Projective Planes of Small Order

This site is intended to provide a current list of known projective planes of small order. See also my page of other generalised polygons of small order.)

The completeness of this list is known only for planes of order n at most 10 [C.W.H. Lam, G. Kolesova and L. Thiel (1988); C.W.H. Lam, L. Thiel and S. Swiercz (1988)]. There is also a substantial literature classifying (or showing nonexistence of) planes of certain small orders (such as 11, 12, 15) admitting automorphisms of certain orders, or containing certain embedded configurations.

For basic definitions and results on the subject of projective planes, please refer to

P. Dembowski, Finite Geometries, Springer-Verlag, Berlin, 1968
D.R. Hughes and F.C. Piper, Projective Planes, Springer-Verlag, New York, 1973
A. Beutelspacher, ‘Projective planes’, pp.107-136 in Handbook of Incidence Geometry, ed. F. Buekenhout, North-Holland, 1995

For each plane listed, I have provided

a list of point-line incidences, and
a list of generators of the automorphism group.

These lists are provided as text files assessible through links found in the columns headed “Name” and “|Aut. Gp.|” respectively. Format: A projective plane of order n has N=n²+n+1 points and the same number of lines. The first text file lists, for each point, all n+1 lines (as integers in the range 0,1,...,N−1) incident with it. The second text file lists generators of the automorphism group as permutations of {0,1,2,...,2N−1}, in which the integers 0,1,...,N−1 represent points (in the same order as they appear in the first file) and the integers N,N+1,...,2N−1 represent lines (in the same order as they appear in the first file, but with N added to each index).

I have made extensive use of Brendan McKay's celebrated software package nauty for computing graph automorphisms.

If you are aware of small planes which I have overlooked in my list, I would appreciate an email message () from you. My list below includes all known projective planes of order n < 34 that I am aware of.

I am also currently working on a list of the thousands of known planes of order 49, similar to my list in the case of smaller orders. Check back soon!

order n	name	elementary divisors	\|Aut. Gp.\|	Point orbit lengths	Line orbit lengths	Remarks
2	PG₂(2)	1⁴2²6¹	168	7	7	self-dual
3	PG₂(3)	1⁷3⁵12¹	5616	13	13	self-dual
4	PG₂(4)	1¹⁰2²4⁸20¹	120960	21	21	self-dual
5	PG₂(5)	1¹⁶5¹⁴30¹	372000	31	31	self-dual
7	PG₂(7)	1²⁹7²⁷56¹	5630688	57	57	self-dual
8	PG₂(8)	1²⁸2⁹4⁹8²⁶72¹	49448448	73	73	self-dual
9	PG₂(9)	1³⁷3¹⁸9³⁵90¹	84913920	91	91	self-dual
9	Hall(9)	1⁴¹3¹⁰9³⁹90¹	311040	10, 81	1, 90
9	dual Hall(9)	1⁴¹3¹⁰9³⁹90¹	311040	1, 90	10, 81
9	Hughes(9)	1⁴¹3¹⁰9³⁹90¹	33696	13, 78	13, 78	self-dual
11	PG₂(11)	1⁶⁷11⁶⁵132¹	212427600	133	133	self-dual
13	PG₂(13)	1⁹²13⁹⁰182¹	810534816	183	183	self-dual
16	(22 planes)			273	273	G. Royle
17	PG₂(17)	1¹⁵⁴17¹⁵²306¹	6950204928	307	307	self-dual
19	PG₂(19)	1¹⁹¹19¹⁸⁹380¹	16934047920	381	381	self-dual
23	PG₂(23)	1²⁷⁷23²⁷⁵552¹	78156525216	553	553	self-dual
25	(193 planes)
27	(13 planes)
29	PG₂(29)	1⁴³⁶29⁴³⁴870¹	499631102880	871	871	self-dual
31	PG₂(31)	1⁴⁹⁷31⁴⁹⁵992¹	851974934400	993	993	self-dual
32	(12 planes)					U. Dempwolff
...	...	...	...	...	...	...
49	(hundreds of thousands of planes)

/ revised November, 2017