The finite group is one of the three nonAbelian 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 nonAbelian 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)
