Sylow basis ($\{P_p:p\mid |G|\},P_p{P}_q=P_q{P}_p$)

A Sylow basis consists of one Sylow subgroup for each prime dividing the group order, with any two of them permutable as subgroups. Permutable means $P_p{P}_q=P_q{P}_p$, not elementwise commutation. Their product over any subset of primes is a Hall subgroup. Every finite soluble group has such a basis, and all bases are simultaneously conjugate. Induction over an elementary abelian minimal normal subgroup proves these facts using coprime splitting over an elementary abelian normal subgroup: lift the other-prime quotient basis into a complement, and lift the characteristic-prime Sylow subgroup by taking its full preimage.

Let $V$ be an elementary abelian $r$-group and let $r\nmid |H|$, with $H$ soluble. In each Sylow basis of $V⋊H$, $V$ 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 $5^3\cdot 12=1500$.