= Counting Sylow bases in a coprime elementary abelian extension
{title2=$b(V\rtimes H)=|V:C_V(H)|\,b(H)$}
Let $V$ be an elementary abelian $r$-group and let $r\nmid|H|$, with $H$ soluble. In each Sylow basis of $V\rtimes H$, $V$ is the unique Sylow $r$-subgroup and the product of the remaining Sylow subgroups is a complement to $V$. All complements are $V$-conjugate, and the stabilizer of $H$ under this conjugation is its fixed-vector space $C_V(H)$. Each complement has $b(H)$ Sylow bases, proving the formula. For the permutation module $V=\mathbb F_5^4$ of $S_4$, the fixed-vector space is the one-dimensional constant-coordinate subspace, so the number is $5^3\cdot12=1500$.
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