A numerical example is provided by 
. The set of proper
partitions characterizing the conjugacy classes of 
is
the set of corresponding cycle types is
Thus the types of
turn out to be
The orders of the conjugacy classes are 6,18,12,1,9,4,6,4,12.
We now describe an interesting action of 
which
is in fact an action of the form 
.
.
Example
The action of on 
is obviously
similar to the following
action of
on the set 
:
The corresponding permutation group on will be denoted by
and called the composition
of 
and 
, while
will be used for the permutation group on 
, induced by the
natural action of 
on .
The action of the wreath product on
induces a natural action of on the set
i.e. on the set of 0-1-matrices consisting of 
rows and 
columns:
Since 
,
we can do this in two steps:
Hence we can first of all permute the columns of 
in
such a way that the numbers of 1's in the columns of the resulting matrix
is nonincreasing from left to right:
And after having carried out
this permutation with a suitable 
, we can find
a 
that moves the 1's of each column in flush top
position. This proves that the orbit of
under is characterized by an element of the form
i.e. by a proper partition of 
. Hence the orbits of
on 
are characterized by the
proper partitions 
, where each part 
and where the total number of parts is 
:
.
Corollary
There exists a natural bijection
Hence an application of the Cauchy-Frobenius Lemma yields the following formula for the number of partitions of this form:
which can be made more explicit by an application of 
.
