The dihedral group 
gives the group of symmetries of a regular hexagon. The
group generators are given by a counterclockwise
rotation through 
radians and reflection in a line joining the midpoints of two opposite edges. If
denotes rotation
and 
reflection, we have
 |
(1)
|
From this, the group elements can be listed as
 |
(2)
|
The conjugacy classes of 
are given by
 |
(3)
|
The set of elements which by themselves make up conjugacy classes are in the center of 
,
denoted 
,
so
 |
(4)
|
The commutator subgroup is given by
 |
(5)
|
which can be used to find the Abelianization. The set 
of all left cosets of 
is given by
 |  |  |
(6)
|
 |  |  |
(7)
|
Thus we appear to have two generators for this group, namely 
and 
. Therefore, Abelianization
gives 
.
It is also known that 
where 
is the symmetric group. Furthermore 
where 
is the dihedral group
with 6 elements, i.e., the group of symmetries of an equilateral triangle.
There are thus two ways to produce the character table, either inducing from 
and using the orthogonality relations or simply by finding the character tables for
and 
and taking their group direct
sum.
 | 1 |  |  |  |  |  |
 | 1 | 1 | 1 | 1 | 1 | 1 |
 | 1 | 1 | 1 | 1 |  |  |
 | 1 |  |  | 1 | 1 |  |
 | 1 |  |  | 1 |  | 1 |
 | 2 |  | 1 |  | 0 | 0 |
 | 2 | 2 |  |  | 0 | 0 |
