\begin{figure}[htp]
    \centering
  \begin{tikzpicture}[scale = 0.7, line width=1.2]
    \coordinate (1) at (0, 3+4);
    
    \coordinate (2) at (3/1.71, 4);
    \coordinate (3) at (-3/1.71, 4);
    
    \coordinate (4) at (-4, 3);
    
    \coordinate (5) at (3/1.71 - 4, 0);
    \coordinate (6) at (-3/1.71 - 4, 0);
    
    \coordinate (7) at (4, 3);
    
    \coordinate (8) at (3/1.71 + 4, 0);
    \coordinate (9) at (-3/1.71 + 4, 0);
    
\draw[fill] (1) circle [radius=0.1] ;
    \draw[fill] (2) circle [radius=0.1];
    \draw[fill] (3) circle [radius=0.1];
    \draw[fill] (4) circle [radius=0.1];
    \draw[fill] (5) circle [radius=0.1];
    \draw[fill] (6) circle [radius=0.1];
    \draw[fill] (7) circle [radius=0.1];
    \draw[fill] (8) circle [radius=0.1];
    \draw[fill] (9) circle [radius=0.1];
    
    \node at (0, 7.5) {$1^1$};
    
    \node at (3/1.71, 3.6) {$2^1$};
    \node at (-3/1.71 -0.4, 4) {$3^1$};
    
    \node at (-4 + 0.5, 3+0.3) {$1^2$};
    
    \node at (3/1.71 - 4, -0.5) {$2^2$};
    \node at (-3/1.71 - 4 -0.5, 0) {$3^2$};
    
    \node at (4 + 0.5, 3) {$1^3$};
    
    \node at (3/1.71 + 4 + 0.5, 0) {$2^3$};
    \node at (-3/1.71 + 4, -0.5) {$3^3$};
    
    \path[draw = black, postaction = {on each segment = {mid arrow = red}}]
    
    (1) to (7)
    (7) to [bend right] (1);
    
    \draw (4) arc(-45:360-45:0.4);
    
    \path[draw = green, postaction = {on each segment = {mid arrow = red}}]
    (5) -- (6)
    (8) to (9)
    (2) to [bend left] (3);
    
    \path[draw = red, postaction = {on each segment = {mid arrow = red}}]
    (1)  -- (5)
    (4) -- (8)
    (7) -- (2);
    
    \path[draw = blue, postaction = {on each segment = {mid arrow = red}}]
    (3) -- (1)
    (6) -- (4)
    (9) -- (7);
    
    \path[draw = purple, postaction = {on each segment = {mid arrow = red}}]
    (3) to [bend left] (2)
    (6) to [bend right] (8)
    (9) -- (5);
    
    \node at (1.5, 5.1) {$a$};
    \node at (2.8, 5.8) {$a$};
    \node at (-4.2, 4) {$a$};
    
    \node at (0.6, 1.9) {$b$};
    \node at (-1, 2.9) {$b$};
    \node at (2.7, 3.2) {$b$};
    
    \node at (0, 4-0.3) {$c$};
    \node at (-4, 0.3) {$c$};\node at (4, 0.3) {$c$};
    
    \node at (-3/1.71/2 - 0.5, 3/2+4) {$d$};
    \node at (-3/1.71/2 - 0.5-4, 3/2) {$d$};
    \node at (-3/1.71/2 + 0.4+4, 3/2) {$d$};
    
    \node at (0, 4+0.3) {$e$};
    \node at (0, -1.3) {$e$};
    \node at (0, -0.3) {$e$};
    
    \draw[fill] (1) circle [radius=0.1] ;
    \draw[fill] (2) circle [radius=0.1];
    \draw[fill] (3) circle [radius=0.1];
    \draw[fill] (4) circle [radius=0.1];
    \draw[fill] (5) circle [radius=0.1];
    \draw[fill] (6) circle [radius=0.1];
    \draw[fill] (7) circle [radius=0.1];
    \draw[fill] (8) circle [radius=0.1];
    \draw[fill] (9) circle [radius=0.1];
    \end{tikzpicture}
    \caption{The derived covering graph $\tilde{\Gamma}$ of $\Gamma$ in \Cref{fig:perm-voltage-graph}. Edge colors denote correspondence to the edges of $\Gamma$ via the projection map.}
\label{fig:perm-derived-graph}
\end{figure}