To transcribe what is given in levineds's linked answer and remove the parts only relevant to linear molecules, here are the steps you can apply to determine the reducible representation of the vibrations.
- Determine how the molecule is changed by all symmetry operations. If an atom is moved by a symmetry operation, it doesn't contribute to $\Gamma_\text{atoms}$. If the atom remains in place, add +1 to the column of that symmetry operation.
For hexafluoroethane, this gives:
$$\small \begin{array}{c|cccccc} \hline
D_{3\mathrm{d}} & E & 2C_3 & 3C_2' & i & 2S_6 & 3\sigma_d & \\ \hline
\Gamma_\text{atoms} & 8 & 2 & 0 & 0 & 0 & 4 &\\
\end{array}$$
- Multiply each column of $\Gamma_\text{atoms}$ by the trace of that operation's matrix representation.
$$\begin{array}{c|c}E&3\\\hline C_2&-1\\\hline \sigma&1\\\hline i&-3\\\hline C_n&1+2\cos(\frac{2\pi}{n})=1+2\cos\theta\\\hline S_n&-1+2\cos(\frac{2\pi}{n})=-1+2\cos\theta\end{array}$$
$$\small \begin{array}{c|cccccc} \hline
D_{3\mathrm{d}} & E & 2C_3 & 3C_2' & i & 2S_6 & 3\sigma_d & \\ \hline
\Gamma_\text{cart} & 24 & 0 & 0 & 0 & 0 & 4 &\\
\end{array}$$
- Use the reduction formula to determine irreps. $n(i)=\frac{1}{h}\sum_R N\cdot\chi_r(R)\cdot\chi_i(R)$ where $n(i)$ is the number of the $i^{\text{th}}$ irrep, $R$ is a symmetry operation, $h$ is the order of the group, $N$ is the coefficient in front of the operation, and $\chi(R)$ is the character of the operation in the reducible/irreducible representation.
I won't go through the calculation (there are online calculators that you can use to do this), but this gives. $$\Gamma_{xyz}=3A_{1g}+1A_{2g}+4E_g+1A_{1u}+3A_{2u}+4E_u$$
which if we subtract off the translational and rotational irreps you have already obtained gives:
$$\Gamma_{vib}=3A_{1g}+3E_g+1A_{1u}+2A_{2u}+3E_u$$
You had small errors with your rotational and translational representations; the $E$ irreps are degenerate and so count for 2 degrees of freedom. This means in each case you would only have a single $E$ irrep.
