Projective Planes of Order 49 Related to t37

I am currently compiling a list of known projective planes of order 49. As part of this enumeration, here are listed the plane t37 and all known planes of order 49 obtained from it by dualizing and deriving. Coming soon: also planes related by the method of lifting quotients. This list is currently incomplete; check back later for a complete enumeration.

Following the table is a key to the table.

Known Projective Planes of Order 49 Related to t37

Entry	Plane	\|Autgp\|	Point Orbits	Line Orbits	7-rank
1	Translation Plane t37, dual dt37	460992	2,4²,8,16²,2401	1,98,196²,392,784²	941
2	t37_0_0, dt37_0_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
3	t37_0_1, dt37_0_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
4	t37_0_2, dt37_0_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
5	t37_0_3, dt37_0_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
6	t37_1_0, dt37_1_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
7	t37_1_1, dt37_1_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
8	t37_1_2, dt37_1_2	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	985
9	t37_1_3, dt37_1_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
10	t37_1_4, dt37_1_4	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
11	t37_2_0, dt37_2_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
12	t37_2_1, dt37_2_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
13	t37_3_0, dt37_3_0	4116	1,7⁸,14²⁴,2058	1⁸,42,49,98²⁴	985
14	t37_3_1, dt37_3_1	4116	1,7⁸,14²⁴,2058	1⁸,42,49,98²⁴	987
15	t37_4_0, dt37_4_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
16	t37_5_0, dt37_5_0	8232	1,14⁴,28¹²,2058	2⁴,42,49,196¹²	987
17	t37_5_1, dt37_5_1	8232	1,14⁴,28¹²,2058	2⁴,42,49,196¹²	987

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 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.gz 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.)
7-rank The rank of the (0,1)-incidence matrix of the projective plane, over a field of characteristic 7.

/ revised June, 2010