(See the 2x3 calculation for more details of the method.)
Kjell Fredrik Pettersen has observed that representatives for the conjugacy classes can be taken to be formed from a permutation of the rows and a permutation of the columns. This simplifies the book-keeping hugely!
| Size | Row representative | Column representative | Number of invariant grids |
|---|---|---|---|
| 1 | 1 | 1 | 524640665777288616345600 |
| 1 | 1 | (1,2)(3,4)(5,6)(7,8)(9,10) | 227289669304320 |
| 20 | 1 | (1,2)(3,4)(5,6)(7,9)(8,10) | 164379135836160 |
| 60 | 1 | (1,2)(3,5)(4,6)(7,9)(8,10) | 108651431854080 |
| 384 | 1 | (1,3,5,7,9)(2,4,6,8,10) | 24883200 |
| 384 | 1 | (1,3,5,7,9,2,4,6,8,10) | 46080 |
| 20 | (9,10) | (1,2)(3,4)(5,6)(7,8)(9,10) | 93351114178560 |
| 400 | (9,10) | (1,2)(3,4)(5,6)(7,9)(8,10) | 47445631303680 |
| 1200 | (9,10) | (1,2)(3,5)(4,6)(7,9)(8,10) | 23716398366720 |
| 7680 | (9,10) | (1,3,5,7,9,2,4,6,8,10) | 23040 |
| 100 | (4,5)(9,10) | (1,2)(3,4)(5,6)(7,8)(9,10) | 38381102825472 |
| 2000 | (4,5)(9,10) | (1,2)(3,4)(5,6)(7,9)(8,10) | 14021843484672 |
| 6000 | (4,5)(9,10) | (1,2)(3,5)(4,6)(7,9)(8,10) | 5370871283712 |
| 38400 | (4,5)(9,10) | (1,3,5,7,9,2,4,6,8,10) | 12672 |
| 30 | (7,8)(9,10) | (1,2)(3,4)(5,6)(7,8)(9,10) | 36528696852480 |
| 600 | (7,8)(9,10) | (1,2)(3,4)(5,6)(7,9)(8,10) | 14069708881920 |
| 1800 | (7,8)(9,10) | (1,2)(3,5)(4,6)(7,9)(8,10) | 5575488307200 |
| 11520 | (7,8)(9,10) | (1,3,5,7,9,2,4,6,8,10) | 11520 |
| 300 | (4,5)(7,8)(9,10) | (1,2)(3,4)(5,6)(7,8)(9,10) | 15038794432512 |
| 6000 | (4,5)(7,8)(9,10) | (1,2)(3,4)(5,6)(7,9)(8,10) | 4232879013888 |
| 18000 | (4,5)(7,8)(9,10) | (1,2)(3,5)(4,6)(7,9)(8,10) | 1292658278400 |
| 115200 | (4,5)(7,8)(9,10) | (1,3,5,7,9,2,4,6,8,10) | 6912 |
| 225 | (2,3)(4,5)(7,8)(9,10) | (1,2)(3,4)(5,6)(7,8)(9,10) | 5953025998848 |
| 4500 | (2,3)(4,5)(7,8)(9,10) | (1,2)(3,4)(5,6)(7,9)(8,10) | 1304218828800 |
| 13500 | (2,3)(4,5)(7,8)(9,10) | (3,5)(4,6)(7,9)(8,10) | 1321425960960 |
| 13500 | (2,3)(4,5)(7,8)(9,10) | (1,2)(3,5)(4,6)(7,9)(8,10) | 319026102272 |
| 86400 | (2,3)(4,5)(7,8)(9,10) | (1,3,5,7,9,2,4,6,8,10) | 4608 |
| 32000 | (3,4,5)(8,9,10) | (5,7,9)(6,8,10) | 2356936704 |
| 32000 | (3,4,5)(8,9,10) | (1,2)(3,4)(5,7,9,6,8,10) | 884736 |
| 64000 | (3,4,5)(8,9,10) | (1,3)(2,4)(5,7,9,6,8,10) | 331776 |
| 64000 | (3,4,5)(6,7)(8,9,10) | (1,2)(3,4)(5,7,9,6,8,10) | 221184 |
| 128000 | (3,4,5)(6,7)(8,9,10) | (1,3)(2,4)(5,7,9,6,8,10) | 55296 |
| 32000 | (1,2)(3,4,5)(6,7)(8,9,10) | (1,2)(3,4)(5,7,9,6,8,10) | 147456 |
| 64000 | (1,2)(3,4,5)(6,7)(8,9,10) | (1,3)(2,4)(5,7,9,6,8,10) | 36864 |
| 216000 | (2,3,4,5)(7,8,9,10) | (3,5,7,9)(4,6,8,10) | 5242880 |
| 216000 | (2,3,4,5)(7,8,9,10) | (1,2)(3,5,7,9)(4,6,8,10) | 532480 |
| 18432 | (6,7,8,9,10) | (1,3,5,7,9)(2,4,6,8,10) | 619200 |
| 18432 | (6,7,8,9,10) | (1,3,5,7,9,2,4,6,8,10) | 5760 |
| 184320 | (4,5)(6,7,8,9,10) | (1,3,5,7,9,2,4,6,8,10) | 2880 |
| 276480 | (2,3)(4,5)(6,7,8,9,10) | (1,3,5,7,9,2,4,6,8,10) | 1440 |
| 576 | (1,2,3,4,5)(6,7,8,9,10) | 1 | 132710400 |
| 576 | (1,2,3,4,5)(6,7,8,9,10) | (1,2)(3,4)(5,6)(7,8)(9,10) | 46080 |
| 11520 | (1,2,3,4,5)(6,7,8,9,10) | (1,2)(3,4)(5,6)(7,9)(8,10) | 23040 |
| 34560 | (1,2,3,4,5)(6,7,8,9,10) | (1,2)(3,5)(4,6)(7,9)(8,10) | 11520 |
| 221184 | (1,2,3,4,5)(6,7,8,9,10) | (1,3,5,7,9)(2,4,6,8,10) | 114700 |
| 221184 | (1,2,3,4,5)(6,7,8,9,10) | (1,3,5,7,9,2,4,6,8,10) | 840 |
| 120 | (1,6)(2,7)(3,8)(4,9)(5,10) | 1 | 13580303155200 |
| 120 | (1,6)(2,7)(3,8)(4,9)(5,10) | (1,2)(3,4)(5,6)(7,8)(9,10) | 958060338610176 |
| 2400 | (1,6)(2,7)(3,8)(4,9)(5,10) | (7,9)(8,10) | 3171955138560 |
| 2400 | (1,6)(2,7)(3,8)(4,9)(5,10) | (1,2)(3,4)(5,6)(7,9)(8,10) | 22201474572288 |
| 7200 | (1,6)(2,7)(3,8)(4,9)(5,10) | (3,5)(4,6)(7,9)(8,10) | 770505216000 |
| 7200 | (1,6)(2,7)(3,8)(4,9)(5,10) | (1,2)(3,5)(4,6)(7,9)(8,10) | 4232109871104 |
| 46080 | (1,6)(2,7)(3,8)(4,9)(5,10) | (1,3,5,7,9)(2,4,6,8,10) | 8640 |
| 46080 | (1,6)(2,7)(3,8)(4,9)(5,10) | (1,3,5,7,9,2,4,6,8,10) | 9216 |
| 432000 | (1,6)(2,7,3,8)(4,9,5,10) | (3,5,7,9)(4,6,8,10) | 1300480 |
| 432000 | (1,6)(2,7,3,8)(4,9,5,10) | (1,2)(3,5,7,9)(4,6,8,10) | 632832 |
| 192000 | (1,6)(2,7)(3,8,4,9,5,10) | (5,7,9)(6,8,10) | 331776 |
| 192000 | (1,6)(2,7)(3,8,4,9,5,10) | (1,2)(3,4)(5,7,9,6,8,10) | 552960 |
| 384000 | (1,6)(2,7)(3,8,4,9,5,10) | (1,3)(2,4)(5,7,9)(6,8,10) | 82944 |
| 384000 | (1,6)(2,7)(3,8,4,9,5,10) | (1,3)(2,4)(5,7,9,6,8,10) | 175104 |
| 2880 | (1,6,2,7,3,8,4,9,5,10) | 1 | 46080 |
| 2880 | (1,6,2,7,3,8,4,9,5,10) | (1,2)(3,4)(5,6)(7,8)(9,10) | 9216 |
| 57600 | (1,6,2,7,3,8,4,9,5,10) | (7,9)(8,10) | 23040 |
| 57600 | (1,6,2,7,3,8,4,9,5,10) | (1,2)(3,4)(5,6)(7,9)(8,10) | 4608 |
| 172800 | (1,6,2,7,3,8,4,9,5,10) | (3,5)(4,6)(7,9)(8,10) | 11520 |
| 172800 | (1,6,2,7,3,8,4,9,5,10) | (1,2)(3,5)(4,6)(7,9)(8,10) | 2304 |
| 1105920 | (1,6,2,7,3,8,4,9,5,10) | (1,3,5,7,9)(2,4,6,8,10) | 320 |
| 1105920 | (1,6,2,7,3,8,4,9,5,10) | (1,3,5,7,9,2,4,6,8,10) | 2556 |
| Total (red × green) = 524641104917993005056000 | |||
| Total (red × green / 110592000) = 4743933602050718 | |||