The conjugacy class of 
will be denoted by 
,
the centralizer by 
, so that we obtain the following descriptions
and properties of conjugacy classes and centralizers of elements of 
:
There are some examples to compute the orders of the
conjugacy classes and centralizers
in 
.
Corollary
Let 
and 
denote elements of . Then
.
, i.e. is ambivalent
, which means that each element is a conjugate of its
inverse.
.
.
.
occurs as the cycle
partition
of some .
(The first, second, fourth and fifth item is clear from the foregoing, while the
third one follows from the fact that there are exactly 
mappings
which map a set of 
-tuples onto this same set up to cyclic permutations
inside each -tuple.)
For the sake of simplicity we can therefore parametrize
the conjugacy classes
of elements in (and correspondingly in 
)
by partitions or cycle types putting
