I'm trying to understand the Latin square in picture 2.1 of the paper "On Projective Planes of Order Nine" by Marshall Hall, Dean Swift, and Raymond Killgrove (https://www.ams.org/journals/mcom/1959-13-068/S0025-5718-1959-0107208-8/S0025-5718-1959-0107208-8.pdf). This is regarding a plane of order 9 with three pencils of finite lines x=c, y=c, and y=x+c. "Specifically this last pencil may be represented by the Latin square:"
$\begin{matrix}0&1&2&3&4&5&6&7&8 \\ 1&2&0&4&5&3&7&8&6 \\ 2&0&1&5&3&4&8&6&7 \\ 3&4&5&6&7&8&0&1&2 \\ 4&5&3&7&8&6&1&2&0 \\ 5&3&4&8&6&7&2&0&1 \\ 6&7&8&0&1&2&3&4&5 \\ 7&8&6&1&2&0&4&5&3 \\ 8&6&7&2&0&1&5&3&4 \\ \end{matrix}$
The rows represent permutations of $S_9$ and are understood as lines with 9 finite points.
I can't for the life of me figure out the operation (possibly ternary) of this "elementary abelian group." Any insights would be appreciated!