= Sylow counts in a faithful degree-seven action with S4 point stabilizers
{c}
Suppose a group acts faithfully and transitively on seven points, every point stabilizer is $S_4$, and every two-point stabilizer is a Klein four-group. Then the group has order $168$ and
$$
n_2=21,
\qquad n_3=28,
\qquad n_7=8.
$$
Count pairs consisting of a point and a Sylow subgroup fixing it for $p=2,3$; the two-point stabilizer ensures uniqueness. For $p=7$, a normal Sylow subgroup would force the group into its order-$42$ normalizer in $S_7$.
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