>What  type  of  compact  manifolds,   can be acted  freely  by symmetric  group $S_{m}$  for  some  $m>2$?

>Is  there  a  compact manifold  which  can be  act  freely by  all  symmetric  groups $S_{m}$?


This  question have  been  asked  already [here](https://math.stackexchange.com/questions/981053/free-action-of-symmetric-group) 
  and  is  indirectly related to [this](https://mathoverflow.net/questions/184814/a-possible-generalization-of-the-borsuk-ulam-theorem-via-action-of-symmetric-gro).