# SmallGroup(16,3)

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 using the following presentation:

Note that is generated by alone, because the final relation allows us to write in terms of and .

The subgroup is isomorphic to the direct product of Z4 and Z2, and the element is an element of order two that acts on the subgroup by conjugation by fixing and sending to .

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions

## Group properties

Want to compare with other groups of the same order? Check out groups of order 16#Group properties.

Property | Satisfied | Explanation |
---|---|---|

abelian group | No | don't commute |

group of prime power order | Yes | |

nilpotent group | Yes | prime power order implies nilpotent |

group of nilpotency class two | Yes | |

supersolvable group | Yes | |

T-group | No | The subgroup is 2-subnormal, not normal |

monolithic group | No | |

one-headed group | No | |

ambivalent group | No | |

rational group | No | |

rational-representation group | No |

## Subgroups

`Further information: subgroup structure of SmallGroup(16,3)`

- The trivial group. (1)
- The subgroup , which is the unique characteristic subgroup of order . Isomorphic to cyclic group:Z2. It is the commutator subgroup, and can also be described as the unique group of order two containing an element that is
*not*a square but is a product of squares. The quotient group is isomorphic to direct product of Z4 and Z2. (1) - The subgroups and , which are both normal subgroups related by an outer automorphism. Isomorphic to cyclic group:Z2. The quotient group for each is isomorphic to dihedral group:D8. (2)
- The subgroups , , and . Neither is normal, and they come in two conjugacy classes of size two each. Isomorphic to cyclic group:Z2. (4)
- The subgroup , which is the center, first agemo subgroup, and Frattini subgroup. Isomorphic to Klein four-group. The quotient group is isomorphic to Klein four-group. (1)
- The subgroups and . Both are normal subgroups and are related by an outer automorphism. Isomorphic to Klein four-group. The quotient group is isomorphic to cyclic group:Z4. (2)
- The subgroups , , , and . Two conjugacy classes of size two each. Isomorphic to Klein four-group. The quotient group is also isomorphic to a Klein four-group. (4)
- The subgroups , , , and . All of them are related by outer automorphisms, and they form two conjugacy classes of subgroups of size two each: is conjugate to , while is conjugate to . Isomorphic to cyclic group:Z4. (4)
- The subgroups and . These are both normal subgroups related by an outer automorphism. Isomorphic to direct product of Z4 and Z2. The quotient group is isomorphic to cyclic group:Z2. (2)
- The subgroup . Isomorphic to elementary abelian group:E8. The quotient group is isomorphic to cyclic group:Z2. (1)
- The whole group. (1)

## Subgroup-defining functions

Subgroup-defining function | Subgroup type in list | Page on subgroup embedding | Isomorphism class | Comment |
---|---|---|---|---|

Center | (4) | center of SmallGroup(16,3) | Klein four-group | |

Commutator subgroup | (2) | commutator subgroup of SmallGroup(16,3) | cyclic group:Z2 | |

Frattini subgroup | (4) | center of SmallGroup(16,3) | Klein four-group | |

Socle | (4) | center of SmallGroup(16,3) | Klein four-group | |

first omega subgroup | (7) | elementary abelian group:E8 |

## GAP implementation

### Group ID

This finite group has order 16 and has ID 3 among the groups of order 16 in GAP's SmallGroup library. For context, there are 14 groups of order 16. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(16,3)`

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

`gap> G := SmallGroup(16,3);`

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

`IdGroup(G) = [16,3]`

or just do:

`IdGroup(G)`

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

### Other descriptions

The group can be constructed using the following GAP commands:

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