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.

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.

order 2 (1)

order 3 (1)

order 4 (2)

order 5 (2)

order 6 (22)

order 7 (564)

order 8 (gzipped; 28.5MB) (1676267)

order 2 (1)

order 3 (1)

order 4 (2)

order 5 (2)

order 6 (12)

order 7 (147)

order 8 (gzipped; 5.1MB) (283657)

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

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

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)

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)

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)

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, brendan.mckay@anu.edu.au