# M64

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]

## Definition

This group, denoted , is defined as the semidirect product of cyclic group:Z32 by cyclic group:Z2 where the latter acts on the former by the power map. Explicitly, it has the following presentation:

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 64#Arithmetic functions

## 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 | do not commute | |

metacyclic group | Yes | Quotient by is cyclic | |

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

## GAP implementation

### Group ID

This finite group has order 64 and has ID 51 among the groups of order 64 in GAP's SmallGroup library. For context, there are groups of order 64. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(64,51)`

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

`gap> G := SmallGroup(64,51);`

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

`IdGroup(G) = [64,51]`

or just do:

`IdGroup(G)`

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