Projective Planes of Order 49 Related to t113

I am currently compiling a list of known projective planes of order 49. As part of this enumeration, here are listed the plane t113 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 t113

Entry	Plane	\|Autgp\|	Point Orbits	Line Orbits	7-rank
1	Translation Plane t113, dual dt113	172872	1²,2,4,6³,12²,2401	1,49²,98,196,294³,588²	941
2	t113_0_0, dt113_0_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
3	t113_0_1, dt113_0_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
4	t113_0_2, dt113_0_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
5	t113_0_3, dt113_0_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
6	t113_0_4, dt113_0_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
7	t113_0_5, dt113_0_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
8	t113_0_6, dt113_0_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
9	t113_0_7, dt113_0_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
10	t113_1_0, dt113_1_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
11	t113_1_1, dt113_1_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
12	t113_1_2, dt113_1_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
13	t113_1_3, dt113_1_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
14	t113_1_4, dt113_1_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
15	t113_1_5, dt113_1_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
16	t113_1_6, dt113_1_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
17	t113_1_7, dt113_1_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
18	t113_2_0, dt113_2_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
19	t113_2_1, dt113_2_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
20	t113_2_2, dt113_2_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
21	t113_2_3, dt113_2_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
22	t113_3_0, dt113_3_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
23	t113_3_1, dt113_3_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
24	t113_3_2, dt113_3_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
25	t113_3_3, dt113_3_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
26	t113_4_0, dt113_4_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
27	t113_4_1, dt113_4_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
28	t113_4_2, dt113_4_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
29	t113_4_3, dt113_4_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
30	t113_5_0, dt113_5_0	6174	1,7⁸,21¹⁶,2058	1²,3²,42,49⁷,147¹⁴	987
31	t113_5_1, dt113_5_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
32	t113_5_2, dt113_5_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
33	t113_5_3, dt113_5_3	6174	1,7⁸,21¹⁶,2058	1²,3²,42,49⁷,147¹⁴	987
34	t113_6_0, dt113_6_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
35	t113_6_1, dt113_6_1	6174	1,7⁸,21¹⁶,2058	1²,3²,42,49⁷,147¹⁴	985
36	t113_7_0, dt113_7_0	12348	1,7²,14³,21²,42⁷,2058	1²,3²,42,49,98³,294⁷	983
37	t113_7_1, dt113_7_1	4116	1,7⁸,14²⁴,2058	1⁸,42,49,98²⁴	987
38	t113_8_0, dt113_8_0	4116	1,7⁸,14²⁴,2058	1⁸,42,49,98²⁴	987
39	t113_8_1, dt113_8_1	12348	1,7²,14³,21²,42⁷,2058	1²,3²,42,49,98³,294⁷	985

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