Projective Planes of Order 49 Related to t51

I am currently compiling a list of known projective planes of order 49. As part of this enumeration, here are listed the plane t51 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 t51

Entry	Plane	\|Autgp\|	Point Orbits	Line Orbits	7-rank
1	Translation Plane t51, dual dt51	115248	1⁴,2⁷,4²,8³,2401	1,49⁴,98⁷,196²,392³	941
2	t51_0_0, dt51_0_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
3	t51_0_1, dt51_0_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
4	t51_0_2, dt51_0_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
5	t51_0_3, dt51_0_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
6	t51_0_4, dt51_0_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
7	t51_0_5, dt51_0_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
8	t51_0_6, dt51_0_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
9	t51_0_7, dt51_0_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
10	t51_1_0, dt51_1_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
11	t51_1_1, dt51_1_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
12	t51_1_2, dt51_1_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
13	t51_1_3, dt51_1_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
14	t51_1_4, dt51_1_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
15	t51_1_5, dt51_1_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
16	t51_1_6, dt51_1_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
17	t51_1_7, dt51_1_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
18	t51_2_0, dt51_2_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
19	t51_2_1, dt51_2_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
20	t51_2_2, dt51_2_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
21	t51_2_3, dt51_2_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
22	t51_2_4, dt51_2_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
23	t51_2_5, dt51_2_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
24	t51_2_6, dt51_2_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
25	t51_2_7, dt51_2_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
26	t51_3_0, dt51_3_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
27	t51_3_1, dt51_3_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
28	t51_3_2, dt51_3_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
29	t51_3_3, dt51_3_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
30	t51_4_0, dt51_4_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
31	t51_4_1, dt51_4_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
32	t51_4_2, dt51_4_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
33	t51_4_3, dt51_4_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
34	t51_5_0, dt51_5_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
35	t51_5_1, dt51_5_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
36	t51_6_0, dt51_6_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
37	t51_6_1, dt51_6_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
38	t51_7_0, dt51_7_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
39	t51_7_1, dt51_7_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
40	t51_8_0, dt51_8_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
41	t51_8_1, dt51_8_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
42	t51_9_0, dt51_9_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
43	t51_9_1, dt51_9_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
44	t51_10_0, dt51_10_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
45	t51_10_1, dt51_10_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
46	t51_11_0, dt51_11_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
47	t51_11_1, dt51_11_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
48	t51_12_0, dt51_12_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
49	t51_13_0, dt51_13_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
50	t51_14_0, dt51_14_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
51	t51_15_0, dt51_15_0	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