It is well known that the symmetric group $S_n$ admits presentation with $\{(ij) \mid i\neq j\}$ as the set of generators and the following list of relations (in every formula distinct letters denote distinct indices): \begin{align} (ij) =&\, (ji), \label{Sym0} \tag{S0} \\ (ij)^2 = &\, 1, \label{Sym1} \tag{S1} \\ (jk)(ij)(jk) = &\, (ik), \label{Sym2} \tag{S2} \\ [(ij), (kl)] = &\,1. \label{Sym3} \tag{S3} \end{align}

If for $n\geq 3$ we drop relation \ref{Sym0} from this list, we will get an extension of $S_n$, denote it by $\widetilde{S}_n$. My question is: is there a standard name for this group, has it already been studied by anyone in any context? Is there any standard representation for it.

Using GAP I was able to compute $\widetilde{S_3}$ and $\widetilde{S_4}$ explicitly, the answers are [48, 41]; [384, 20069], respectively. So it looks like $\widetilde{S}_n$ is a nontrivial extension of $S_n$ by some finite nonabelian group of order $2^n$. As for the kernels $K_n=\mathrm{Ker}(\widetilde{S}_n\to S_n)$ the answers for $n=3,4,5$ are as follows: [8, 4], [16, 12], [32, 50]. In other words, $K_3 \cong Q_8$, $K_4 \cong Q_8 \times C_2$, and $K_5$ is isomorphic to the central product of $Q_8$ and $D_8$ over $C_2$.