= Solution
Use the <D3h molecular point group> for <phosphorus pentafluoride>, with $z$ along the two axial P–F bonds. For each <symmetry> operation, only atoms left in place contribute to the character of the Cartesian displacement representation; each contributes the trace of the corresponding $3\times3$ spatial <matrix>. In the class order $E,2C_3,3C_2',\sigma_h,2S_3,3\sigma_v$ this gives
$$
\chi_{3N}=(18,0,-2,4,-2,4).
$$
For example a $C_2'$ fixes P and one equatorial F, and each fixed atom has Cartesian trace $-1$, giving $-2$. <reflection> in the equatorial plane fixes four atoms, each with trace $1$, giving $4$. Reduction with the <character table> gives
$$
\Gamma_{3N}=2A_1'+A_2'+4E'+3A_2''+2E''.
$$
Subtract translations $E'+A_2''$ and rotations $A_2'+E''$ to obtain
$$
\boxed{\Gamma_{\rm vib}=2A_1'+3E'+2A_2''+E''.}
$$
Its dimension is $2+6+2+2=12=3(6)-6$, as required for a nonlinear six-atom molecule. Thus there are eight symmetry-distinct fundamental <frequencies> in the harmonic picture, with each E mode doubly degenerate. This analysis concerns vibrations about one trigonal-bipyramidal minimum; it does not treat large-amplitude rearrangements between minima.
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