Projective Planes of Order 49 Related to t59

I am currently compiling a list of known projective planes of order 49. As part of this enumeration, here are listed the plane t59 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 t59

Entry	Plane	\|Autgp\|	Point Orbits	Line Orbits	7-rank
1	Translation Plane t59, dual dt59	115248	2⁵,4⁴,8³,2401	1,98⁵,196⁴,392³	941
2	t59_0_0, dt59_0_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
3	t59_0_1, dt59_0_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
4	t59_0_2, dt59_0_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
5	t59_0_3, dt59_0_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
6	t59_0_4, dt59_0_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
7	t59_0_5, dt59_0_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
8	t59_0_6, dt59_0_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
9	t59_0_7, dt59_0_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
10	t59_1_0, dt59_1_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
11	t59_1_1, dt59_1_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
12	t59_1_2, dt59_1_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
13	t59_1_3, dt59_1_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
14	t59_1_4, dt59_1_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
15	t59_1_5, dt59_1_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
16	t59_1_6, dt59_1_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
17	t59_1_7, dt59_1_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
18	t59_2_0, dt59_2_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
19	t59_2_1, dt59_2_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
20	t59_2_2, dt59_2_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
21	t59_2_3, dt59_2_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
22	t59_2_4, dt59_2_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
23	t59_2_5, dt59_2_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
24	t59_2_6, dt59_2_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
25	t59_2_7, dt59_2_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
26	t59_3_0, dt59_3_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
27	t59_3_1, dt59_3_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
28	t59_3_2, dt59_3_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
29	t59_3_3, dt59_3_3	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
30	t59_3_4, dt59_3_4	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
31	t59_4_0, dt59_4_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
32	t59_4_1, dt59_4_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
33	t59_4_2, dt59_4_2	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
34	t59_4_3, dt59_4_3	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
35	t59_4_4, dt59_4_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
36	t59_5_0, dt59_5_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
37	t59_5_1, dt59_5_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
38	t59_5_2, dt59_5_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
39	t59_5_3, dt59_5_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
40	t59_6_0, dt59_6_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
41	t59_6_1, dt59_6_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
42	t59_6_2, dt59_6_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
43	t59_6_3, dt59_6_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
44	t59_7_0, dt59_7_0	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
45	t59_7_1, dt59_7_1	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
46	t59_7_2, dt59_7_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
47	t59_8_0, dt59_8_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
48	t59_8_1, dt59_8_1	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
49	t59_8_2, dt59_8_2	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
50	t59_9_0, dt59_9_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
51	t59_9_1, dt59_9_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
52	t59_10_0, dt59_10_0	4116	1,7⁸,14²⁴,2058	1⁸,42,49,98²⁴	987
53	t59_10_1, dt59_10_1	4116	1,7⁸,14²⁴,2058	1⁸,42,49,98²⁴	987
54	t59_10_2, dt59_10_2	4116	1,7⁸,14²⁴,2058	1⁸,42,49,98²⁴	987
55	t59_10_3, dt59_10_3	4116	1,7⁸,14²⁴,2058	1⁸,42,49,98²⁴	985
56	t59_11_0, dt59_11_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
57	t59_11_1, dt59_11_1	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