Projective Planes of Order 49 Related to t72

I am currently compiling a list of known projective planes of order 49. As part of this enumeration, here are listed the plane t72 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 t72

Entry	Plane	\|Autgp\|	Point Orbits	Line Orbits	7-rank
1	Translation Plane t72, dual dt72	518616	1²,6²,9²,18,2401	1,49²,294²,441²,882	941
2	t72_0_0, dt72_0_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
3	t72_0_1, dt72_0_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
4	t72_0_2, dt72_0_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
5	t72_0_3, dt72_0_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
6	t72_1_0, dt72_1_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
7	t72_1_1, dt72_1_1	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
8	t72_1_2, dt72_1_2	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
9	t72_2_0, dt72_2_0	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	985
10	t72_2_1, dt72_2_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
11	t72_2_2, dt72_2_2	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
12	t72_3_0, dt72_3_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
13	t72_3_1, dt72_3_1	6174	1,7⁸,21¹⁶,2058	1²,3²,42,49⁷,147¹⁴	987
14	t72_4_0, dt72_4_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
15	t72_4_1, dt72_4_1	6174	1,7⁸,21¹⁶,2058	1²,3²,42,49⁷,147¹⁴	987
16	t72_5_0, dt72_5_0	12348	1,7²,21²,42⁸,2058	1²,6,42,49,147²,294⁷	985
17	t72_5_1, dt72_5_1	37044	1,14,21²,42²,63²,126,2058	2,3²,42,49,294²,441²,882	983
18	t72_6_0, dt72_6_0	12348	1,7²,21²,42⁸,2058	1²,6,42,49,147²,294⁷	987
19	t72_6_1, dt72_6_1	37044	1,14,21²,42²,63²,126,2058	2,3²,42,49,294²,441²,882	983

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 February, 2011