The finite group
is one of the three non-Abelian groups of order 12 (out of a total of fives groups
of order 12), the other two being the alternating
group
and the dihedral group
. However, it is highly unfortunate that the symbol
is used to refer this particular group,
since the symbol
is also used to denote the point group
that constitutes the pure rotational subgroup of the full
tetrahedral group
and is isomorphic to
.
Thus, of the three distinct non-Abelian groups of order 12, two different
ones are each known as
under some circumstances. Extreme caution is therefore needed.
is the semidirect product of
by
by the map
given by
, where
is the automorphism
. The group can be constructed
from the generators
(1)
| |||
(2)
|
where
as the group elements 1,
,
,
,
,
,
,
,
,
,
, and
. The multiplication table is illustrated above.
has conjugacy classes
,
,
,
,
, and
. There are 8 subgroups, and their lengths are 1,
2, 3, 4, 4, 4, 6, and 12. Of these, the following five are normal:
,
,
,
, and the entire group.
The cycle graph of is illustrated above. The numbers of elements for with
for
, 2, ... are 1, 2, 3, 8, 1, 6, 1, 8, 3, 2, 1, and 12.
The finite group
has the presentations
(3)
|
and
(4)
|