Let $S_n$ be the symmetric group on $\{1, \ldots, n\}$. Let \begin{align} T=\sum_{g\in S_n} g. \end{align}

Are there some references about the factorization of $T$?

In the case of $n=3$, we have \begin{align} & T=1 + (12) + (23) + (12)(23) + (23)(12) + (12)(23)(12) \\ & = 1 + (12) + (23) + (12)(23) + (23)(12) + (23)(12)(23) \\ & = (1 + (12))((12) + (23) + (23)(12)) \\ & = (1 + (23))((12) + (23) + (12)(23)). \end{align} Has this problem been studied in some references?

Thank you very much.

Edit: the group algebra I consider is $\mathbb{C} S_n$.