Find the table of characters of the symmetry group $C_{6v}\times\Z_2$ of benzene $\chem{C_6H_6}$.
$C_{6v}$ is generated by two elements: $C,S$, which satisfy the relations: $C^6=S^2=CSCS=E$; it has six characters $\chi_1,\ldots,\chi_6$ and the following representatives of the conjugacy classes (cf. subsection):
$$
\{E\},\{C,C^5\},\{C^2,C^4\},\{C^3\},\{\s,\s C^4,\s C^2\},\{\s C^1,\s C^5,\s C^3\}
$$
$\Z_2$ has two characters $\psi_1,\psi_2$, representatives $e,a$:
$$
\begin{array}{c|rrrrrr}
C_{6v}&1E&1C^3&3\s&2C&2C^2&3\s C\\
\hline
\chi_1&1 & 1 & 1 & 1 & 1 & 1 \\
\chi_2&1 & -1 & -1 & -1 & 1 & 1 \\
\chi_3&1 & -1 & 1 & -1 & 1 & -1 \\
\chi_4&1 & 1 & -1 & 1 & 1 & -1 \\
\chi_5&2 & -2 & 0 & 1 &-1 & 0 \\
\chi_6&2 & 2 & 0 & -1 &-1 & 0
\end{array}
\qquad\qquad
\begin{array}{c|rr}
\Z_2&1e&1a\\
\hline
\psi_1&1 & 1\\
\psi_2&1 & -1
\end{array}
$$
Thus $C_{6v}\times\Z_2$ has 12 characters: $\chi_j\otimes\psi_k$.
$E=(123456)$,
$C=(234561)$,
$S=(165432)$,
$C^3=(456123)$,
$C^4=(561234)$,
$C^3S=(432165)$,
CSF.$\la$C,S,F$\ra$=FreeGroup()
G=CSF/[C^6,S^2,F^2,C*S*C*S,C*F*C^5*F,S*F*S*F]
Reps=G.conjugacy_classes_representatives()
for g in Reps:
cg=G.conjugacy_class(g)
print(g, len(cg))
print G.character_table()
$$
\begin{array}{c|rrrrrrrrrrrr}
C_{6v}\times\Z_2&1E&1C^3&3S&2C&2C^4&3C^3S&1F&1C^3F&3SF&2CF&2C^4F&3C^3SF\\
\hline
\vp_1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 &1& 1 & 1 \\
\vp_2 & 1 & -1& -1& -1& 1 & 1 & -1& 1 & 1 &1& -1& -1\\
\vp_3 & 1 & -1& -1&-1 & 1 & 1 & 1 & -1& -1&-1& 1 & 1 \\
\vp_4 & 1 & -1& 1 & -1& 1 & -1& -1& 1 & -1&1& -1& 1 \\
\vp_5 & 1 & -1& 1 &-1 & 1 & -1& 1 & -1& 1 &-1& 1 & -1\\
\vp_6 & 1 & 1 & -1& 1 & 1 & -1& -1& -1& 1 &-1& -1& 1 \\
\vp_7 & 1 & 1 & -1& 1 & 1 & -1& 1 & 1 & -1&1& 1 & -1\\
\vp_8 & 1 & 1 & 1 & 1 & 1 & 1 & -1& -1& -1&-1& -1& -1\\
\vp_9 & 2 & -2& 0 & 1 & -1& 0 & -2& 2 & 0 &-1& 1 & 0 \\
\vp_{10}& 2 & -2& 0 & 1 & -1& 0 & 2 & -2& 0 &1& -1& 0 \\
\vp_{11}& 2 & 2 & 0 & -1& -1& 0 & -2& -2& 0 &1& 1 & 0 \\
\vp_{12}& 2 & 2 & 0 & -1& -1& 0 & 2 & 2 & 0 &-1& -1& 0
\end{array}
$$