The dicyclic group is a non-abelian group of order $4n$ which contains a cyclic subgroup of order $2n$ for $n\>1$. It is defined by a presentation of the form

If $n$ is a power of $2$, the resulting group is a generalized quaternion group.

The DicyclicGroup( n ) command returns a dicyclic group, either as a permutation group (the default) or as a finitely presented group.

You can specify the form of the group returned explicitly by passing one of the options 'form' = "permgroup" or 'form' = "fpgroup".

If the parameter n is not a positive integer, then a symbolic group representing the dicyclic group of order 4*n is returned.

In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$\mathrm{DicyclicGroup}\left(6\right)$

$⟨\left(1\,2\,3\,4\right)\left(5\,6\,7\,8\right)\left(9\,10\,11\right)\,\left(1\,7\,3\,5\right)\left(2\,6\,4\,8\right)\left(9\,11\right)⟩$

$\mathrm{DicyclicGroup}\left(6\,'\mathrm{form}'=''permgroup''\right)$

$⟨\left(1\,2\,3\,4\right)\left(5\,6\,7\,8\right)\left(9\,10\,11\right)\,\left(1\,7\,3\,5\right)\left(2\,6\,4\,8\right)\left(9\,11\right)⟩$

$\mathrm{DicyclicGroup}\left(6\,'\mathrm{form}'=''fpgroup''\right)$

$⟨a\,b\mid b^{-1}aba\,a^6{b}^2\,a^{12}⟩$

$\mathrm{IsNilpotent}\left(\mathrm{DicyclicGroup}\left(82^k\right)\right)\mathbf{assuming}k::\mathrm{posint}$

$\mathrm{true}$

$G≔\mathrm{DicyclicGroup}\left(6\right)$

$G≔⟨\left(1\,2\,3\,4\right)\left(5\,6\,7\,8\right)\left(9\,10\,11\right)\,\left(1\,7\,3\,5\right)\left(2\,6\,4\,8\right)\left(9\,11\right)⟩$

$Z≔\mathrm{Center}\left(G\right)$

$Z≔Z\left(⟨\left(1\,2\,3\,4\right)\left(5\,6\,7\,8\right)\left(9\,10\,11\right)\,\left(1\,7\,3\,5\right)\left(2\,6\,4\,8\right)\left(9\,11\right)⟩\right)$

$\mathrm{Generators}\left(Z\right)$

$\left[\left(1\,3\right)\left(2\,4\right)\left(5\,7\right)\left(6\,8\right)\right]$

$S≔\mathrm{SylowSubgroup}\left(2\,G\right)$

$S≔⟨\left(1\,6\,3\,8\right)\left(2\,5\,4\,7\right)\left(9\,10\right)\,\left(1\,2\,3\,4\right)\left(5\,6\,7\,8\right)⟩$

$\mathrm{IsCyclicSylowGroup}\left(\mathrm{DicyclicGroup}\left(7\right)\right)$

$\mathrm{true}$

$\mathrm{IsQuaternionGroup}\left(\mathrm{SylowSubgroup}\left(2\,\mathrm{DicyclicGroup}\left(12\right)\right)\right)$

$\mathrm{true}$

$\mathrm{Display}\left(\mathrm{CharacterTable}\left(\mathrm{DicyclicGroup}\left(5\right)\right)\right)$

The GroupTheory[DicyclicGroup] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.

The GroupTheory[DicyclicGroup] command was updated in Maple 2021.