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
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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
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
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.|
Here, the is as in the parametrization. The order of the group is .
|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|
|Abelian group||No for .|
|Nilpotent group||Yes only for a power of two.|
|Ambivalent group||Yes for even, no for odd|
|Rational group||Yes only for , i.e., the quaternion group|
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|
|16||4||Generalized quaternion group:Q16|
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.