# Dicyclic group

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with metacyclic group

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with dihedral group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

*This is a family of groups parametrized by the natural numbers, viz, for each natural number, there is a unique group (upto isomorphism) in the family corresponding to the natural number. The natural number is termed the parameter for the group family*

## Contents

## Definition

The **dicyclic group**, also called the **binary dihedral group** with parameter is defined in the following equivalent ways:

- It is given by the presentation:

Here, is the identity element.

- It has the following faithful representation as a subgroup of the quaternions: .
- It is the binary von Dyck group with parameters , i.e., it has the presentation:

.

The dicyclic group with parameter has order , and it is an extension of a cyclic group of order by a cyclic group of order 2.

### Equivalence of definitions

`Further information: equivalence of presentations of dicyclic group`

## Cases

The dicyclic group has some alternate descriptions in specific cases.

Case on | Examples | Description of dicyclic group |
---|---|---|

Odd number | , so dicyclic group:Dic12 | Semidirect product of cyclic normal subgroup of order (generated by ) and group of order 4 generated by (in the first presentation) or (in the second). The latter element conjugates to its inverse. |

## Arithmetic functions

Here, the is as in the parametrization. The order of the group is .

Function | Value | Explanation |
---|---|---|

order | ||

exponent | least common multiple of and | |

nilpotency class | if , undefined otherwise. | |

derived length | 2 for | |

number of conjugacy classes | ||

number of subgroups | where is the divisor sum function and is the divisor count function |

## Group properties

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

Abelian group | No for . | |

Nilpotent group | Yes only for a power of two. | |

Solvable group | Yes | |

Supersolvable group | Yes | |

Metacyclic group | Yes | |

Ambivalent group | Yes for even, no for odd | |

Rational group | Yes only for , i.e., the quaternion group |

## Particular cases

### For small values

Note that all dicyclic groups are metacyclic and hence supersolvable. A dicyclic group is nilpotent if and only if it is of order for some . It is abelian only if it has order 4.

Order of group | Degree | Common name for the group | Comment |
---|---|---|---|

4 | 1 | Cyclic group:Z4 | Not typically considered a dicyclic group |

8 | 2 | Quaternion group | |

12 | 3 | Dicyclic group:Dic12 | |

16 | 4 | Generalized quaternion group:Q16 |

## Elements

`Further information: Element structure of dicyclic groups`

The dicyclic group of order has conjugacy classes. In the discussion below, we use the presentation:

The elements are:

- The identity element. (1)
- The unique central non-identity element, which is given by . (1)
- The remaining elements in . There are of these elements, and they occur in conjugacy classes of size two: each element is conjugate to its inverse. There are thus conjugacy classes of size each.
- The elements outside come in two conjugacy classes: the conjugacy class of , which contains all elements of the form , and the conjugacy class of . These two conjugacy classes are related by an outer automorphism and each has elements.