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
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 | PG_{2}(2) | 1^{4}2^{2}6^{1} | 168 | 7 | 7 | self-dual |
3 | PG_{2}(3) | 1^{7}3^{5}12^{1} | 5616 | 13 | 13 | self-dual |
4 | PG_{2}(4) | 1^{10}2^{2}4^{8}20^{1} | 120960 | 21 | 21 | self-dual |
5 | PG_{2}(5) | 1^{16}5^{14}30^{1} | 372000 | 31 | 31 | self-dual |
7 | PG_{2}(7) | 1^{29}7^{27}56^{1} | 5630688 | 57 | 57 | self-dual |
8 | PG_{2}(8) | 1^{28}2^{9}4^{9}8^{26}72^{1} | 49448448 | 73 | 73 | self-dual |
9 | PG_{2}(9) | 1^{37}3^{18}9^{35}90^{1} | 84913920 | 91 | 91 | self-dual |
9 | Hall(9) | 1^{41}3^{10}9^{39}90^{1} | 311040 | 10, 81 | 1, 90 | |
9 | dual Hall(9) | 1^{41}3^{10}9^{39}90^{1} | 311040 | 1, 90 | 10, 81 | |
9 | Hughes(9) | 1^{41}3^{10}9^{39}90^{1} | 33696 | 13, 78 | 13, 78 | self-dual |
11 | PG_{2}(11) | 1^{67}11^{65}132^{1} | 212427600 | 133 | 133 | self-dual |
13 | PG_{2}(13) | 1^{92}13^{90}182^{1} | 810534816 | 183 | 183 | self-dual |
16 | (22 planes) | 273 | 273 | G. Royle | ||
17 | PG_{2}(17) | 1^{154}17^{152}306^{1} | 6950204928 | 307 | 307 | self-dual |
19 | PG_{2}(19) | 1^{191}19^{189}380^{1} | 16934047920 | 381 | 381 | self-dual |
23 | PG_{2}(23) | 1^{277}23^{275}552^{1} | 78156525216 | 553 | 553 | self-dual |
25 | (193 planes) | |||||
27 | (13 planes) | |||||
29 | PG_{2}(29) | 1^{436}29^{434}870^{1} | 499631102880 | 871 | 871 | self-dual |
31 | PG_{2}(31) | 1^{497}31^{495}992^{1} | 851974934400 | 993 | 993 | self-dual |
... | ... | ... | ... | ... | ... | ... |
49 | (hundreds of thousands of planes) |