Projective Planes of Order 49 Related to t42

I am currently compiling a list of known projective planes of order 49. As part of this enumeration, here are listed the plane t42 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 t42

Entry	Plane	\|Autgp\|	Point Orbits	Line Orbits	7-rank
1	Translation Plane t42, dual dt42	230496	2³,4,8⁵,2401	1,98³,196,392⁵	939
2	t42_0_0, dt42_0_0	2058	1
3	t42_0_1, dt42_0_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
4	t42_0_2, dt42_0_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
5	t42_0_3, dt42_0_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
6	t42_1_0, dt42_1_0	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
7	t42_1_1, dt42_1_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
8	t42_1_2, dt42_1_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
9	t42_1_3, dt42_1_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
10	t42_1_4, dt42_1_4	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
11	t42_2_0, dt42_2_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
12	t42_2_1, dt42_2_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
13	t42_2_2, dt42_2_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
14	t42_2_3, dt42_2_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
15	t42_3_0, dt42_3_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
16	t42_3_1, dt42_3_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
17	t42_3_2, dt42_3_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
18	t42_3_3, dt42_3_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
19	t42_4_0, dt42_4_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
20	t42_4_1, dt42_4_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
21	t42_4_2, dt42_4_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
22	t42_4_3, dt42_4_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
23	t42_5_0, dt42_5_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
24	t42_5_1, dt42_5_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
25	t42_6_0, dt42_6_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
26	t42_7_0, dt42_7_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
27	t42_8_0, dt42_8_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	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 February, 2011