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

• On projective planes of order less than 32, in Finite Geometries, Groups, and Computation, ed. A. Hulpke et. al.; de Gruyter, Berlin, 2006, pp.149-162.   (172 KB)

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

• a key to the table;
• a diagram illustrating connections between the planes listed;
• a brief description of the constructions used;
• a note concerning subplanes;
• a summary of what's known for other small orders.

Table of Known Projective Planes of Order 25

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.
• 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