # is a non-Desarguesian projective plane of odd square prime power order of Lenz-Barlotti type I.1, # has a Desarguesian Baer subplane H0, # is a self-dual plane in which every orthogonal polarity of H0 can be extended to a polarity of H, # every central collineation of H0 extends to a central collineation of H, and # the full collineation group of H has two point orbits (one of which is H0), two line orbits, and four flag orbits.The smallest Hughes Plane (order 9)
The Hughes plane of order 9 was actually found earlier by Veblen and Wedderburn in 1907. A construction of this plane can be found in where it is called the plane Ψ.Notes
References
* * * T. G. Room & P.B. Kirkpatrick (1971) ''Miniquaternion geometry'', Part III Miniquaternion planes, chapter V The Plane Ψ, pp 130–68,