# Projective Planes of Order 25

This site is intended to provide a current list of known projective planes of order 25. I have listed the 193 planes of which I am aware (5 self-dual planes plus 94 dual pairs). The planes are listed in increasing order of 5-rank, the most readily computable isomorphism invariant. For basic definitions and results on the subject of projective planes, please refer to

• P. Dembowski, Finite Geometries, Springer-Verlag, Berlin, 1968; or
• D.R. Hughes and F.C. Piper, Projective Planes, Springer-Verlag, New York, 1973.

More details on the new planes w1 and w2 can be found in

I have made extensive use of Brendan McKay's celebrated software package nauty for computing graph automorphisms; also the computational algebra package GAP (Graphs, Algorithms and Programming) for some of the group computations (e.g. computing conjugacy classes of involutions in groups).

If you are aware of planes which I have overlooked in my list, I would appreciate an email message () from you.

I have also provided

## Table of Known Projective Planes of Order 25

No. Plane Description 5-rank |Autgp| Point orbit lengths Line orbit lengths Derived Planes  Subplanes  Fingerprint
1* s1 Desarguesian 226 304668000000 651 651 s2 5409500 s1
2 b4, b4Dual Hering 239 1800000 6,20,625 1,150,500 none ; b4a,b4b 257300000 519500 b4, b4Dual
3 s2, s2Dual Hall 251 3600000 6,20,625 1,150,500 s1,s3,s4 ; s2a,s2b 242600000 519500 s2, s2Dual
4 a3, a3Dual Walker 253 1500000 1,25,625 1,25,625 a7 ; a3a,a3b 263000000 519500 a3, a3Dual
5 a2, a2Dual Dickson nearfield 255 2880000 2,24,625 1,50,600 a2 ; a2a,a2b 258080000 519500 a2, a2Dual
6 a4, a4Dual Rao 256 180000 2,6,18,625 1,50,150,450 a2,a8 ; a4a,a4b,a4c,a4d 259070000 519500 a4, a4Dual
7 a8, a8Dual Czerwinski & Oakden 257 120000 1,1,24,625 1,25,25,600 a4 ; a8a,a8b,a8c,a8d,a8e,a8f 256640000 519500 a8, a8Dual
8 s4, s4Dual subregular (exceptional nearfield) 258 1440000 2,24,625 1,50,600 s2,s5 ; s4a,s4b 255680000 519500 s4, s4Dual
9 b1, b1Dual Foulser 258 130000 26,625 1,650 none ; b1a,b1b,b1c 255510000 519500 b1, b1Dual
10 b6, b6Dual Czerwinski & Oakden 258 120000 2,122,625 1,50,3002 none ; b6a,b6b,b6c,b6d,b6e 252200000 519500 b6, b6Dual
11 s5, s5Dual subregular 259 720000 8,18,625 1,200,450 s4 ; s5a,s5b,s5c 255980000 519500 s5, s5Dual
12 a6, a6Dual Czerwinski & Oakden 259 360000 6,8,12,625 1,150,200,300 a1 ; a6a,a6b,a6c 252110000 519500 a6, a6Dual
13 a5, a5Dual Rao 259 60000 42,6,12,625 1,1002,150,300 a1 ; a5a,a5b,a5c,a5d,a5e,a5f,a5g 254180000 519500 a5, a5Dual
14 s3, s3Dual subregular 260 720000 2,24,625 1,50,600 s2 ; s3a,s3b 250880000 519500 s3, s3Dual
15 a7, a7Dual Rao 260 300000 6,20,625 1,150,500 a3 ; a7a,a7b,a7c 246800000 519500 a7, a7Dual
16 b5, b5Dual Walker 261 4800000 10,16,625 1,250,400 none ; b5a,b5b 2110800000 519500 b5, b5Dual
17 b7, b7Dual Czerwinski & Oakden 261 240000 4,6,16,625 1,100,150,400 none ; b7a,b7b,b7c,b7d 258860000 519500 b7, b7Dual
18 a1, a1Dual Czerwinski & Oakden 262 360000 6,8,12,625 1,150,200,300 a5,a6 ; a1a,a1b,a1c 253400000 519500 a1, a1Dual
19 b2, b2Dual Foulser 262 130000 26,625 1,650 none ; b2a,b2b,b2c 255510000 519500 b2, b2Dual
20 b8, b8Dual Czerwinski & Oakden 262 80000 2,8,16,625 1,50,200,400 none ; b8a,b8b,b8c,b8d,b8e,b8f,b8g 257120000 519500 b8, b8Dual
22 b4a, b4aDual   262 10000 1,5,20,125,2502 1,252,502,500 b4Dual ; none 240620000 51125 b4a, b4aDual
23 b3, b3Dual Czerwinski & Oakden 264 90000 2,6,18,625 1,50,150,450 none ; b3a,b3b,b3c,b3d,b3e 253190000 519500 b3, b3Dual
24 b4b, b4bDual   264 3000 32,20,25,754,1502 1,156,302,500 b4Dual ; none 240531500 5975 b4b, b4bDual
25 a8a, a8aDual   266 12000 6,20,25,600 1,30,120,500 a8Dual ; none 243080000 5900 a8a, a8aDual
26* h1 (Ordinary) Hughes 268 744000 31,620 31,620 h1a 238812000 331000 54620 h1
27 a2a, a2aDual   268 48000 6,20,25,600 1,30,120,500 a2Dual ; none 242348000 51500 a2a, a2aDual
28* s2a   268 20000 1,5,20,25,502,500 1,5,20,25,502,500 s2Dual 235110000 52000 s2a
31* h2 Exceptional Hughes 272 1488000 31,620 31,620 h2a 247616000 54620 h2
32 s2b, s2bDual   272 6000 32,20,25,1504 1,152,304,500 s2Dual ; s3bDual,s4bDual 239453000 51950 s2b, s2bDual
33 s3a, s3aDual   273 12000 32,20,25,3002 1,152,602,500 s3Dual ; none 240926000 5900 s3a, s3aDual
34 b5a, b5aDual   274 16000 2,4,20,25,200,400 1,10,20,40,80,500 b5Dual ; none 251432000 51100 b5a, b5aDual
35 a8b, a8bDual   274 6000 32,20,25,3002 1,152,602,500 a8Dual ; none 240716000 51050 a8b, a8bDual
36* h1b   274 4000 1,5,20,25,502,500 1,5,20,25,502,500 h1aDual 238446000 34000 51120 h1b
37 h1a, h1aDual   275 24000 12,22,20,25,600 12,24,25,502,500 h1 ; h1b 238644000 33000 51020 h1a, h1aDual
38 b8a, b8aDual   275 4000 12,22,20,25,1002,2002 1,52,102,202,402,500 b8Dual ; none 240760000 51100 b8a, b8aDual
39 a4a, a4aDual   275 3000 32,20,25,1504 1,152,304,500 a4Dual ; none 241691000 51125 a4a, a4aDual
40 b6a, b6aDual   275 2000 12,22,20,25,1006 1,52,102,206,500 b6Dual ; none 240308000 5850 b6a, b6aDual
41 b7a, b7aDual   276 4000 12,22,20,25,1002,2002 1,52,102,202,402,500 b7Dual ; none 240084000 5900 b7a, b7aDual
43 a2b, a2bDual   276 4000 12,4,20,25,502,100,2002 1,52,102,202,402,500 a2Dual ; a4cDual 243569000 51600 a2b, a2bDual
44 a8c, a8cDual   276 4000 23,20,25,2003 1,103,403,500 a8Dual ; none 240996000 51000 a8c, a8cDual
45 b7b, b7bDual   276 2000 12,22,20,25,1006 1,52,102,206,500 b7Dual ; none 239706000 51050 b7b, b7bDual
48 a4b, a4bDual   277 1000 12,22,20,255,5010 1,56,1012,500 a4Dual ; a8dDual 242115500 51075 a4b, a4bDual
49 a4c, a4cDual   277 1000 12,22,20,255,5010 1,56,1012,500 a4Dual ; a2bDual 240980000 51000 a4c, a4cDual
50 b6b, b6bDual   277 1000 12,22,20,255,5010 1,56,1012,500 b6Dual ; none 240240500 5825 b6b, b6bDual
51 b8b, b8bDual   277 500 16,20,2525 1,530,500 b8Dual ; none 240395000 5925 b8b, b8bDual
52 a4d, a4dDual   277 500 16,20,2525 1,530,500 a4Dual ; a8eDual,a8fDual 242190500 51150 a4d, a4dDual
53 b5b, b5bDual   278 10000 1,5,20,125,2502 1,252,502,500 b5Dual ; none 250682500 51000 b5b, b5bDual
54 b7c, b7cDual   278 2000 12,4,20,25,502,1005 1,52,102,206,500 b7Dual ; none 239690500 51000 b7c, b7cDual
55 a3b, a3bDual   278 2000 12,4,20,25,502,1005 1,52,102,206,500 a3Dual ; a7bDual 240428000 51075 a3b, a3bDual
57 a6b, a6bDual   278 1500 32,20,25,758 1,1510,500 a6Dual ; none 239174000 51050 a6b, a6bDual
58 a6c, a6cDual   278 1000 12,22,20,255,5010 1,56,1012,500 a6Dual ; none 238948500 51075 a6c, a6cDual
59 a5b, a5bDual   278 500 16,20,2525 1,530,500 a5Dual ; none 239882000 5850 a5b, a5bDual
60 a5c, a5cDual   278 500 16,20,2525 1,530,500 a5Dual ; none 239749500 51025 a5c, a5cDual
61 a5d, a5dDual   278 500 16,20,2525 1,530,500 a5Dual ; none 240096000 5975 a5d, a5dDual
62 a8d, a8dDual   278 500 16,20,2525 1,530,500 a8Dual ; a4bDual 241253000 5950 a8d, a8dDual
63 a8e, a8eDual   278 500 16,20,2525 1,530,500 a8Dual ; a4dDual 241176500 5950 a8e, a8eDual
64 a8f, a8fDual   278 500 16,20,2525 1,530,500 a8Dual ; a4dDual 241358500 5925 a8f, a8fDual
65 h2a, h2aDual   279 48000 12,4,20,25,600 12,24,25,100,500 h2, h2b ; none 241160000 51620 h2a, h2aDual
66 w1, w1Dual Wyoming 279 19200 12,24,25,600 12,24,25,600 w2 ; none 242390400 32400 51100 w1, w1Dual
67 b8c, b8cDual   279 2000 12,22,20,25,1006 1,52,102,206,500 b8Dual ; none 241082000 51100 b8c, b8cDual
68 s4b, s4bDual   279 2000 12,22,20,25,504,1004 1,52,106,204,500 s4Dual ; s2bDual,s5cDual 240268000 51300 s4b, s4bDual
69 a7b, a7bDual   279 2000 12,22,20,25,1006 1,52,102,206,500 a7Dual ; a3bDual 238834000 51175 a7b, a7bDual
70 b3a, b3aDual   279 1500 32,20,25,758 1,1510,500 b3Dual ; none 239174000 51050 b3a, b3aDual
71 a1b, a1bDual   279 1500 32,20,25,758 1,1510,500 a1Dual ; a5gDual 240705500 51275 a1b, a1bDual
72 a1c, a1cDual   279 1000 12,22,20,255,5010 1,56,1012,500 a1Dual ; none 239467000 51050 a1c, a1cDual
73 b6c, b6cDual   279 1000 12,22,20,255,5010 1,56,1012,500 b6Dual ; none 240192500 5775 b6c, b6cDual
74 b8d, b8dDual   279 1000 12,22,20,255,5010 1,56,1012,500 b8Dual ; none 240465500 5975 b8d, b8dDual
75 s3b, s3bDual   279 1000 12,22,20,255,5010 1,56,1012,500 s3Dual ; s2bDual 237683500 51200 s3b, s3bDual
76 b3b, b3bDual   279 500 16,20,2525 1,530,500 b3Dual ; none 239136000 5925 b3b, b3bDual
77 b3c, b3cDual   279 500 16,20,2525 1,530,500 b3Dual ; none 239433500 5900 b3c, b3cDual
78 b3d, b3dDual   279 500 16,20,2525 1,530,500 b3Dual ; none 239758500 51025 b3d, b3dDual
79 b3e, b3eDual   279 500 16,20,2525 1,530,500 b3Dual ; none 239707000 5975 b3e, b3eDual
80 a7c, a7cDual   279 500 16,20,2525 1,530,500 a7Dual ; none 238175500 51000 a7c, a7cDual
81 b1a, b1aDual   279 500 16,20,2525 1,530,500 b1Dual ; none 240329500 5875 b1a, b1aDual
82 b1b, b1bDual   279 500 16,20,2525 1,530,500 b1Dual ; none 240605500 5850 b1b, b1bDual
83 b1c, b1cDual   279 500 16,20,2525 1,530,500 b1Dual ; none 240532500 5900 b1c, b1cDual
84 b2a, b2aDual   279 500 16,20,2525 1,530,500 b2Dual ; none 240417000 5875 b2a, b2aDual
85 b2b, b2bDual   279 500 16,20,2525 1,530,500 b2Dual ; none 240542500 5925 b2b, b2bDual
86 b2c, b2cDual   279 500 16,20,2525 1,530,500 b2Dual ; none 240415000 5825 b2c, b2cDual
87 b6d, b6dDual   279 500 16,20,2525 1,530,500 b6Dual ; none 240587000 5850 b6d, b6dDual
88 b6e, b6eDual   279 500 16,20,2525 1,530,500 b6Dual ; none 240354500 5825 b6e, b6eDual
89 b8e, b8eDual   279 500 16,20,2525 1,530,500 b8Dual ; none 240196000 5950 b8e, b8eDual
90 b8f, b8fDual   279 500 16,20,2525 1,530,500 b8Dual ; none 240494000 5925 b8f, b8fDual
91 b8g, b8gDual   279 500 16,20,2525 1,530,500 b8Dual ; none 240138000 5875 b8g, b8gDual
92 s5b, s5bDual   280 3000 32,20,25,754,1502 1,156,302,500 s5Dual ; none 240216500 51125 s5b, s5bDual
93 s5c, s5cDual   280 2000 12,22,20,25,504,1004 1,52,106,204,500 s5Dual ; s4bDual 240359000 51100 s5c, s5cDual
94 a5e, a5eDual   280 500 16,20,2525 1,530,500 a5Dual ; none 239469500 51000 a5e, a5eDual
95 a5f, a5fDual   280 500 16,20,2525 1,530,500 a5Dual ; none 239553000 5875 a5f, a5fDual
96 a5g, a5gDual   280 500 16,20,2525 1,530,500 a5Dual ; a1bDual 239942000 51000 a5g, a5gDual
97 b7d, b7dDual   280 500 16,20,2525 1,530,500 b7Dual ; none 240349500 5975 b7d, b7dDual
98 w2, w2Dual Wyoming 286 3200 2,42,16,25,200,400 1,10,20,40,80,100,400 w1 ; none 238559200 5420 w2, w2Dual
99 h2b, h2bDual   300 9600 4,6,16,25,600 1,30,100,120,400 h2a ; none 239224800 5420 h2b, h2bDual

## Key to the table

Only one line is displayed for both a plane and its dual, an asterisk (*) in the first column indicating that the plane is self-dual. Each line includes the following information and isomorphism invariants for each plane.

• Plane    provides a text file containing the projective plane. The text file has 651 rows, corresponding to the points listed in the order 0,1,2,...,650; each row lists lines incident with the given point, with lines labeled 0,1,2,...,650. For non-self-dual planes, a second file is also given, containing the dual of the first.
• 5-rank    The rank of the (0,1)-incidence matrix of the projective plane, over a field of characteristic 5.
• |Autgp|    The order of the full automorphism group (collineation group) of the projective plane. This table entry is linked to a file providing generators of the automorphism group, listed as permutations of the integer list 0,1,2,...,1301 where 0,1,2,...,650 are labels for the points and 651,...,1301 are labels for the lines. (I've added 651 to each of the previously used line labels 0,1,...,650, to distinguish lines from points.) Generators for the automorphism group of the dual plane are not provided since these are trivially obtained from the generators given for the original plane.
• Point Orbits    The lengths of the full automorphism group on the points. (These are the lengths of the line orbits for the dual plane.)
• Line Orbits    The lengths of the full automorphism group on the lines. (These are the lengths of the point orbits for the dual plane.)
• Derived Planes    A list of all planes obtained by derivation from the planes listed in this line. When a line in the table lists a dual pair of planes, I list all the planes derivable from the first plane in the pair, then a colon, followed by all the planes derivable from the second plane in the pair.
• Subplanes    The orders of the proper subplanes, and the total number of each (work in progress).
• Fingerprint    The fingerprint of each plane is provided through a link to an accompanying table of fingerprints. See this accompanying document for the definition of this useful isomorphism invariant.

## Connections between the planes

The image below illustrates connections between the planes we have listed. Each circle represents a dual pair of planes (or a self-dual plane), and most of these have not been labeled. Yellow bonds between planes represent the derivability of one plane from another (a symmetric relation). Blue bonds indicate pairs of planes which share a semibiplane.

## Constructions

I've included the exhaustive list of all 21 translations planes of order 25 due to T. Czerwinski and D. Oakden, ‘The translation planes of order twenty-five’ J. Comb. Theory A 59 (1992), 193-217. Following their notation, I denote these planes a1-a8, b1-b8, s1-s5. Also included are the ordinary Hughes plane h1 and the exceptional Hughes plane h2, constructed using the regular and irregular nearfields of order 25 respectively; see H. Luneburg, ‘Characterizations of the generalized Hughes planes’ Canad. J. Math. 28 (1976), 376-402. With the exception of the Wyoming planes w1 and w2, all remaining planes listed are constructed by repeatedly dualising and deriving the translation planes and Hughes planes. This list is closed under the processes of dualisation and derivation.

The Wyoming planes w1 and w2 are related to each other by derivation, but not to any other planes. They are produced by lifting a homology semibiplane obtained from a2, and a Baer semibiplane obtained from a2a, respectively. Our list is also closed under the process of producing and lifting semibiplanes. Another pair of planes related by the sharing of a homology semibiplane is: s2 and the dual of h1a. Further pairs of planes related by the sharing of Baer semibiplanes are: s1 and h1 (the Desarguesian and ordinary Hughes planes); and s2a and h1b. For more information on semibiplanes and their use in constructing new planes from old, see e.g. my preprint Planes, semibiplanes and related complexes.

## Subplanes

In April-May, 2010 I exhaustively enumerated all subplanes of the planes listed here. Notes:

• All planes of order 25 in this list have subplanes of order 5. Roughly speaking, the lower the 5-rank, the more subplanes of order 5.
• All, except for the Desarguesian plane s1, have large numbers of subplanes of order 2. (It has been conjectured that all finite projective planes, other than Desarguesian planes of odd order, have subplanes of order 2.)
• Subplanes of order 3 were found in the Hughes plane h1 and its relatives h1a and h1b, and in the first Wyoming plane w1 (and in their duals) but in no other planes. It is known that Hughes planes of order q2 contain subplanes of order 3 whenever q is 5 mod 6.
• None of the known planes of order 25 (as listed above) have any subplanes of order 4. In other words, all subplanes of order 2 are maximal.
• The planes b1 and b2 have the same number of subplanes of each order; likewise the planes a6b and b3a. Among the known planes of order 25, these are the only instances of pairs of distinct planes (nonisomorphic and not dual) having the same number of subplanes of each order.

/ revised April, 2010