2 added 21 characters in body

I'd like to see a combinatorial/group theoretic proof. For instance, if one could exhibit an injective homomorphism of direct (or just semidirect) products of symmetric groups:

$\phi:S_{3n} \times S_{4n} \times S_{6n} \to S_{n}\times S_{12n},$

then the index of Im($\phi$) would be that ratio. This is just a vague hint.

1

I'd like to see a combinatorial/group theoretic proof. For instance, if one could exhibit an injective homomorphism of direct products of symmetric groups:

$\phi:S_{3n} \times S_{4n} \times S_{6n} \to S_{n}\times S_{12n},$

then the index of Im($\phi$) would be that ratio. This is just a vague hint.