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Peres/Mermin arguments about no-hidden variables in quantum mechanics areused for displaying a pair (R, S) of entangling Clifford quantum gates, actingon two qubits. From them, a natural unitary representation of Coxeter/Weylgroups W(D5) and W(F4) emerges, which is also reflected into the splitting ofthe n-qubit Clifford group Cn into dipoles C$\pm$n . The union of thethree-qubit real Clifford group C+ 3 and the Toffoli gate ensures a orthogonalrepresentation of the Weyl/Coxeter group W(E8), and of its relatives. Otherconcepts involved are complex reflection groups, BN pairs, unitary groupdesigns and entangled states of the GHZ family.

Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates