I am trying to construct the complete graph $K_4$ as a quotient of the Bruhat-Tits tree $\mathcal{B}$ of $G=\operatorname{GL}_2(\mathbb{Q}_2)$, which is the universal cover of $K_4$. So I'm trying to find a cocompact, torsion-free subgroup $\Gamma$ of $G$ such that $\Gamma \setminus \mathcal{B} \widetilde{=} K_4$. I tried the following: I think $\Gamma$ should be isomorphic to the fundamental group of $K_4$ which is $\pi_1(K_4) = \mathbb{Z}*\mathbb{Z}*\mathbb{Z}$. I used the fact that the standard lattice gets stabilized by $K = \operatorname{GL}_n(\mathbb{Z}_p)$, so the vertices of the tree can be identified with the cosets $G/K$. One system of representatives of the vertices of distance $k$ from the standard lattice is given by $$\begin{pmatrix} 1 & 0\\ \sum_{j=1}^k i_j p^{j-1} & p^k \end{pmatrix}, \begin{pmatrix} p^k & \sum_{j=2}^k i_j p^{j-1}\\ 0 & 1 \end{pmatrix}, i_j \in \{ 0, \dots, p-1 \}.$$ Now I drew a big picture of the tree and labeled the vertices using these representatives. I spent a lot of time trying to identify the right cosets to get $K_4$ and hoped to "see" $\Gamma$. But this seems to be really hard.
Can someone give me an adivice how to approach this problem?