Projective Planes of Order 49 Related to t34

I am currently compiling a list of known projective planes of order 49. As part of this enumeration, here are listed the plane t34 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 t34

Entry	Plane	\|Autgp\|	Point Orbits	Line Orbits	7-rank
1	Translation Plane t34, dual dt34	86436	1²,3¹²,6²,2401	1,49²,147¹²,294²	941
2	t34_0_0, dt34_0_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
3	t34_0_1, dt34_0_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
4	t34_0_2, dt34_0_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
5	t34_0_3, dt34_0_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
6	t34_0_4, dt34_0_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
7	t34_0_5, dt34_0_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
8	t34_0_6, dt34_0_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
9	t34_0_7, dt34_0_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
10	t34_1_0, dt34_1_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
11	t34_1_1, dt34_1_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
12	t34_1_2, dt34_1_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
13	t34_1_3, dt34_1_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
14	t34_1_4, dt34_1_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
15	t34_1_5, dt34_1_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
16	t34_1_6, dt34_1_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
17	t34_1_7, dt34_1_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
18	t34_2_0, dt34_2_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
19	t34_2_1, dt34_2_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
20	t34_2_2, dt34_2_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
21	t34_2_3, dt34_2_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
22	t34_3_0, dt34_3_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
23	t34_3_1, dt34_3_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
24	t34_3_2, dt34_3_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
25	t34_3_3, dt34_3_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
26	t34_4_0, dt34_4_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
27	t34_4_1, dt34_4_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
28	t34_4_2, dt34_4_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
29	t34_4_3, dt34_4_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
30	t34_5_0, dt34_5_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
31	t34_5_1, dt34_5_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
32	t34_5_2, dt34_5_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
33	t34_5_3, dt34_5_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
34	t34_6_0, dt34_6_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
35	t34_6_1, dt34_6_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
36	t34_6_2, dt34_6_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
37	t34_6_3, dt34_6_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
38	t34_7_0, dt34_7_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
39	t34_7_1, dt34_7_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
40	t34_7_2, dt34_7_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
41	t34_7_3, dt34_7_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
42	t34_8_0, dt34_8_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
43	t34_8_1, dt34_8_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
44	t34_8_2, dt34_8_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
45	t34_8_3, dt34_8_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
46	t34_9_0, dt34_9_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
47	t34_9_1, dt34_9_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
48	t34_9_2, dt34_9_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
49	t34_9_3, dt34_9_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
50	t34_10_0, dt34_10_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
51	t34_10_1, dt34_10_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
52	t34_10_2, dt34_10_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
53	t34_10_3, dt34_10_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
54	t34_11_0, dt34_11_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
55	t34_11_1, dt34_11_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
56	t34_11_2, dt34_11_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
57	t34_11_3, dt34_11_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
58	t34_12_0, dt34_12_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
59	t34_12_1, dt34_12_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
60	t34_12_2, dt34_12_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
61	t34_12_3, dt34_12_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
62	t34_13_0, dt34_13_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
63	t34_13_1, dt34_13_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
64	t34_13_2, dt34_13_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
65	t34_13_3, dt34_13_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
66	t34_14_0, dt34_14_0	6174	1,7²,21¹⁸,2058	1²,3²,42,49,147¹⁶	987
67	t34_14_1, dt34_14_1	6174	1,7²,21¹⁸,2058	1²,3²,42,49,147¹⁶	987
68	t34_14_2, dt34_14_2	6174	1,7²,21¹⁸,2058	1²,3²,42,49,147¹⁶	987
69	t34_14_3, dt34_14_3	6174	1,7²,21¹⁸,2058	1²,3²,42,49,147¹⁶	987
70	t34_15_0, dt34_15_0	6174	1,7⁸,21¹⁶,2058	1⁸,42,49,147¹⁶	987
71	t34_15_1, dt34_15_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