I've been studying finite projective geometry for several weeks and I came across the fact that the most studied planes are the translation planes. Is there any known example (of minimum order if possible) of a finite projective plane which is not a translation plane ?
2026-03-25 06:05:29.1774418729
Example of a finite projective plane which is not a translation plane
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All projective planes of order $\leq 8$ are Desarguesian (and hence translation planes). Up to isomorphism, there are four projective planes of order $9$, among them the Desarguesian plane and a single further translation plane. Hence the smallest order of a projective plane which is not a translation plane is $9$, and up to isomorphism, there are two such planes.