Groups of order 54

Statistics at a glance
The number 54 has prime factors 2 and 3. The prime factorization is:

$$54 =2^1 \cdot 3^3 = 2 \cdot 27$$

The 3-Sylow subgroup of a group of order 54 is a normal Sylow subgroup and has order 27. The 2-Sylow subgroup is isomorphic to cyclic group:Z2 and the whole group is an internal semidirect product of these. There are two possibilities:


 * The group is a finite nilpotent group, in which case it is an internal direct product of its 3-Sylow subgroup and 2-Sylow subgroup.
 * The group is an internal semidirect product of its 3-Sylow subgroup and a 2-Sylow subgroup whose non-identity element acts via conjugation as a non-identity automorphism of order two.

GAP implementation
gap> SmallGroupsInformation(54);

There are 15 groups of order 54. They are sorted by their Frattini factors. 1 has Frattini factor [ 6, 1 ]. 2 has Frattini factor [ 6, 2 ]. 3 - 6 have Frattini factor [ 18, 3 ]. 7 - 8 have Frattini factor [ 18, 4 ]. 9 - 11 have Frattini factor [ 18, 5 ]. 12 - 15 have trivial Frattini subgroup.

For the selection functions the values of the following attributes are precomputed and stored: IsAbelian, IsNilpotentGroup, IsSupersolvableGroup, IsSolvableGroup, LGLength, FrattinifactorSize and FrattinifactorId.

This size belongs to layer 2 of the SmallGroups library. IdSmallGroup is available for this size.