Projective Planes of Order 49 Related to t81

I am currently compiling a list of known projective planes of order 49. As part of this enumeration, here are listed the plane t81 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 t81

Entry	Plane	\|Autgp\|	Point Orbits	Line Orbits	7-rank
1	Translation Plane t81, dual dt81	57624	1²,2⁸,4⁸,2401	1,49²,98⁸,196⁸	941
2	t81_0_0, dt81_0_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
3	t81_0_1, dt81_0_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
4	t81_0_2, dt81_0_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
5	t81_0_3, dt81_0_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
6	t81_0_4, dt81_0_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
7	t81_0_5, dt81_0_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
8	t81_0_6, dt81_0_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
9	t81_0_7, dt81_0_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
10	t81_1_0, dt81_1_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
11	t81_1_1, dt81_1_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
12	t81_1_2, dt81_1_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
13	t81_1_3, dt81_1_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
14	t81_1_4, dt81_1_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
15	t81_1_5, dt81_1_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
16	t81_1_6, dt81_1_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
17	t81_1_7, dt81_1_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
18	t81_2_0, dt81_2_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
19	t81_2_1, dt81_2_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
20	t81_2_2, dt81_2_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
21	t81_2_3, dt81_2_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
22	t81_2_4, dt81_2_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
23	t81_2_5, dt81_2_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
24	t81_2_6, dt81_2_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
25	t81_2_7, dt81_2_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
26	t81_3_0, dt81_3_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
27	t81_3_1, dt81_3_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
28	t81_3_2, dt81_3_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
29	t81_3_3, dt81_3_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
30	t81_3_4, dt81_3_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
31	t81_3_5, dt81_3_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
32	t81_3_6, dt81_3_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
33	t81_3_7, dt81_3_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
34	t81_4_0, dt81_4_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
35	t81_4_1, dt81_4_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
36	t81_4_2, dt81_4_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
37	t81_4_3, dt81_4_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
38	t81_4_4, dt81_4_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
39	t81_4_5, dt81_4_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	985
40	t81_4_6, dt81_4_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
41	t81_4_7, dt81_4_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
42	t81_5_0, dt81_5_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
43	t81_5_1, dt81_5_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
44	t81_5_2, dt81_5_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	985
45	t81_5_3, dt81_5_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
46	t81_5_4, dt81_5_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
47	t81_5_5, dt81_5_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
48	t81_5_6, dt81_5_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
49	t81_5_7, dt81_5_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
50	t81_6_0, dt81_6_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
51	t81_6_1, dt81_6_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
52	t81_6_2, dt81_6_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
53	t81_6_3, dt81_6_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
54	t81_6_4, dt81_6_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
55	t81_6_5, dt81_6_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
56	t81_6_6, dt81_6_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
57	t81_6_7, dt81_6_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
58	t81_7_0, dt81_7_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
59	t81_7_1, dt81_7_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
60	t81_7_2, dt81_7_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
61	t81_7_3, dt81_7_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
62	t81_7_4, dt81_7_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
63	t81_7_5, dt81_7_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
64	t81_7_6, dt81_7_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
65	t81_7_7, dt81_7_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
66	t81_8_0, dt81_8_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
67	t81_8_1, dt81_8_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
68	t81_8_2, dt81_8_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
69	t81_8_3, dt81_8_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
70	t81_9_0, dt81_9_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
71	t81_9_1, dt81_9_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
72	t81_9_2, dt81_9_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
73	t81_9_3, dt81_9_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
74	t81_10_0, dt81_10_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
75	t81_10_1, dt81_10_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
76	t81_10_2, dt81_10_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
77	t81_10_3, dt81_10_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
78	t81_11_0, dt81_11_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
79	t81_11_1, dt81_11_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
80	t81_11_2, dt81_11_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
81	t81_11_3, dt81_11_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
82	t81_12_0, dt81_12_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
83	t81_12_1, dt81_12_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
84	t81_12_2, dt81_12_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
85	t81_12_3, dt81_12_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
86	t81_13_0, dt81_13_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
87	t81_13_1, dt81_13_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
88	t81_13_2, dt81_13_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
89	t81_13_3, dt81_13_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
90	t81_14_0, dt81_14_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
91	t81_14_1, dt81_14_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
92	t81_14_2, dt81_14_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
93	t81_14_3, dt81_14_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
94	t81_15_0, dt81_15_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
95	t81_15_1, dt81_15_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
96	t81_15_2, dt81_15_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
97	t81_15_3, dt81_15_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
98	t81_16_0, dt81_16_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
99	t81_16_1, dt81_16_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
100	t81_17_0, dt81_17_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
101	t81_17_1, dt81_17_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