User:Jeremydover/sandbox/Desarguesian plane of order 3

Projective plane of order 3
Order	3
Lenz-Barlotti Class	VII.2
Automorphisms	$2^{4}\times 3^{3}\times 13$ $PGL(3,3)$
Point Orbit Lengths	13
Line Orbit Lengths	13
Properties	Desarguesian Self-dual

Construction

This plane can be constructed using the standard vector space construction with a three-dimensional vector space over the finite field GF(3).

Ovals and Hyperovals

This plane has odd order, so by Qvist's theorem it does not have a hyperoval. Being a Desarguesian, this plane does have ovals, and by Segré's theorem, all ovals in this plane are conics, being the set of absolute points of an orthogonal polarity.

Subplanes

Since this plane is a Desarguesian plane of prime order, it has no proper subplanes.

Unitals

Since the order of this plane is not square, it does not have a unital.