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

For each plane listed, I have provided

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. Here I have listed all known projective planes of order n < 32

I am also currently (2009-2010) 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
... ... ... ... ... ... ...
49 (hundreds of thousands of planes)          


/ revised June, 2010