= Solution
In the class order $(E,8C_3,3C_2,6S_4,6\sigma_d)$, the 15 Cartesian nuclear displacements of <methane> have <characters> $(15,0,-1,-1,3)$. Only nuclei left fixed by an operation contribute to that operation's <trace>. Subtract translations $T_2$ and rotations $T_1$ to obtain the <methane normal-mode representation>
$$
\boxed{\Gamma_{\rm vib}=A_1+E+2T_2}.
$$
Its dimension is $1+2+3+3=9=3(5)-6$. There are four frequency families: a nondegenerate symmetric stretch, a doubly degenerate bend, and triply degenerate stretch and bend modes.
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