Question: Does N variables caylay table have N permutation group

Does N variables caylay table have N permutation group so that can generate N functions?

for exmaple 3 variables cayley table have 3 permutation group, for 1, it has a permutation group , for 2 has a permutation group etc.

then does it mean that it has 3 functions, do it need to composite 3 functions in order to get a function belong to this cayley table?

1 1 1

1 2 2

1 2 3

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