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@alcinnz Yup, exhaustive search gives the following four-element example:

2 4 3 1
1 3 4 2
4 2 1 3
3 1 2 4

This satisfies (a * b) * c = (a * c) * b, but
• 1 * 2 = 4 ≠ 1 = 2 * 1; and
• (1 * 2) * 3 = 2 ≠ 1 = 1 * (2 * 3).

I'm not claiming to know a non-group structure where you can usefully do Diffie–Hellman, but I wouldn't rule it out just yet.