  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.
