There are 49 essentially different Sudoku 2x3 grids

... and the 2x3 Sudoku symmetry group

Ed Russell and Frazer Jarvis


In the computation of the number 28200960 (by Kjell Fredrik Pettersen) of 2x3 Sudoku grids, we treat two grids as different even when they might be closely related, perhaps simply by relabelling the numbers, or by a reflection or rotation. It would be nice to know the number of essentially different Sudoku grids, where we regard two grids as "essentially different" if one cannot be transformed into another by some sequence of symmetries, or by relabelling.

Let us label the squares as follows:

1 2 3 4 5 6
7 8 9 10 11 12
13 14 15 16 17 18
19 20 21 22 23 24
25 26 27 28 29 30
31 32 33 34 35 36

We regard these in 2x3 boxes; here, the boxes will be taken to be vertical, so that squares 1, 2, 7, 8, 13 and 14 form a box.

The permissible symmetries which we allow are:

Apart from relabelling, the others can be regarded as fixed permutations of the underlying 36 squares, and so the group G they form can be regarded as a subgroup of the symmetric group S36.

In fact, using GAP, we can input this group as a permutation group on the 36 squares, and its order is 3456. Using GAP we can say more. The group has centre consisting of the identity element, and the exchange of the top three rows with the bottom three rows. The exponent of the group is 12. The group is solvable, but not nilpotent. It has derived length 3.

Most importantly, for the Sudoku problem, there are 90 conjugacy classes of elements of G. To solve the problem above, we have to work out, for each representative in a conjugacy class, exactly how many grids are invariant up to relabelling under this permutation. In other words, given a representative, we have to work out how many grids there are, which when we apply the permutation, give a relabelled version of the original grid. The final table on this page gives a representative for each class, the size of the class and the number of grids invariant up to relabelling under this representative (divided by 6!).

The table that follows contains all the relevant data for this problem. Column 1 gives a label to each conjugacy class (this is arbitrary, and is just in the order given by the program). Column 2 gives the size of the class, column 3 gives a representative element, and column 4 gives the computation of the number of grids fixed up to relabelling. We must add up the red numbers multiplied by the green numbers, and divide by 3456, the group order. The result is that there are 49 essentially different grids.

Here are the representatives for the classes:

Class Size Representative Number of invariant grids
1 1 1 39168
2 4 (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36) 576
3 4 (19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36) 0
4 3 (3,4)(5,6)(9,10)(11,12)(15,16)(17,18)(21,22)(23,24)(27,28)(29,30)(33,34)(35,36) 0
5 12 (1,7,13)(2,8,14)(3,10,15,4,9,16)(5,12,17,6,11,18)(19,25,31)(20,26,32)(21,28,33,22,27,34)(23,30,35,24,29,36) 0
6 12 (3,4)(5,6)(9,10)(11,12)(15,16)(17,18)(19,25,31)(20,26,32)(21,28,33,22,27,34)(23,30,35,24,29,36) 0
7 8 (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,33,35)(32,34,36) 288
8 32 (1,9,17)(2,10,18)(3,11,13)(4,12,14)(5,7,15)(6,8,16)(19,27,35)(20,28,36)(21,29,31)(22,30,32)(23,25,33)(24,26,34) 126
9 32 (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,27,35)(20,28,36)(21,29,31)(22,30,32)(23,25,33)(24,26,34) 72
10 36 (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19) 560
11 72 (1,24,13,30,7,36)(2,23,14,29,8,35)(3,22,15,28,9,34)(4,21,16,27,10,33)(5,20,17,26,11,32)(6,19,18,25,12,31) 8
12 36 (1,35,2,36)(3,33)(4,34)(5,32,6,31)(7,29,8,30)(9,27)(10,28)(11,26,12,25)(13,23,14,24)(15,21)(16,22)(17,20,18,19) 0
13 72 (1,23,14,30,7,35,2,24,13,29,8,36)(3,21,15,27,9,33)(4,22,16,28,10,34)(5,20,18,25,11,32,6,19,17,26,12,31) 0
14 9 (7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36) 0
15 27 (3,4)(5,6)(7,13)(8,14)(9,16)(10,15)(11,18)(12,17)(21,22)(23,24)(25,31)(26,32)(27,34)(28,33)(29,36)(30,35) 0
16 72 (1,3,5)(2,4,6)(7,15,11,13,9,17)(8,16,12,14,10,18)(19,21,23)(20,22,24)(25,33,29,31,27,35)(26,34,30,32,28,36) 0
17 6 (25,31)(26,32)(27,33)(28,34)(29,35)(30,36) 0
18 12 (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30) 0
19 18 (3,4)(5,6)(9,10)(11,12)(15,16)(17,18)(21,22)(23,24)(25,31)(26,32)(27,34)(28,33)(29,36)(30,35) 0
20 36 (1,7,13)(2,8,14)(3,10,15,4,9,16)(5,12,17,6,11,18)(19,25)(20,26)(21,28)(22,27)(23,30)(24,29)(33,34)(35,36) 0
21 48 (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,33,29,31,27,35)(26,34,30,32,28,36) 0
22 96 (1,9,17)(2,10,18)(3,11,13)(4,12,14)(5,7,15)(6,8,16)(19,27,23,25,21,29)(20,28,24,26,22,30)(31,33,35)(32,34,36) 0
23 108 (1,36,7,30)(2,35,8,29)(3,34,9,28)(4,33,10,27)(5,32,11,26)(6,31,12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19) 0
24 108 (1,35,8,30)(2,36,7,29)(3,33,9,27)(4,34,10,28)(5,32,12,25)(6,31,11,26)(13,23,14,24)(15,21)(16,22)(17,20,18,19) 0
25 54 (3,5,4,6)(7,13)(8,14)(9,17,10,18)(11,16,12,15)(21,23,22,24)(25,31)(26,32)(27,35,28,36)(29,34,30,33) 0
26 54 (1,2)(3,6)(4,5)(7,14)(8,13)(9,18)(10,17)(11,16)(12,15)(19,20)(21,24)(22,23)(25,32)(26,31)(27,36)(28,35)(29,34)(30,33) 96
27 96 (1,36,10,19,18,28)(2,35,9,20,17,27)(3,32,11,21,14,29)(4,31,12,22,13,30)(5,33,8,23,15,26)(6,34,7,24,16,25) 12
28 48 (1,24,4,19,6,22)(2,23,3,20,5,21)(7,30,10,25,12,28)(8,29,9,26,11,27)(13,36,16,31,18,34)(14,35,15,32,17,33) 24
29 36 (1,32,7,20,13,26)(2,31,8,19,14,25)(3,34,9,22,15,28)(4,33,10,21,16,27)(5,35,11,23,17,29)(6,36,12,24,18,30) 0
30 18 (1,20)(2,19)(3,22)(4,21)(5,23)(6,24)(7,26)(8,25)(9,28)(10,27)(11,29)(12,30)(13,32)(14,31)(15,34)(16,33)(17,35)(18,36) 0
31 12 (1,31,7,19,13,25)(2,32,8,20,14,26)(3,33,9,21,15,27)(4,34,10,22,16,28)(5,35,11,23,17,29)(6,36,12,24,18,30) 48
32 6 (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36) 672
33 6 (3,5,4,6)(9,11,10,12)(15,17,16,18)(21,23,22,24)(27,29,28,30)(33,35,34,36) 0
34 24 (1,7,13)(2,8,14)(3,11,16,6,9,17,4,12,15,5,10,18)(19,25,31)(20,26,32)(21,29,34,24,27,35,22,30,33,23,28,36) 0
35 24 (3,5,4,6)(9,11,10,12)(15,17,16,18)(19,25,31)(20,26,32)(21,29,34,24,27,35,22,30,33,23,28,36) 0
36 6 (1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,24)(22,23)(25,26)(27,30)(28,29)(31,32)(33,36)(34,35) 1344
37 24 (1,8,13,2,7,14)(3,12,15,6,9,18)(4,11,16,5,10,17)(19,26,31,20,25,32)(21,30,33,24,27,36)(22,29,34,23,28,35) 24
38 24 (1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,14)(15,18)(16,17)(19,26,31,20,25,32)(21,30,33,24,27,36)(22,29,34,23,28,35) 0
39 36 (3,5,4,6)(7,13)(8,14)(9,17,10,18)(11,16,12,15)(21,23,22,24)(27,29,28,30)(33,35,34,36) 0
40 72 (1,7)(2,8)(3,11,4,12)(5,10,6,9)(15,17,16,18)(19,25,31)(20,26,32)(21,29,34,24,27,35,22,30,33,23,28,36) 0
41 36 (1,2)(3,6)(4,5)(7,14)(8,13)(9,18)(10,17)(11,16)(12,15)(19,20)(21,24)(22,23)(25,26)(27,30)(28,29)(31,32)(33,36)(34,35) 288
42 72 (1,8)(2,7)(3,12)(4,11)(5,10)(6,9)(13,14)(15,18)(16,17)(19,26,31,20,25,32)(21,30,33,24,27,36)(22,29,34,23,28,35) 0
43 144 (1,36,4,31,6,34)(2,35,3,32,5,33)(7,24,16,25,12,22,13,30,10,19,18,28)(8,23,15,26,11,21,14,29,9,20,17,27) 0
44 54 (1,32)(2,31)(3,34)(4,33)(5,35)(6,36)(7,20,13,26)(8,19,14,25)(9,22,15,28)(10,21,16,27)(11,23,17,29)(12,24,18,30) 0
45 18 (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,19,13,25)(8,20,14,26)(9,21,15,27)(10,22,16,28)(11,23,17,29)(12,24,18,30) 0
46 3 (5,6)(11,12)(17,18)(23,24)(29,30)(35,36) 0
47 12 (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,12,17,6,11,18)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,30,35,24,29,36) 0
48 12 (5,6)(11,12)(17,18)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,30,35,24,29,36) 0
49 1 (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36) 1536
50 4 (1,8,13,2,7,14)(3,10,15,4,9,16)(5,12,17,6,11,18)(19,26,31,20,25,32)(21,28,33,22,27,34)(23,30,35,24,29,36) 48
51 4 (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,26,31,20,25,32)(21,28,33,22,27,34)(23,30,35,24,29,36) 0
52 8 (1,3,5,2,4,6)(7,9,11,8,10,12)(13,15,17,14,16,18)(19,21,23,20,22,24)(25,27,29,26,28,30)(31,33,35,32,34,36) 48
53 32 (1,9,17,2,10,18)(3,11,14,4,12,13)(5,8,16,6,7,15)(19,27,35,20,28,36)(21,29,32,22,30,31)(23,26,34,24,25,33) 18
54 32 (1,3,5,2,4,6)(7,9,11,8,10,12)(13,15,17,14,16,18)(19,27,35,20,28,36)(21,29,32,22,30,31)(23,26,34,24,25,33) 24
55 36 (1,36,2,35)(3,34)(4,33)(5,31,6,32)(7,30,8,29)(9,28)(10,27)(11,25,12,26)(13,24,14,23)(15,22)(16,21)(17,19,18,20) 0
56 72 (1,24,14,29,7,36,2,23,13,30,8,35)(3,22,15,28,9,34)(4,21,16,27,10,33)(5,19,18,26,11,31,6,20,17,25,12,32) 0
57 36 (1,35)(2,36)(3,33)(4,34)(5,31)(6,32)(7,29)(8,30)(9,27)(10,28)(11,25)(12,26)(13,23)(14,24)(15,21)(16,22)(17,19)(18,20) 240
58 72 (1,23,13,29,7,35)(2,24,14,30,8,36)(3,21,15,27,9,33)(4,22,16,28,10,34)(5,19,17,25,11,31)(6,20,18,26,12,32) 24
59 27 (5,6)(7,13)(8,14)(9,15)(10,16)(11,18)(12,17)(23,24)(25,31)(26,32)(27,33)(28,34)(29,36)(30,35) 0
60 9 (1,2)(3,4)(5,6)(7,14)(8,13)(9,16)(10,15)(11,18)(12,17)(19,20)(21,22)(23,24)(25,32)(26,31)(27,34)(28,33)(29,36)(30,35) 448
61 72 (1,3,5,2,4,6)(7,15,11,14,10,18)(8,16,12,13,9,17)(19,21,23,20,22,24)(25,33,29,32,28,36)(26,34,30,31,27,35) 16
62 18 (5,6)(11,12)(17,18)(23,24)(25,31)(26,32)(27,33)(28,34)(29,36)(30,35) 0
63 36 (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,12,17,6,11,18)(19,25)(20,26)(21,27)(22,28)(23,30)(24,29)(35,36) 0
64 6 (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,32)(26,31)(27,34)(28,33)(29,36)(30,35) 768
65 12 (1,8,13,2,7,14)(3,10,15,4,9,16)(5,12,17,6,11,18)(19,26)(20,25)(21,28)(22,27)(23,30)(24,29)(31,32)(33,34)(35,36) 0
66 48 (1,3,5,2,4,6)(7,9,11,8,10,12)(13,15,17,14,16,18)(19,21,23,20,22,24)(25,33,29,32,28,36)(26,34,30,31,27,35) 24
67 96 (1,9,17,2,10,18)(3,11,14,4,12,13)(5,8,16,6,7,15)(19,27,23,26,22,30)(20,28,24,25,21,29)(31,33,35,32,34,36) 12
68 108 (1,36,8,29)(2,35,7,30)(3,34,9,28)(4,33,10,27)(5,31,12,26)(6,32,11,25)(13,24,14,23)(15,22)(16,21)(17,19,18,20) 0
69 108 (1,35,7,29)(2,36,8,30)(3,33,9,27)(4,34,10,28)(5,31,11,25)(6,32,12,26)(13,23)(14,24)(15,21)(16,22)(17,19)(18,20) 0
70 54 (3,5)(4,6)(7,13)(8,14)(9,17)(10,18)(11,15)(12,16)(21,23)(22,24)(25,31)(26,32)(27,35)(28,36)(29,33)(30,34) 512
71 54 (1,2)(3,6,4,5)(7,14)(8,13)(9,18,10,17)(11,15,12,16)(19,20)(21,24,22,23)(25,32)(26,31)(27,36,28,35)(29,33,30,34) 0
72 96 (1,36,9,20,17,28)(2,35,10,19,18,27)(3,32,11,22,13,30)(4,31,12,21,14,29)(5,34,7,24,15,26)(6,33,8,23,16,25) 40
73 48 (1,24,3,20,5,22)(2,23,4,19,6,21)(7,30,9,26,11,28)(8,29,10,25,12,27)(13,36,15,32,17,34)(14,35,16,31,18,33) 16
74 12 (1,32,7,20,13,26)(2,31,8,19,14,25)(3,34,9,22,15,28)(4,33,10,21,16,27)(5,36,11,24,17,30)(6,35,12,23,18,29) 16
75 6 (1,20)(2,19)(3,22)(4,21)(5,24)(6,23)(7,26)(8,25)(9,28)(10,27)(11,30)(12,29)(13,32)(14,31)(15,34)(16,33)(17,36)(18,35) 1504
76 36 (1,32,7,20,13,26)(2,31,8,19,14,25)(3,33,9,21,15,27)(4,34,10,22,16,28)(5,35,11,23,17,29)(6,36,12,24,18,30) 0
77 18 (1,20)(2,19)(3,21)(4,22)(5,23)(6,24)(7,26)(8,25)(9,27)(10,28)(11,29)(12,30)(13,32)(14,31)(15,33)(16,34)(17,35)(18,36) 0
78 6 (3,5)(4,6)(9,11)(10,12)(15,17)(16,18)(21,23)(22,24)(27,29)(28,30)(33,35)(34,36) 0
79 24 (1,7,13)(2,8,14)(3,11,15,5,9,17)(4,12,16,6,10,18)(19,25,31)(20,26,32)(21,29,33,23,27,35)(22,30,34,24,28,36) 0
80 24 (3,5)(4,6)(9,11)(10,12)(15,17)(16,18)(19,25,31)(20,26,32)(21,29,33,23,27,35)(22,30,34,24,28,36) 0
81 6 (1,2)(3,6,4,5)(7,8)(9,12,10,11)(13,14)(15,18,16,17)(19,20)(21,24,22,23)(25,26)(27,30,28,29)(31,32)(33,36,34,35) 0
82 24 (1,8,13,2,7,14)(3,12,16,5,9,18,4,11,15,6,10,17)(19,26,31,20,25,32)(21,30,34,23,27,36,22,29,33,24,28,35) 0
83 24 (1,2)(3,6,4,5)(7,8)(9,12,10,11)(13,14)(15,18,16,17)(19,26,31,20,25,32)(21,30,34,23,27,36,22,29,33,24,28,35) 0
84 36 (3,5)(4,6)(7,13)(8,14)(9,17)(10,18)(11,15)(12,16)(21,23)(22,24)(27,29)(28,30)(33,35)(34,36) 0
85 72 (1,7)(2,8)(3,11)(4,12)(5,9)(6,10)(15,17)(16,18)(19,25,31)(20,26,32)(21,29,33,23,27,35)(22,30,34,24,28,36) 0
86 36 (1,2)(3,6,4,5)(7,14)(8,13)(9,18,10,17)(11,15,12,16)(19,20)(21,24,22,23)(25,26)(27,30,28,29)(31,32)(33,36,34,35) 0
87 72 (1,8)(2,7)(3,12,4,11)(5,9,6,10)(13,14)(15,18,16,17)(19,26,31,20,25,32)(21,30,34,23,27,36,22,29,33,24,28,35) 0
88 144 (1,36,3,32,5,34)(2,35,4,31,6,33)(7,24,15,26,11,22,13,30,9,20,17,28)(8,23,16,25,12,21,14,29,10,19,18,27) 0
89 18 (1,32)(2,31)(3,34)(4,33)(5,36)(6,35)(7,20,13,26)(8,19,14,25)(9,22,15,28)(10,21,16,27)(11,24,17,30)(12,23,18,29) 0
90 54 (1,32)(2,31)(3,33)(4,34)(5,35)(6,36)(7,20,13,26)(8,19,14,25)(9,21,15,27)(10,22,16,28)(11,23,17,29)(12,24,18,30) 0
Total (red green) = 169344
Total (red green / 3456) = 49