Projective Planes of Order 49 Related to t82

I am currently compiling a list of known projective planes of order 49. As part of this enumeration, here are listed the plane t82 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 t82

Entry	Plane	\|Autgp\|	Point Orbits	Line Orbits	7-rank
1	Translation Plane t82, dual dt82	57624	2⁹,4⁸,2401	1,98⁹,196⁸	941
2	t82_0_0, dt82_0_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
3	t82_0_1, dt82_0_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
4	t82_0_2, dt82_0_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
5	t82_0_3, dt82_0_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
6	t82_0_4, dt82_0_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
7	t82_0_5, dt82_0_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
8	t82_0_6, dt82_0_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
9	t82_0_7, dt82_0_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
10	t82_1_0, dt82_1_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
11	t82_1_1, dt82_1_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
12	t82_1_2, dt82_1_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
13	t82_1_3, dt82_1_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
14	t82_1_4, dt82_1_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
15	t82_1_5, dt82_1_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
16	t82_1_6, dt82_1_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
17	t82_1_7, dt82_1_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
18	t82_2_0, dt82_2_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
19	t82_2_1, dt82_2_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
20	t82_2_2, dt82_2_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
21	t82_2_3, dt82_2_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
22	t82_2_4, dt82_2_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
23	t82_2_5, dt82_2_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
24	t82_2_6, dt82_2_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
25	t82_2_7, dt82_2_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
26	t82_3_0, dt82_3_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
27	t82_3_1, dt82_3_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
28	t82_3_2, dt82_3_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
29	t82_3_3, dt82_3_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
30	t82_3_4, dt82_3_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
31	t82_3_5, dt82_3_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
32	t82_3_6, dt82_3_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
33	t82_3_7, dt82_3_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
34	t82_4_0, dt82_4_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
35	t82_4_1, dt82_4_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
36	t82_4_2, dt82_4_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
37	t82_4_3, dt82_4_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
38	t82_4_4, dt82_4_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
39	t82_4_5, dt82_4_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
40	t82_4_6, dt82_4_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
41	t82_4_7, dt82_4_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
42	t82_5_0, dt82_5_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
43	t82_5_1, dt82_5_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
44	t82_5_2, dt82_5_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
45	t82_5_3, dt82_5_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
46	t82_5_4, dt82_5_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
47	t82_5_5, dt82_5_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
48	t82_5_6, dt82_5_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
49	t82_5_7, dt82_5_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
50	t82_6_0, dt82_6_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
51	t82_6_1, dt82_6_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
52	t82_6_2, dt82_6_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
53	t82_6_3, dt82_6_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
54	t82_6_4, dt82_6_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
55	t82_6_5, dt82_6_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
56	t82_6_6, dt82_6_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
57	t82_6_7, dt82_6_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
58	t82_7_0, dt82_7_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
59	t82_7_1, dt82_7_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
60	t82_7_2, dt82_7_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
61	t82_7_3, dt82_7_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
62	t82_7_4, dt82_7_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
63	t82_7_5, dt82_7_5	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
64	t82_7_6, dt82_7_6	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
65	t82_7_7, dt82_7_7	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
66	t82_8_0, dt82_8_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
67	t82_8_1, dt82_8_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
68	t82_8_2, dt82_8_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
69	t82_8_3, dt82_8_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
70	t82_9_0, dt82_9_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
71	t82_9_1, dt82_9_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
72	t82_9_2, dt82_9_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
73	t82_9_3, dt82_9_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
74	t82_10_0, dt82_10_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
75	t82_10_1, dt82_10_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
76	t82_10_2, dt82_10_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
77	t82_10_3, dt82_10_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
78	t82_11_0, dt82_11_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
79	t82_11_1, dt82_11_1	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
80	t82_11_2, dt82_11_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
81	t82_11_3, dt82_11_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
82	t82_11_4, dt82_11_4	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
83	t82_12_0, dt82_12_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
84	t82_12_1, dt82_12_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
85	t82_12_2, dt82_12_2	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
86	t82_12_3, dt82_12_3	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
87	t82_12_4, dt82_12_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
88	t82_13_0, dt82_13_0	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
89	t82_13_1, dt82_13_1	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	985
90	t82_13_2, dt82_13_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
91	t82_13_3, dt82_13_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
92	t82_13_4, dt82_13_4	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
93	t82_14_0, dt82_14_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
94	t82_14_1, dt82_14_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
95	t82_14_2, dt82_14_2	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
96	t82_14_3, dt82_14_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
97	t82_14_4, dt82_14_4	4116	1,7⁸,14²⁴,2058	1²,2³,42,49⁷,98²¹	987
98	t82_15_0, dt82_15_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
99	t82_15_1, dt82_15_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
100	t82_15_2, dt82_15_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
101	t82_15_3, dt82_15_3	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
102	t82_16_0, dt82_16_0	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
103	t82_16_1, dt82_16_1	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
104	t82_16_2, dt82_16_2	2058	1,7⁵⁶,2058	1⁸,42,49⁴⁹	987
105	t82_16_3, dt82_16_3	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