Groups of order 108

Contents

See pages on algebraic structures of order 108| See pages on groups of a particular order

Statistics at a glance

The number 108 has 2 and 3 as its only prime factors. It has prime factorization:

$\! 108 = 2^2 \cdot 3^3 = 4 \cdot 27$

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.

GAP implementation

The order 108 is part of GAP's SmallGroup library. Hence, any group of order 108 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 108 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(108);

There are 45 groups of order 108.
They are sorted by their Frattini factors.
1 has Frattini factor [ 6, 1 ].
2 has Frattini factor [ 6, 2 ].
3 has Frattini factor [ 12, 3 ].
4 has Frattini factor [ 12, 4 ].
5 has Frattini factor [ 12, 5 ].
6 - 9 have Frattini factor [ 18, 3 ].
10 - 11 have Frattini factor [ 18, 4 ].
12 - 14 have Frattini factor [ 18, 5 ].
15 has Frattini factor [ 36, 9 ].
16 - 17 have Frattini factor [ 36, 10 ].
18 - 22 have Frattini factor [ 36, 11 ].
23 - 26 have Frattini factor [ 36, 12 ].
27 - 28 have Frattini factor [ 36, 13 ].
29 - 31 have Frattini factor [ 36, 14 ].
32 has Frattini factor [ 54, 12 ].
33 has Frattini factor [ 54, 13 ].
34 has Frattini factor [ 54, 14 ].
35 has Frattini factor [ 54, 15 ].
36 - 45 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.