Projective Planes of Order 49


Under Construction

This site, currently under construction, is intended to provide a current list of known projective planes of order 49, similar to my lists for other small orders. This work starts with the classification of the 1347 translation planes of order 49, due to Mathon and Royle, and independently verified by Dempwolff and Charnes; also the two Hughes planes of order 49. See

From these 1349 planes and their duals, hundreds of thousands of others sare obtained by repeatedly deriving and dualizing; also by the process of lifting quotients. This work, currently in progress, is inspired by the similar project which I completed years ago for known projective planes of order 25. Currently I have found over 309,000 nonisomorphic planes of order 49; but this number is growing by several planes every hour and is expected to reach at least half a million planes before completion sometime within the next year (2011). Check back later for updates!

Who needs hundreds of thousands of planes of order 49?
A very good question. My motivation in this work is to provide a large database of examples for testing of various conjectures on projective planes. For example


The Known Projective Planes of Order 49

We list the 1347 translation planes (including the Desarguesian plane) and the two Hughes planes, with links to the related planes obtained by derivation and lifting of quotients. Following the table is a key to the table.

No. Plane |Autgp| Point orbit lengths Line orbit lengths 7-rank Related Planes
1
Translation Plane t1, dual dt1
2304960
10,40,2401 1,490,1960
941
10
2
Translation Plane t2, dual dt2
77446656
8,42,2401 1,392,2058
855
5
3
Translation Plane t3, dual dt3
7375872
2,16,32,2401 1,98,784,1568
917
8
4
Translation Plane t4, dual dt4
5531904
2,16,32,2401 1,98,784,1568
915
10
5
Translation Plane t5, dual dt5
4148928
6,8,36,2401 1,294,392,1764
931
12
6
Translation Plane t6, dual dt6
5531904
2,16,32,2401 1,98,784,1568
927
10
7
Translation Plane t7, dual dt7
691488
2,8,122,16,2401 1,98,392,5882,784
941
26
8
Translation Plane t8, dual dt8
691488
12,2,6,82,122,2401 1,492,98,294,3922,5882
939
26
9
Translation Plane t9, dual dt9
460992
2,82,162,2401 1,98,3922,7842
941
34
10
Translation Plane t10, dual dt10
1037232
2,12,182,2401 1,98,588,8822
941
20
11
Translation Plane t11, dual dt11
2074464
2,12,36,2401 1,98,588,1764
927
16
12
Translation Plane t12, dual dt12
230496
23,4,8,162,2401 1,983,196,392,7842
941
52
13
Translation Plane t13, dual dt13
3687936
2,82,162,2401 1,98,3922,7842
899
12
14
Translation Plane t14, dual dt14
1440600
50,2401 1,2450
925
10
15
Translation Plane t15, dual dt15
460992
2,42,83,16,2401 1,98,1962,3923,784
941
30
16
Translation Plane t16, dual dt16
691488
63,8,24,2401 1,2943,392,1176
935
20
17
Translation Plane t17, dual dt17
460992
2,42,83,16,2401 1,98,1962,3923,784
941
34
18
Translation Plane t18, dual dt18
115248
12,22,45,83,2401 1,492,982,1965,3923
939
102
19
Translation Plane t19, dual dt19
57624
12,24,410,2401 1,492,984,19610
941
210
20
Translation Plane t20, dual dt20
691488
63,8,24,2401 1,2943,392,1176
935
20
21
Translation Plane t21, dual dt21
10372320
20,30,2401 1,980,1470
911
8
22
Translation Plane t22, dual dt22
1843968
2,16,32,2401 1,98,784,1568
907
12
23
Translation Plane t23, dual dt23
691488
12,22,62,8,122,2401 1,492,982,2942,392,5882
937
26
24
Translation Plane t24, dual dt24
3687936
2,163,2401 1,98,7843
917
10
25
Translation Plane t25, dual dt25
921984
2,82,32,2401 1,98,3922,1568
941
16
26
Translation Plane t26, dual dt26
172872
22,34,4,63,12,2401 1,982,1474,196,2943,588
939
74
27
Translation Plane t27, dual dt27
3457440
10,40,2401 1,490,1960
935
10
28
Translation Plane t28, dual dt28
2074464
2,12,36,2401 1,98,588,1764
931
16
29
Translation Plane t29, dual dt29
115248
25,48,8,2401 1,985,1968,392
941
106
30
Translation Plane t30, dual dt30
172872
24,34,63,12,2401 1,984,1474,2943,588
941
74
31
Translation Plane t31, dual dt31
2074464
32,8,36,2401 1,1472,392,1764
939
14
32
Translation Plane t32, dual dt32
115248
2,44,84,2401 1,98,1964,3924
941
102
33
Translation Plane t33, dual dt33
1382976
2,122,24,2401 1,98,5882,1176
935
14
34
Translation Plane t34, dual dt34
86436
12,312,62,2401 1,492,14712,2942
941
142
35
Translation Plane t35, dual dt35
115248
14,25,43,83,2401 1,494,985,1963,3923
939
102
36
Translation Plane t36, dual dt36
115248
23,45,83,2401 1,983,1965,3923
941
104
37
Translation Plane t37, dual dt37
460992
2,42,8,162,2401 1,98,1962,392,7842
941
34
38
Translation Plane t38, dual dt38
2765952
6,12,32,2401 1,294,588,1568
929
10
39
Translation Plane t39, dual dt39
230496
23,45,83,2401 1,983,1965,3923
941
62
40
Translation Plane t40, dual dt40
460992
2,42,8,162,2401 1,98,1962,392,7842
941
32
41
Translation Plane t41, dual dt41
28812
12,224,2401 1,492,9824
941
402
42
Translation Plane t42, dual dt42
230496
23,4,85,2401 1,983,196,3925
939
54
43
Translation Plane t43, dual dt43
4840416
8,42,2401 1,392,2058
921
8
44
Translation Plane t44, dual dt44
230496
2,42,83,16,2401 1,98,1962,3923,784
941
62
45
Translation Plane t45, dual dt45
230496
2,42,85,2401 1,98,1962,3925
937
66
46
Translation Plane t46, dual dt46
230496
2,46,83,2401 1,98,1966,3923
941
58
47
Translation Plane t47, dual dt47
230496
23,45,8,16,2401 1,983,1965,392,784
933
54
48
Translation Plane t48, dual dt48
57624
14,27,48,2401 1,494,987,1968
941
202
49
Translation Plane t49, dual dt49
172872
22,4,65,12,2401 1,982,196,2945,588
939
78
50
Translation Plane t50, dual dt50
1382976
6,12,162,2401 1,294,588,7842
941
14
51
Translation Plane t51, dual dt51
115248
14,27,42,83,2401 1,494,987,1962,3923
941
102
52
Translation Plane t52, dual dt52
691488
6,8,12,24,2401 1,294,392,588,1176
931
22
53
Translation Plane t53, dual dt53
1382976
6,8,12,24,2401 1,294,392,588,1176
935
14
54
Translation Plane t54, dual dt54
28812
14,223,2401 1,494,9823
941
402
55
Translation Plane t55, dual dt55
172872
12,66,12,2401 1,492,2946,588
937
78
56
Translation Plane t56, dual dt56
115248
12,24,42,84,2401 1,492,984,1962,3924
941
102
57
Translation Plane t57, dual dt57
57624
12,26,49,2401 1,492,986,1969
941
202
58
Translation Plane t58, dual dt58
115248
2,46,83,2401 1,98,1966,3923
941
110
59
Translation Plane t59, dual dt59
115248
25,44,83,2401 1,985,1964,3923
941
114
60
Translation Plane t60, dual dt60
115248
12,22,45,83,2401 1,492,982,1965,3923
939
106
61
Translation Plane t61, dual dt61
115248
16,28,43,82,2401 1,496,988,1963,3922
941
102
62
Translation Plane t62, dual dt62
115248
14,25,43,83,2401 1,494,985,1963,3923
941
102
63
Translation Plane t63, dual dt63
1382976
2,8,122,16,2401 1,98,392,5882,784
927
18
64
Translation Plane t64, dual dt64
230496
12,22,43,82,16,2401 1,492,982,1963,3922,784
941
54
65
Translation Plane t65, dual dt65
460992
12,46,8,16,2401 1,492,1966,392,784
939
30
66
Translation Plane t66, dual dt66
172872
24,34,63,12,2401 1,984,1474,2943,588
939
74
67
Translation Plane t67, dual dt67
57624
12,24,410,2401 1,492,984,19610
941
202
68
Translation Plane t68, dual dt68
115248
12,46,83,2401 1,492,1966,3923
941
102
69
Translation Plane t69, dual dt69
57624
14,27,48,2401 1,494,987,1968
941
202
70
Translation Plane t70, dual dt70
57624
25,410,2401 1,985,19610
941
202
71
Translation Plane t71, dual dt71
115248
2,44,84,2401 1,98,1964,3924
941
118
72
Translation Plane t72, dual dt72
518616
12,62,92,18,2401 1,492,2942,4412,882
941
38
73
Translation Plane t73, dual dt73
115248
23,45,83,2401 1,983,1965,3923
939
102
74
Translation Plane t74, dual dt74
57624
12,26,49,2401 1,492,986,1969
941
202
75
Translation Plane t75, dual dt75
115248
2,44,84,2401 1,98,1964,3924
937
110
76
Translation Plane t76, dual dt76
28812
14,223,2401 1,494,9823
941
402
77
Translation Plane t77, dual dt77
806736
72,8,28,2401 1,3432,392,1372
921
18
78
Translation Plane t78, dual dt78
921984
2,44,82,16,2401 1,98,1964,3922,784
925
24
79
Translation Plane t79, dual dt79
57624
14,27,48,2401 1,494,987,1968
941
202
80
Translation Plane t80, dual dt80
115248
14,25,45,82,2401 1,494,985,1965,3922
939
102
81
Translation Plane t81, dual dt81
57624
12,28,48,2401 1,492,988,1968
941
202
82
Translation Plane t82, dual dt82
57624
29,48,2401 1,989,1968
941
210
83
Translation Plane t83, dual dt83
28812
16,222,2401 1,496,9822
941
402
84
Translation Plane t84, dual dt84
16595712
2,48,2401 1,98,2352
941
6
85
Translation Plane t85, dual dt85
345744
42,6,123,2401 1,1962,294,5883
935
38
86
Translation Plane t86, dual dt86
28812
12,224,2401 1,492,9824
941
402
87
Translation Plane t87, dual dt87
172872
22,4,65,12,2401 1,982,196,2945,588
941
74
88
Translation Plane t88, dual dt88
57624
29,48,2401 1,989,1968
941
210
89
Translation Plane t89, dual dt89
460992
12,82,162,2401 1,492,3922,7842
921
32
90
Translation Plane t90, dual dt90
345744
42,6,12,24,2401 1,1962,294,588,1176
941
42
91
Translation Plane t91, dual dt91
57624
25,410,2401 1,985,19610
941
210
92
Translation Plane t92, dual dt92
57624
14,29,47,2401 1,494,989,1967
941
202
93
Translation Plane t93, dual dt93
57624
27,49,2401 1,987,1969
941
202
94
Translation Plane t94, dual dt94
57624
12,28,48,2401 1,492,988,1968
941
218
95
Translation Plane t95, dual dt95
28812
12,224,2401 1,492,9824
941
402
96
Translation Plane t96, dual dt96
28812
14,223,2401 1,494,9823
941
402
97
Translation Plane t97, dual dt97
14406
150,2401 1,4950
941
802
98
Translation Plane t98, dual dt98
1382976
2,64,24,2401 1,98,2944,1176
925
18
99
Translation Plane t99, dual dt99
115248
23,45,83,2401 1,983,1965,3923
937
118
100
Translation Plane t100, dual dt100
57624
14,25,49,2401 1,494,985,1969
941
202
101
Translation Plane t101, dual dt101
1037232
2,12,182,2401 1,98,588,8822
941
20
102
Translation Plane t102, dual dt102
57624
12,24,410,2401 1,492,984,19610
941
210
103
Translation Plane t103, dual dt103
57624
27,49,2401 1,987,1969
941
218
104
Translation Plane t104, dual dt104
57624
12,210,47,2401 1,492,9810,1967
941
202
105
Translation Plane t105, dual dt105
57624
14,27,48,2401 1,494,987,1968
937
218
106
Translation Plane t106, dual dt106
1843968
2,16,32,2401 1,98,784,1568
907
12
107
Translation Plane t107, dual dt107
28812
14,223,2401 1,494,9823
941
402
108
Translation Plane t108, dual dt108
230496
23,43,84,2401 1,983,1963,3924
941
54
109
Translation Plane t109, dual dt109
28812
16,222,2401 1,496,9822
941
402
110
Translation Plane t110, dual dt110
28812
110,220,2401 1,4910,9820
941
410
111
Translation Plane t111, dual dt111
28812
112,219,2401 1,4912,9819
941
402
112
Translation Plane t112, dual dt112
115248
23,45,83,2401 1,983,1965,3923
937
110
113
Translation Plane t113, dual dt113
172872
12,2,4,63,122,2401 1,492,98,196,2943,5882
941
78
114
Translation Plane t114, dual dt114
230496
2,42,83,16,2401 1,98,1962,3923,784
941
54
115
Translation Plane t115, dual dt115
28812
16,222,2401 1,496,9822
941
402
116
Translation Plane t116, dual dt116
28812
16,222,2401 1,496,9822
941
402
117
Translation Plane t117, dual dt117
2765952
4,6,16,24,2401 1,196,294,784,1176
935
12
118
Translation Plane t118, dual dt118
115248
14,25,43,83,2401 1,494,985,1963,3923
941
102
Most rows will not appear until major editing is complete...
1342
Translation Plane t1342, dual dt1342
28812
16,222,2401 1,496,9822
941
...
1343
Translation Plane t1343, dual dt1343
57624
12,28,48,2401 1,492,988,1968
939
...
1344
Translation Plane t1344, dual dt1344
86436
12,34,66,2401 1,492,1474,2946
941
...
1345
Translation Plane t1345, dual dt1345
14406
150,2401 1,4950
941
...
1346
Translation Plane t1346, dual dt1346
86436
12,34,66,2401 1,492,1474,2946
941
...
1347
Translation Plane t1347, dual dt1347
230496
12,84,16,2401 1,492,3924,784
917
...
h1*
Ordinary Hughes plane h1
11261376
57,2394 57,2394
965
 
h2*
Exceptional Hughes Plane h2
16892064
57,2394 57,2394
1005
 

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 for each plane.

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/ revised November, 2010