Latin squares

A Latin square is reduced (also called "normalized") if the first row and first column are in natural order. Any Latin square can be reduced by sorting the rows and columns.

The two most common equivalence classes defined for Latin squares are isotopy classes and main classes. Two squares are in the same isotopy class if one can be obtained from the other by permuting rows, columns and symbols. To be in the same main class, one is in addition permitted to permute the three roles "row", "column" and "symbol" (for example, symbol s in row r and column c might become symbol r in row c and column s).

In the following files, numbering starts with 0. There is one square per line in an obvious format.

Reduced Latin squares

order 2 (1)
order 3 (1)
order 4 (4)
order 5 (56)
order 6 (9408)
order 7 (gzipped) part 1 (6000000), part 2 (6000000), part 3 (4942080)

Watch out, the files for order 7 unzip to a total of 948,756,480 bytes.

Isotopy class representatives

order 2 (1)
order 3 (1)
order 4 (2)
order 5 (2)
order 6 (22)
order 7 (564)
order 8 (gzipped; 28.5MB) (1676267)

Main class representatives

order 2 (1)
order 3 (1)
order 4 (2)
order 5 (2)
order 6 (12)
order 7 (147)
order 8 (gzipped; 5.1MB) (283657)

Isotopy classes with nontrivial groups

Here we give one member of each isotopy class which has a non-trivial isotopy. Up to order 6, all isotopy classes have this property.

order 7 (149)
order 8 (gzipped) (31833)
Order 9 (2393407) in 3 gzipped files (about 28MB each): part 1  part 2  part 3

Main classes with nontrivial groups

Here we give one member of each main class which has a non-trivial main-class automorphism. Up to order 6, all main classes have this property.

order 7 (103)
order 8 (gzipped) (13046)
Order 9 (2523159) in 4 gzipped files (about 18MB each): part 1  part 2  part 3  part 4

Main classes of Latin squares without intercalates

An intercalate is a 2x2 Latin subsquare; that is, two rows and two columns whose intersection includes only two symbols. This property is preserved under the main class equivalences.

order 5 (1)
order 6 (1)
order 7 (2)
order 8 (3)
order 9 (1707)

Loops

A loop is a quasigroup with identity. In these files we give the multiplication tables of the small non-isomorphic loops. Element 0 is an identity in each case.

order 2 (1)
order 3 (1)
order 4 (2)
order 5 (6)
order 6 (109)
order 7 (23746)

Main classes of Graeco-Latin squares

A Graeco-Latin square consists of a pair of orthogonal Latin squares. Main classes are the equivalence classes defined by permutation of the rows, permutations of the columns, permutation of the symbols in the first square, permutation of the symbols in the second square, and permutations of the roles (row, column, symbol1, symbol2). There are no Graeco-Latin squares of order 1, 2 or 6.

order 3 (1)
order 4 (1)
order 5 (1)
order 7 (7)
order 8 (2165)

Isotopy classes of Latin rectangles

order 3x2 (1)
order 4x2 (2)
order 4x3 (2)
order 5x2 (2)
order 5x3 (3)
order 5x4 (3)
order 6x2 (4)
order 6x3 (16)
order 6x4 (56)
order 6x5 (40)
order 7x2 (4)
order 7x3 (56)
order 7x4 (1398)
order 7x5 (6941)
order 7x6 (3479)
order 8x2 (7)
order 8x3 (370)
order 8x4 (93561)
order 8x5 (gzipped; 54 MB) (4735238)
order 9x2 (8)
order 9x3 (2877)
order 9x4 (gzipped; 90 MB) (8024046)
order 10x2 (12)
order 10x3 (27841)

 


Page Master: Brendan McKay, bdm@cs.anu.edu.au and http://cs.anu.edu.au/~bdm.

Up to the combinatorial data page