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GroupTheory

 DicyclicGroup
 construct a dicyclic group as a permutation group or a finitely presented group

 Calling Sequence DicyclicGroup( n ) DicyclicGroup( n, s )

Parameters

 n - algebraic; understood to be a positive integer s - (optional) equation of the form form = "fpgroup" or form = "permgroup" (default)

Description

 • The dicyclic group is a non-abelian group of order $4n$ which contains a cyclic subgroup of order $2n$ for $n>1$.
 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{DicyclicGroup}\left(6\right)$
 $⟨\left({1}{,}{2}{,}{3}{,}{4}\right)\left({5}{,}{6}{,}{7}{,}{8}\right)\left({9}{,}{10}{,}{11}\right){,}\left({1}{,}{7}{,}{3}{,}{5}\right)\left({2}{,}{6}{,}{4}{,}{8}\right)\left({9}{,}{11}\right)⟩$ (1)
 > $\mathrm{DicyclicGroup}\left(6,'\mathrm{form}'="permgroup"\right)$
 $⟨\left({1}{,}{2}{,}{3}{,}{4}\right)\left({5}{,}{6}{,}{7}{,}{8}\right)\left({9}{,}{10}{,}{11}\right){,}\left({1}{,}{7}{,}{3}{,}{5}\right)\left({2}{,}{6}{,}{4}{,}{8}\right)\left({9}{,}{11}\right)⟩$ (2)
 > $\mathrm{DicyclicGroup}\left(6,'\mathrm{form}'="fpgroup"\right)$
 $⟨{}{a}{,}{b}{}{\mid }{}{{b}}^{{-1}}{}{a}{}{b}{}{a}{,}{{a}}^{{6}}{}{{b}}^{{2}}{,}{{a}}^{{12}}{}⟩$ (3)
 > $\mathrm{IsNilpotent}\left(\mathrm{DicyclicGroup}\left(8{2}^{k}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}k::'\mathrm{posint}'$
 ${\mathrm{true}}$ (4)
 > $G≔\mathrm{DicyclicGroup}\left(6\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}{,}{4}\right)\left({5}{,}{6}{,}{7}{,}{8}\right)\left({9}{,}{10}{,}{11}\right){,}\left({1}{,}{7}{,}{3}{,}{5}\right)\left({2}{,}{6}{,}{4}{,}{8}\right)\left({9}{,}{11}\right)⟩$ (5)
 > $Z≔\mathrm{Center}\left(G\right)$
 ${Z}{≔}{Z}{}\left(⟨\left({1}{,}{2}{,}{3}{,}{4}\right)\left({5}{,}{6}{,}{7}{,}{8}\right)\left({9}{,}{10}{,}{11}\right){,}\left({1}{,}{7}{,}{3}{,}{5}\right)\left({2}{,}{6}{,}{4}{,}{8}\right)\left({9}{,}{11}\right)⟩\right)$ (6)
 > $\mathrm{Generators}\left(Z\right)$
 $\left[\left({1}{,}{3}\right)\left({2}{,}{4}\right)\left({5}{,}{7}\right)\left({6}{,}{8}\right)\right]$ (7)

Compatibility

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