Find the table of characters of $\Gl(3,\Z_2)$. Find a cyclic subgroup of order $7$.
Enter the following commands in sage:
G=GL(3,2)
Reps=G.conjugacy_classes_representatives()
for g in Reps:
cg=G.conjugacy_class(g)
print(cg,len(cg))
latex(G.character_table())
This gives you $6$ conjugacy classes, representatives thereof are:
$$
E=\left(\begin{array}{ccc}
1&0&0\\
0&1&0\\
0&0&1
\end{array}\right),
A=\left(\begin{array}{ccc}
0&1&0\\
1&0&0\\
0&0&1
\end{array}\right),
B=\left(\begin{array}{ccc}
0&0&1\\
1&0&1\\
0&1&1
\end{array}\right)
$$
and
$$
C=\left(\begin{array}{ccc}
0&0&1\\
1&0&0\\
0&1&1
\end{array}\right),
D=\left(\begin{array}{ccc}
0&0&1\\
1&0&1\\
0&1&0
\end{array}\right),
F=\left(\begin{array}{ccc}
1&0&0\\
0&0&1\\
0&1&1
\end{array}\right)~.
$$
The sizes of the corresponding conjugacy classes: $1,21,42,24,24,56$ and the character table:
$$
\begin{array}{c|rrrccr}
\Gl(2,\Z_3)&1E&21A&42B&24C&24D&56F\\
\hline
\chi_1&1 & 1 & 1 & 1 & 1 & 1 \\
\chi_2&3 & -1 & 1 & \z_7^2 + \z_7^2 + \z_7 & -\z_7^4 - \z_7^2 - \z_7 - 1 & 0 \\
\chi_3&3 & -1 & 1 & -\z_7^4 - \z_7^2 - \z_7 - 1 & \z_7^4 + \z_7^2 + \z_7 & 0 \\
\chi_4&6 & 2 & 0 & -1 & -1 & 0 \\
\chi_5&7 & -1 & -1 & 0 & 0 & 1 \\
\chi_6&8 & 0 & 0 & 1 & 1 & -1
\end{array}
$$
where $\z_7\colon=e^{2\pi i/7}$. Finally the commands
i=0
while(order(G[i]) < 7): i += 1
a=G[i]
print(a)
H=G.subgroup([a])
H.cayley_table()
show the Cayley table of the cyclic group generated by 'a', which is given by
$$
\left(\begin{array}{ccc}
0&0&1\\
0&1&1\\
1&1&0
\end{array}\right)
$$