Where does this permutation product live?

This is from Loring Tu's An introduction to manifolds.

The coefficient $1/(k!ℓ!)$ in the definition of the wedge product compensates for repetitions in the sum: for every permutation $σ ∈ S_{k+ℓ}$ , there are k! permutations τ in $S_k$ that permute the first k arguments $v_{σ (1)} , . . . , v_{σ (k)}$ and leave the arguments of g alone; for all τ in $S_k$ , the resulting permutations σ τ in $S_{k+ℓ}$ contribute the same term to the sum, since $$ \begin{aligned}(\mathrm{sgn}\,\sigma\tau)f\left(\nu_{\sigma\tau(1)},\cdots,\nu_{\sigma\tau(k)}\right) &=(\mathrm{sgn}\,\sigma\tau)(\mathrm{sgn}\,\tau)f\left(\nu_{\sigma(1)},\cdots,\nu_{\sigma(k)}\right)\\ &= (\mathrm{sgn}\,\sigma)f\left(\nu_{\sigma(1)},\cdots,\nu_{\sigma(k)}\right),\end{aligned} $$ where the first equality follows from the fact that ( τ (1), . . . , τ (k)) is a permutation of (1, . . . , k). So we divide by k! to get rid of the k! repeating terms in the sum coming from permutations of the k arguments of f ; similarly, we divide by ℓ! on account of the ℓ arguments of g.

My confusion is this: why are we allowed to multiply $\sigma$ and $\tau$ like that? The text states that $\sigma \in S_{k + l}$ and $\tau \in S_k$, so should this operation even be defined? I feel like the author might mean several things.

It might not be multiplication, but some sort of concatenation-type operation. Or perhaps there is an implicit embedding of $\tau$ into $S_k$? I'm a bit lost by the notation used.

Any pointers appreciated!

I don't have the book in front of me. But it seems clear to me that he is using the inclusion $S_k\to S_{k+l}$ which sends $\tau$ to $\tau\times id_{S_l}$; the permutation which does nothing on the last $l$ elements, and maps $1,\cdots, k$ to $\tau(1),\cdots ,\tau(k)$.