There are exactly 277 seven-element hyperfields, all built on the cyclic multiplicative group of order six, and the paper classifies which of them arise as quotients of fields.
Hypergroupes canoniques values et hypervalues
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On the borderline of fields and hyperfields, part II -- Enumeration and classification of the hyperfields of order 7
There are exactly 277 seven-element hyperfields, all built on the cyclic multiplicative group of order six, and the paper classifies which of them arise as quotients of fields.