In this paper, quadratic algebras on which $\tilde{ H}_4$, the Heisenberg group of order 64, acts as degree-preserving algebra automorphisms are studied. In particular, we show that if $\mathcal{ A}$ is a four-dimensional Artin-Schelter regular quadratic $\tilde{ H}_4$-algebra with the degree one part isomorphic to the Schr\"odinger representation of $\tilde{ H}_4$, then $\mathcal{ A}$ is (a twist of) a four-dimensional Sklyanin algebra or (a twist of) a quantum Clifford algebra of global dimension 4.
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