User:Jeremydover/sandbox/Hall plane of order 9

Hall plane of order 9
Order	9
Lenz-Barlotti Class	IVa.3
Automorphisms	$2^{8}\times 3^{5}\times 5$
Point Orbit Lengths	10,81
Line Orbit Lengths	1,90
Properties	Hall plane André plane Translation plane

Construction

This plane can be constructed from a Hall quasifield with defining polynomial $f(x)=x^{2}-\omega$, where $\omega$ is a primitive element of $GF(9)$.

This plane can be constructed using the André/Bruck-Bose construction with a regular spread of $PG(3,3)$ that has had a single regulus replaced by its opposite.

Ovals and Hyperovals

This plane has odd order, so by Qvist's theorem it does not have a hyperoval. It does have ovals.

Subplanes

The Hall plane of order 9 has subplanes of orders 2 and 3.

Unitals

It has several unitals.