Projective Planes of Order 49 Related to t55

I am currently compiling a list of known projective planes of order 49. As part of this enumeration, here are listed the plane t55 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 t55

Entry	Plane	\|Autgp\|	Point Orbits	Line Orbits	7-rank
1	Translation Plane t55, dual dt55	172872	1²,6⁶,12,2401	1,49²,294⁶,588	937
2	t55_0_0, dt55_0_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
3	t55_0_1, dt55_0_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
4	t55_0_2, dt55_0_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
5	t55_0_3, dt55_0_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
6	t55_0_4, dt55_0_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
7	t55_0_5, dt55_0_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
8	t55_0_6, dt55_0_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
9	t55_0_7, dt55_0_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
10	t55_1_0, dt55_1_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
11	t55_1_1, dt55_1_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
12	t55_1_2, dt55_1_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
13	t55_1_3, dt55_1_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
14	t55_2_0, dt55_2_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
15	t55_2_1, dt55_2_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
16	t55_2_2, dt55_2_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
17	t55_2_3, dt55_2_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
18	t55_3_0, dt55_3_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
19	t55_3_1, dt55_3_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
20	t55_3_2, dt55_3_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
21	t55_3_3, dt55_3_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
22	t55_4_0, dt55_4_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
23	t55_4_1, dt55_4_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
24	t55_4_2, dt55_4_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
25	t55_4_3, dt55_4_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
26	t55_5_0, dt55_5_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
27	t55_5_1, dt55_5_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
28	t55_5_2, dt55_5_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
29	t55_5_3, dt55_5_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
30	t55_6_0, dt55_6_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
31	t55_6_1, dt55_6_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
32	t55_6_2, dt55_6_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
33	t55_6_3, dt55_6_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
34	t55_7_0, dt55_7_0	12348	1,7²,21²,42⁸,2058	1²,3²,42,49,294⁸	987
35	t55_7_1, dt55_7_1	12348	1,7²,21²,42⁸,2058	1²,3²,42,49,294⁸	985
36	t55_7_2, dt55_7_2	12348	1,7²,21²,42⁸,2058	1²,3²,42,49,294⁸	985
37	t55_7_3, dt55_7_3	12348	1,7²,21²,42⁸,2058	1²,3²,42,49,294⁸	987
38	t55_8_0, dt55_8_0	4116	1,7⁸,14²⁴,2058	1⁸,42,49,98²⁴	987
39	t55_8_1, dt55_8_1	12348	1,7⁸,42⁸,2058	1⁸,42,49,294⁸	981

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