Projective Planes of Order 49 Related to t26

I am currently compiling a list of known projective planes of order 49. As part of this enumeration, here are listed the plane t26 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 t26

Entry	Plane	\|Autgp\|	Point Orbits	Line Orbits	7-rank
1	Translation Plane t26, dual dt26	172872	2²,3⁴,4,6³,12,2401	1,98²,147⁴,196,294³,588	939
2	t26_0_0, dt26_0_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
3	t26_0_1, dt26_0_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
4	t26_0_2, dt26_0_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
5	t26_0_3, dt26_0_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
6	t26_0_4, dt26_0_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
7	t26_0_5, dt26_0_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
8	t26_0_6, dt26_0_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
9	t26_0_7, dt26_0_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
10	t26_1_0, dt26_1_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
11	t26_1_1, dt26_1_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
12	t26_1_2, dt26_1_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
13	t26_1_3, dt26_1_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
14	t26_2_0, dt26_2_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
15	t26_2_1, dt26_2_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
16	t26_2_2, dt26_2_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
17	t26_2_3, dt26_2_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
18	t26_3_0, dt26_3_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
19	t26_3_1, dt26_3_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
20	t26_3_2, dt26_3_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
21	t26_3_3, dt26_3_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
22	t26_4_0, dt26_4_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
23	t26_4_1, dt26_4_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
24	t26_4_2, dt26_4_2	6174	1,7⁸,21¹⁶,2058	1²,3²,42,49⁷,147¹⁴	987
25	t26_4_3, dt26_4_3	6174	1,7⁸,21¹⁶,2058	1²,3²,42,49⁷,147¹⁴	987
26	t26_5_0, dt26_5_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
27	t26_5_1, dt26_5_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
28	t26_6_0, dt26_6_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
29	t26_6_1, dt26_6_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
30	t26_7_0, dt26_7_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
31	t26_7_1, dt26_7_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
32	t26_8_0, dt26_8_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
33	t26_8_1, dt26_8_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
34	t26_9_0, dt26_9_0	6174	1,7⁸,21¹⁶,2058	1²,3²,42,49⁷,147¹⁴	987
35	t26_9_1, dt26_9_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
36	t26_10_0, dt26_10_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
37	t26_10_1, dt26_10_1	6174	1,7⁸,21¹⁶,2058	1²,3²,42,49⁷,147¹⁴	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