# SmallGroup(32,2)

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Contents

## Definition

This group can be defined by the following presentation:

where denotes the identity element and stands for the commutator of two elements (the isomorphism type of the group is independent of the choice of commutator map).

See the section #Description by presentation for how to construct the group using this presentation in GAP.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 32#Arithmetic functions

## Group properties

Want to compare and contrast group properties with other groups of the same order? Check out groups of order 32#Group properties

Property | Satisfied? | Explanation | Comment |
---|---|---|---|

group of prime power order | Yes | ||

nilpotent group | Yes | prime power order implies nilpotent | |

supersolvable group | Yes | via nilpotent: finite nilpotent implies supersolvable | |

solvable group | Yes | via nilpotent: nilpotent implies solvable | |

abelian group | No | ||

T-group | No | ||

monolithic group | No | ||

one-headed group | No | ||

group of nilpotency class two | Yes | ||

metabelian group | Yes | ||

finite group that is 1-isomorphic to an abelian group | Yes | via cocycle skew reversal generalization of Baer correspondence |

## GAP implementation

### Group ID

This finite group has order 32 and has ID 2 among the groups of order 32 in GAP's SmallGroup library. For context, there are 51 groups of order 32. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(32,2)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(32,2);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [32,2]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Description by presentation

gap> F := FreeGroup(3); <free group on the generators [ f1, f2, f3 ]> gap> G := F/[F.1^4,F.2^4,F.3^2,Comm(F.1,F.2)*F.3^(-1),Comm(F.1,F.3),Comm(F.2,F.3)]; <fp group on the generators [ f1, f2, f3 ]> gap> IdGroup(G); [ 32, 2 ]

### GAP verification of function values and group properties

Below is a GAP implementation verifying the various function values and group properties as stated in this page. Before beginning, set `G := SmallGroup(32,2);` or set it to this group in any other manner:

gap> Order(G); 32 gap> Exponent(G); 4 gap> NilpotencyClassOfGroup(G); 2 gap> DerivedLength(G); 2 gap> FrattiniLength(G); 2 gap> Rank(G); 2 gap> RankAsPGroup(G); 3 gap> NormalRank(G); 3 gap> CharacteristicRank(G); 3