Projective Planes of Order 49 Related to t112

I am currently compiling a list of known projective planes of order 49. As part of this enumeration, here are listed the plane t112 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 t112

Entry	Plane	\|Autgp\|	Point Orbits	Line Orbits	7-rank
1	Translation Plane t112, dual dt112	115248	2³,4⁵,8³,2401	1,98³,196⁵,392³	937
2	t112_0_0, dt112_0_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
3	t112_0_1, dt112_0_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
4	t112_0_2, dt112_0_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
5	t112_0_3, dt112_0_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
6	t112_0_4, dt112_0_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
7	t112_0_5, dt112_0_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
8	t112_0_6, dt112_0_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
9	t112_0_7, dt112_0_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
10	t112_1_0, dt112_1_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
11	t112_1_1, dt112_1_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
12	t112_1_2, dt112_1_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
13	t112_1_3, dt112_1_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
14	t112_1_4, dt112_1_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
15	t112_1_5, dt112_1_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
16	t112_1_6, dt112_1_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
17	t112_1_7, dt112_1_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
18	t112_2_0, dt112_2_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
19	t112_2_1, dt112_2_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
20	t112_2_2, dt112_2_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
21	t112_2_3, dt112_2_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
22	t112_2_4, dt112_2_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
23	t112_2_5, dt112_2_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
24	t112_2_6, dt112_2_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
25	t112_2_7, dt112_2_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
26	t112_3_0, dt112_3_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
27	t112_3_1, dt112_3_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
28	t112_3_2, dt112_3_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
29	t112_3_3, dt112_3_3	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	985
30	t112_3_4, dt112_3_4	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
31	t112_4_0, dt112_4_0	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
32	t112_4_1, dt112_4_1	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
33	t112_4_2, dt112_4_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
34	t112_4_3, dt112_4_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
35	t112_4_4, dt112_4_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
36	t112_5_0, dt112_5_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
37	t112_5_1, dt112_5_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
38	t112_5_2, dt112_5_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
39	t112_5_3, dt112_5_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
40	t112_6_0, dt112_6_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
41	t112_6_1, dt112_6_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
42	t112_6_2, dt112_6_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
43	t112_6_3, dt112_6_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
44	t112_7_0, dt112_7_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
45	t112_7_1, dt112_7_1	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	985
46	t112_7_2, dt112_7_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
47	t112_7_3, dt112_7_3	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
48	t112_7_4, dt112_7_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
49	t112_8_0, dt112_8_0	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
50	t112_8_1, dt112_8_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
51	t112_8_2, dt112_8_2	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
52	t112_9_0, dt112_9_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
53	t112_9_1, dt112_9_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
54	t112_10_0, dt112_10_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
55	t112_10_1, dt112_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 February, 2011