ligand pi system irreducible representation for trigonal bipyramidal transition metal complex

I am considering a D$\_{3h}$ trigonal bipyramidal transition metal complex MX$\_5$. I have found that the irreducible representation for the ligand s and $\sigma$ systems is 2A$\_1'$+A$\_2''$+E$'$. I have
\begin{array}{|c|c|c|c|c|c|}
\hline
&E& 2C\_3 & 3C\_2 & \sigma\_h & 2S\_3&3\sigma\_v\\ \hline
 \Gamma\_s&5& 2& 1&3&0&3\\ \hline
 \Gamma\_\sigma&5& 2& 1&3&0&3\\ \hline
 \Gamma\_{\pi+\pi'}&10&0&-2&0&0&0\\ \hline
\end{array}
However, I think that my $\Gamma\_{\pi+\pi'}$ is incorrect since I get fractional values for coefficients for the irreducible representations when I try to reduce it to irreducible representations. But from sketching the trigonal pyramidal complex, I cannot see how my $\chi(R)$ are wrong for the pi system representation, $\Gamma\_{\pi+\pi'}$. Any help would be appreciated.




I believe you have the wrong character for the $\hat{C\_3}$ rotation. None of the p orbitals in the equatorial positions remain in place so they all contribute zero. The axial p orbitals are rotated by $120^°$ which means they each contribute $\cos(120^°)=-\frac{1}{2}$. Since there are two of them at each of the two axial position, $\chi(\hat{C\_3})=-2$. Using this new value for the character (assuming all the others are right, which they seemed to be), I obtained: $a\_2',a\_2'',2e',2e''$


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