# Groups of order 54

## Contents

See pages on algebraic structures of order 54| See pages on groups of a particular order

## Statistics at a glance

The number 54 has prime factors 2 and 3. The prime factorization is:

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

There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^aq^b$-theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.

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

The order 54 is part of GAP's SmallGroup library. Hence, any group of order 54 can be constructed using the SmallGroup function by specifying its group ID. Also, IdGroup is available, so the group ID of any group of this order can be queried.

Further, the collection of all groups of order 54 can be accessed as a list using GAP's AllSmallGroups function.

Here is GAP's summary information about how it stores groups of this order, accessed using GAP's SmallGroupsInformation function:

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.