I have found the following structure which curiously satifies all the field axioms except for associativitiy of addition.
$\begin{array}{c|cccccc} \ + & 0 & 1 & 2 & 3 & 4 & 5\\ \hline 0 & 0 & 1 & 2 & 3 & 4 & 5\\ 1 & 1 & 0 & 5 & 4 & 2 & 3\\ 2 & 2 & 5 & 0 & 1 & 3 & 4\\ 3 & 3 & 4 & 1 & 0 & 5 & 2\\ 4 & 4 & 2 & 3 & 5 & 0 & 1\\ 5 & 5 & 3 & 4 & 2 & 1 & 0 \end{array}$
$\begin{array}{c|cccccc} \ \cdot & 0 & 1 & 2 & 3 & 4 & 5\\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 1 & 2 & 3 & 4 & 5\\ 2 & 0 & 2 & 4 & 1 & 5 & 3\\ 3 & 0 & 3 & 1 & 5 & 2 & 4\\ 4 & 0 & 4 & 5 & 2 & 3 & 1\\ 5 & 0 & 5 & 3 & 4 & 1 & 2 \end{array}$
Has anyone seen this type of structure before? If so, could one provide a reference for its use or more information about them?
For the curious, the addition table was derived from a symmetric coloring of the edges of $K_6$, and the multiplication comes from the fact that coloring has a symmetry of order $5$, though I think it would be too much to write up the full details within the question.