Here is one important way in which the Heisenberg group is important:

$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\PGL{PGL}$Let $F$ be a local field of characteristic not equal to 2 (so for example, one of your fields $\mathbb{Q}_p$ or $\mathbb{R}$ or $\mathbb{C}$ above), and let $\psi$ be a nontrivial unitary additive character of $F$. Let $V$ be a symplectic space over $F$ with symplectic form $\langle , \rangle$, and form the Heisenberg group $H(V)$. This is the group which is set theoretically $V \times F$, with group operation $(v_1, x_1)(v_2,x_2) = (v_1+v_2, t_1+t_2 + \frac{1}{2} \langle v_1,v_2 \rangle)$.

By the Stone–Von-Neumann theorem, $H(V)$ has a unique irreducible smooth (or unitary) representation $(\rho_{\psi}, W)$ over $\mathbb{C}$ with central character $\psi$. Note that $\Sp(V)$ acts on $H(V)$ in a natural way through its action on $V$. Then, by Schur's Lemma, for every $g \in \Sp(V)$, there exists an intertwining operator $\phi_g$ between $(\rho_{\psi}, W)$ and $(\rho_{\psi} \circ g, W)$, which is unique up to multiplication by $\mathbb{C}^*$.

We therefore get a projective representation
\begin{align*}
\Sp(V) & {}\rightarrow \PGL(W) \\
g & {}\mapsto \phi_g.
\end{align*}

It can be shown that this projective representation lifts to a representation of $\widetilde{\Sp(V)}$, where $\widetilde{\Sp(V)}$ is a naturally defined double cover of $\Sp(V)$, called the Metaplectic group. This representation, denoted $\omega_{\psi}$, is called the Weil representation. Both the Weil representation and the metaplectic group are important in number theory and representation theory. If you're curious, Wee Teck Gan has a nice survey article on how the metaplectic group arises in the Langlands program: Gan - Representations of metaplectic groups.

youshould think the group important if you want to study its irreps. So let me turn the tables and ask:whydoyouwant to study this group? $\endgroup$