= Numerical group velocity from refraction
The wavelengths specified for incidence from vacuum are $\lambda_0$. With fixed incident angle $\theta_i$, <Snell's law> gives $n=\sin\theta_i/\sin\theta_t$ and $v_p=c\sin\theta_t/\sin\theta_i$. At the central wavelength,
$$
n=\frac{\sin45.00^\circ}{\sin17.80^\circ}=2.3131,\qquad
v_p=1.2961\times10^8\,\mathrm{m\,s^{-1}}.
$$
Use a central difference, converting the angle change to radians:
$$
\frac{d\theta_t}{d\lambda_0}\simeq\frac{0.020\pi/180}{4.0\times10^{-9}}
=8.7266\times10^4\,\mathrm{rad\,m^{-1}}.
$$
The <group index from refracted-angle dispersion> is
$$
n_g=n\left[1+\lambda_0\cot\theta_t\frac{d\theta_t}{d\lambda_0}\right]=2.6275,
\qquad
\boxed{v_g=\frac c{n_g}=1.141\times10^8\,\mathrm{m\,s^{-1}}}.
$$
Directly inserting the incident vacuum wavelengths into $v_p-\lambda\,dv_p/d\lambda$ would instead give $1.120\times10^8\,\mathrm{m\,s^{-1}}$. That is a plausible intended rough answer, but it mixes wavelength coordinates; the boxed result consistently uses the incident data. The two agree only to first order in a sufficiently small dimensionless dispersion correction.
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