Symmetric group:S6

Definition
The symmetric group $$S_6$$, called the symmetric group of degree six, is defined in the following equivalent ways:


 * It is the member of family::symmetric group on a set of size six. In particular, it is a member of family::symmetric group on finite set.
 * It is the member of family::symplectic group $$Sp(4,2)$$, and hence also the member of family::projective symplectic group $$PSp(4,2)$$ (see isomorphism between symplectic and projective symplectic group in characteristic two).

Up to conjugacy
For convenience, we take the underlying set here as $$\{ 1,2,3,4,5,6 \}$$.

There are eleven conjugacy classes, corresponding to the unordered integer partitions of $$6$$ (for more information, refer cycle type determines conjugacy class).

Up to automorphism
The outer automorphism group has order two, and it swaps some conjugacy classes. Below are the equivalence classes up to automorphisms.