We consider the implications of "all to all" coupling of n identical, Z(2) symmetric cells. In particular we demonstrate that how this coupling is achieved with respect to these internal symmetries is important, even in this very simple scenario. The coupling we consider leads to a full symmetry of S-n, Z(2) x S-n or Z(2) \ S-n, where the latter is the "wreath product" symmetry. By considering the generic solutions of each case on their respective irreducible representations, we compare and contrast the three cases, and show how very different solutions could arise in essentially similar systems
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