\chapter{Table of Orbit-length Partition for Small Degree Polynomials}
\label{chap:appD}
This table can be found in L. Soicher and J. McKay's \citep{soicher1985computing} articles.

\begin{table}[htbp]
\centering
\begin{tabular}{cccc}
\hline
$G$ & $x_1+x_2$ & $x_1+x_2+x_3$ & $x_1-x_2$\\
\hline
\textbf{Degree 3} &  &  &  \\
+$A_3$ &  & & $3^2$\\
$S_3$ &  & & 6\\

\textbf{Degree 4} &  &  &  \\
$Z_4$ & 2,4  & & $4^3$\\
+$V_4$ & $2^3$  & & $4^3$\\
$D_4$ & 2,4  & & 4,8\\
+$A_4$ &6  & & 12\\
$S_4$ &6  & & 12\\

\textbf{Degree 6} &  &  &  \\
$Z_6$ & 3,$6^2$  &2,$6^3$ & $6^5$\\
$S_3$ & $3^3$,6  &2,$6^3$ & $6^5$\\
$D_6$ & 3,$6^2$  &2,$6,12$ & 6,$12^2$\\
+$A_4$ &3,12  &$4^2,6^2$ & 6,$12^2$\\
$G_{18}$ &6,9  &2,18 & $6^2$,18\\
$G_{24}$ & 3,12 &$6^2$,8 &6,$12^2$\\
+$S_4/V_4$ & 3,12&$4^2$,12&6,24\\
$S_4/Z_4$ & 3,12 &8,12&6,24\\
$G_{36}^1$ & 6,9 &2,18&12,18\\
+$G_{36}^2$ & 6,9 &2,18&12,18\\
$G_{48}$& 3,12 &8,12 &6,24\\
+$\text{PSL}_2(5)$ & 15 & $10^2$ & 30\\
$G_{72}$ & 6,9 &2,18&12,18\\
$\text{PGL}_2(5)$ & 15&20&30\\
+$A_6$ & 15&20&30\\
$S_6$ & 15&20&30\\
\hline
\end{tabular}
\caption{Orbit-length Partition of Polynomial with Degree 3,4,6}
\label{tab:allorbits}
\end{table}

where ``+" means that the discriminants of the corresponding polynomial is square. Note that we need a non-linear resolvent to determine $PGL_2(5)$ and $S_6$ etc.