Let $F$ be a non-archimedean local field. If $K = \mathrm{GL}(2, \mathcal{O}_{F})$ is a maximal compact subgroup of $G = \mathrm{GL}(2, F)$, then we understand the structure of $\mathcal{H}_{K} = C_{c}^{\infty}(K\backslash G/K)$ of $K$-biinvariant compactly supported smooth (i.e. locally constant) functions with convolution well - this is commutative and generated by elements $T(\mathfrak{p}), R(\mathfrak{p})$ and $R(\mathfrak{p})^{-1}$ where $T(\mathfrak{p}^{k})$ is a characteristic function of the set of all $g\in M_{2}(\mathcal{O}_{F})$ such that ideal generated by $\det(g)$ in $\mathcal{O}_{F}$ is $\mathfrak{p}^{k}$, and $R(\mathfrak{p})$ is a characteristic function of the set $$ K\begin{pmatrix} \varpi & \\ & \varpi \end{pmatrix} K = K\begin{pmatrix} \varpi & \\ & \varpi \end{pmatrix}. $$ Also, we know the relations between such generators well.
I want to know if there's any structural theorem of the algebra $C_{c}^{\infty}(G)$ of all compactly supported smooth functions, which obviously contains $\mathcal{H}_K$. This is not commutative (right?), and I want to know if there's any known result about generators and relations of the algebra.