Projective Planes of Order 49 Related to t36

I am currently compiling a list of known projective planes of order 49. As part of this enumeration, here are listed the plane t36 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 t36

Entry	Plane	\|Autgp\|	Point Orbits	Line Orbits	7-rank
1	Translation Plane t36, dual dt36	115248	2³,4⁵,8³,2401	1,98³,196⁵,392³	941
2	t36_0_0, dt36_0_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
3	t36_0_1, dt36_0_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
4	t36_0_2, dt36_0_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
5	t36_0_3, dt36_0_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
6	t36_0_4, dt36_0_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
7	t36_0_5, dt36_0_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
8	t36_0_6, dt36_0_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
9	t36_0_7, dt36_0_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
10	t36_1_0, dt36_1_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
11	t36_1_1, dt36_1_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
12	t36_1_2, dt36_1_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
13	t36_1_3, dt36_1_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
14	t36_1_4, dt36_1_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
15	t36_1_5, dt36_1_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
16	t36_1_6, dt36_1_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
17	t36_1_7, dt36_1_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
18	t36_2_0, dt36_2_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
19	t36_2_1, dt36_2_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
20	t36_2_2, dt36_2_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
21	t36_2_3, dt36_2_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
22	t36_2_4, dt36_2_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
23	t36_2_5, dt36_2_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
24	t36_2_6, dt36_2_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
25	t36_2_7, dt36_2_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
26	t36_3_0, dt36_3_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
27	t36_3_1, dt36_3_1	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
28	t36_3_2, dt36_3_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
29	t36_3_3, dt36_3_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
30	t36_3_4, dt36_3_4	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
31	t36_4_0, dt36_4_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
32	t36_4_1, dt36_4_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
33	t36_4_2, dt36_4_2	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
34	t36_4_3, dt36_4_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
35	t36_4_4, dt36_4_4	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
36	t36_5_0, dt36_5_0	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
37	t36_5_1, dt36_5_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
38	t36_5_2, dt36_5_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
39	t36_5_3, dt36_5_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
40	t36_5_4, dt36_5_4	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
41	t36_6_0, dt36_6_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
42	t36_6_1, dt36_6_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
43	t36_6_2, dt36_6_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
44	t36_6_3, dt36_6_3	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
45	t36_6_4, dt36_6_4	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	985
46	t36_7_0, dt36_7_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
47	t36_7_1, dt36_7_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
48	t36_7_2, dt36_7_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
49	t36_7_3, dt36_7_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
50	t36_8_0, dt36_8_0	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
51	t36_8_1, dt36_8_1	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
52	t36_8_2, dt36_8_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
53	t36_9_0, dt36_9_0	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
54	t36_9_1, dt36_9_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
55	t36_9_2, dt36_9_2	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
56	t36_10_0, dt36_10_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
57	t36_10_1, dt36_10_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 June, 2010