Nets and Latin Squares of Order 5

Here is an exhaustive list of nets and latin squares of order 5. I realise n=5 is pretty trivial but this site provides a model for the less trivial lists of nets of other small orders.

In addition to my own C++ programs, I have made use of Brendan McKay's graph isomorphism package nauty.

If you are aware of errors or omissions in my list, I would appreciate an email message () from you.

Each isomorphism class of k-net of order 5 is represented by a line in this table, which includes

• A name for the net, consisting of the integer k (the number of parallel classes), followed by a letter distinguishing the isomorphism class among all k-nets of order 5. For k=3, a link is provided to a file containing the lexicographically smallest Latin square representing the 3-net. For k≥4, a link is provided to a set of k−2 MOLS (mutually orthogonal Latin squares of order 5) representing the net; in this case the choice of canonical representative is more complicated to describe.
• The order of automorphism group G of the net.
• The order of the subgroup H≤G consisting of all parallel-class preserving automorphisms of the net.
• A list of isomorphism types of the (k−1)-subnets of the k-net, followed (in parentheses) by a number indicating how many of the k subnets have this type.

/ revised April, 2001