Projective Planes of Order 49

This site, currently under construction, is intended to provide a current list of known projective planes of order 49, similar to my lists for other small orders. This work starts with the classification of the 1347 translation planes of order 49, due to Mathon and Royle, and independently verified by Dempwolff and Charnes; also the two Hughes planes of order 49. See

• R. Mathon, G. Royle, ‘The translation planes of order 49’, Designs, Codes and Cryptography 5 (1995), 52–72. Royle's website
• U. Dempwolff, C. Charnes, ‘The translation planes of order 49 and their automorphism groups’, Math. of Computation 67 (1998), 1207–1224. Dempwolff's website
• P. Dembowski, Finite Geometries, Springer-Verlag, Berlin, 1968.
• D.R. Hughes and F.C. Piper, Projective Planes, Springer-Verlag, New York, 1973.

From these 1349 planes and their duals, hundreds of thousands of others sare obtained by repeatedly deriving and dualizing; also by the process of lifting quotients. This work, currently in progress, is inspired by the similar project which I completed years ago for known projective planes of order 25. Currently I have found over 309,000 nonisomorphic planes of order 49; but this number is growing by several planes every hour and is expected to reach at least half a million planes before completion sometime within the next year (2011). Check back later for updates!

Who needs hundreds of thousands of planes of order 49?
A very good question. My motivation in this work is to provide a large database of examples for testing of various conjectures on projective planes. For example

• Does every finite projective plane of square order have a Baer subplane? and an embedded unital? Does every finite non-Desarguesian projective plane have a projective plane of order 2? I have found no counterexamples among the planes I have generated so far.
• What other orders of subplanes can occur? Based on random searches, it seems that none of the translation planes of order 49 have subplanes of any order other than 2 or 7. However a few (in fact, very few) planes of order 49 do contain subplanes of order 3, as my searches have found. More details soon...
• How often does the method of lifting quotients actually yield new planes from old? Here I have a rather large supply of planes to consider.
• Does every finite non-Desarguesian plane contain an oval? Counterexamples are known among the planes of order 16. I am not aware of any counterexamples to date among planes of odd order; but perhaps with this list of planes, a coutnerexample may be found.
• How small can the full conllineation group be? These planes may not be typical of planes of order 49 (many of which may never be discovered). But already it seems that most of the planes of order 49 found have full collineation group of order 2058=6*7³, just as most of the known planes of order 25 have full collineation group of order 500=4*5³. More details will be appearing soon...
• What are the possible values for the 7-rank of a projective plane of order 49? The 7-rank ranges from 855 to 941 for non-Desarguesian translation planes, and 785 for the Desarguesian plane. For other planes in my list, the 7-rank ranges from *** to ***. It is interesting to try to improve the known bounds for p-rank for planes of order divisible by p.

The Known Projective Planes of Order 49

We list the 1347 translation planes (including the Desarguesian plane) and the two Hughes planes, with links to the related planes obtained by derivation and lifting of quotients. Following the table is a key to the table.

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 for each plane.

• Plane    provides a pzip file containing the projective plane. I assume you have installed pzip under linux, which compresses each plane to about 6 KB. Right-click to save as plane.pz, then type the command punzip plane.pz to recover plane as a text file. This text file has 2451 rows, each row specifying a line of the plane as a subset of the points 0,1,2,...,2450. For non-self-dual planes, a second file is also given, containing the dual of the first.
• |Autgp|    The order of the full collineation group of the projective plane. This table entry is linked to a gzip file providing generators of the automorphism group. Right-click to save as plane.gz, then type the command gunzip plane.pz to recover plane as a text file. This text file lists generators of the full collineation group as permutations of the integers 0,1,2,...,4901 where 0,1,2,...,2450 are labels for the points and 2451,...,4901 are labels for the lines. 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.)
• Related Planes    The number of related planes, including the original plane and those obtained from it by dual derivation, with a link to a list of all such planes. This will be expanded to include planes obtained by repeated dualization and derivation; and by the method of lifting quotients.
• 7-rank    The rank of the (0,1)-incidence matrix of the projective plane, over a field of characteristic 7.

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/ revised November, 2010