In fact I am interested in the case of large $N$

No such a requirement

No, the whole set of integers

Thank you Dave, I didn't think this way. Your method deals with $\bar{\sigma}$'s only and not with their products. Even a finite amount of $\alpha$'s can be good enough for me, so I guess I need to proceed further.

I checked the case of $S_7$, there is no representation of dimension 7 with the required properties, so the assumption of coolpapa doesn't work

Thank you for your answer. I tried for some time to write examples explicitly, but was unable to see any clear pattern. That is why I am asking this question.

What I need is an explicit construction together with a proof that it indeed provides the answer. I actually need it for some numeric computation

Thanks a lot to everybody who gave an answer. In fact what I ment to ask was if there was a way to "naturally" define a representation with dimention equal to a number of conjugacy classes for every symmetric group. So the monster group is not a case of this consideration. Trivial representations may appear in a decomposition to irreps. I am asking for some "natural", meaningful representation possessing some properties, or interesting realizations