\Gamma_0(12) & 12,4,3,3,1,1  \\ \hline
8& \Gamma_0(16) & 16,4,1,1,1,1  \\ \hline
9& \Gamma_1(16;16,2,2) & 16,2,2,2,1,1  \\
\hline \hline
\end{array}$$

Table 2: Torsion-free congruence subgroups of index 24.

Here $$\Gamma_1(8;4,1,2):=\{\pm
\begin{pmatrix} 1+4a & 2b \\ 4c & 1+4d
\end{pmatrix},\, a\equiv c\pmod 2\}$$ and $$\Gamma_1(16;16,2,2):=\{\pm \begin{pmatrix}
1+4a & b \\
8c & 1+4d \end{pmatrix}, \, a\equiv c\pmod 2\}.$$ : [*If we are interested in conjugacy as Fuchsian groups (in $PSL(2,{{\mathbb R}})$), Examples \#1,\#5, and \#8 are conjugate (use the matrix $\begin{pmatrix} 1 & 0
\\ 0 & 2
\end{pmatrix})$, Examples \#2 and \#7 are conjugate (use the same matrix), and Examples \#4, \#6, and \#9 are conjugate (use the same matrix as well as $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix})$.*]{}

[**Proposition 4**]{}:

*Let $\Gamma$ be one of the congruence subgroups in Table 2. Then $\Gamma$ has an explicit congruence lift $\tilde{\Gamma}$ to $SL(2,{{\mathbb Z}})$ with the following properties:*

\(1) $\tilde{\Gamma}$ has no elliptic elements. In particular $-{\rm Id}$ is not in $\tilde{\Gamma}$.

\(2) $\tilde{\Gamma}$ contains no element of trace $-2$.

[**Proof**]{}: We let $\tilde{\Gamma}$ be the subgroup of $SL(2,{{\mathbb Z}})$ consisting of the elements above $\Gamma$. Indeed, the lifts $\tilde{\Gamma}$ are sometimes the same as the groups $\Gamma$ themselves. 