DicyclicGroup - Maple Help
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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 $n>1$. It is defined by a presentation of the form

$⟨xy,|,{x}^{n}={y}^{2},,,{x}^{y}={x}^{-1}⟩$

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{DicyclicGroup}\left(6\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{DicyclicGroup}}{}\left({6}{,}{\mathrm{form}}{=}{"permgroup"}\right)$ (1)
 > $\mathrm{DicyclicGroup}\left(6,'\mathrm{form}'="permgroup"\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{DicyclicGroup}}{}\left({6}{,}{\mathrm{form}}{=}{"permgroup"}\right)$ (2)
 > $\mathrm{DicyclicGroup}\left(6,'\mathrm{form}'="fpgroup"\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{DicyclicGroup}}{}\left({6}{,}{\mathrm{form}}{=}{"fpgroup"}\right)$ (3)
 > $\mathrm{IsNilpotent}\left(\mathrm{DicyclicGroup}\left(8{2}^{k}\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}k::\mathrm{posint}$
 ${\mathrm{true}}$ (4)
 > $G≔\mathrm{DicyclicGroup}\left(6\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{DicyclicGroup}}{}\left({6}{,}{\mathrm{form}}{=}{"permgroup"}\right)$ (5)
 > $Z≔\mathrm{Center}\left(G\right)$
 ${\mathrm{Centre}}{}\left({\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right)$ (6)
 > $\mathrm{Generators}\left(Z\right)$
 $\left[{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right]$ (7)
 > $S≔\mathrm{SylowSubgroup}\left(2,G\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{PermutationGroup}}{}\left(\left\{{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}{,}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right\}{,}{\mathrm{degree}}{=}{11}{,}{\mathrm{supergroup}}{=}{\mathrm{GroupTheory}}{:-}{\mathrm{PermutationGroup}}{}\left(\left\{{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}{,}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right\}{,}{\mathrm{degree}}{=}{11}\right)\right)$ (8)

For odd $n$, the dicyclic group of order $4n$ is a Z-group (all Sylow subgroups are cyclic).

 > $\mathrm{IsCyclicSylowGroup}\left(\mathrm{DicyclicGroup}\left(7\right)\right)$
 ${\mathrm{true}}$ (9)

But, for even $n$, the Sylow $2$-subgroups are generalized quaternion groups.

 > $\mathrm{IsQuaternionGroup}\left(\mathrm{SylowSubgroup}\left(2,\mathrm{DicyclicGroup}\left(12\right)\right)\right)$
 ${\mathrm{true}}$ (10)
 > $\mathrm{Display}\left(\mathrm{CharacterTable}\left(\mathrm{DicyclicGroup}\left(5\right)\right)\right)$

 C 1a 2a 4a 4b 5a 5b 10a 10b |C| 1 1 5 5 2 2 2 2 $\mathrm{χ__1}$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $\mathrm{χ__2}$ $1$ $-1$ $-I$ $I$ $1$ $1$ $-1$ $-1$ $\mathrm{χ__3}$ $1$ $-1$ $I$ $-I$ $1$ $1$ $-1$ $-1$ $\mathrm{χ__4}$ $1$ $1$ $-1$ $-1$ $1$ $1$ $1$ $1$ $\mathrm{χ__5}$ $2$ $-2$ $0$ $0$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ $-{\left(-1\right)}^{3}{5}}+{\left(-1\right)}^{2}{5}}+1$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}$ $\mathrm{χ__6}$ $2$ $-2$ $0$ $0$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}$ $-{\left(-1\right)}^{3}{5}}+{\left(-1\right)}^{2}{5}}+1$ $\mathrm{χ__7}$ $2$ $2$ $0$ $0$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ $\mathrm{χ__8}$ $2$ $2$ $0$ $0$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$

Compatibility

 • 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.