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GroupTheory

 MetacyclicGroup
 construct a finite metacyclic group

 Calling Sequence MetacyclicGroup(m, n, k) MetacyclicGroup(m, n, k, s)

Parameters

 m - a positive integer n - a positive integer k - a positive integer s - (optional) equation of the form form= "fpgroup" or form = "permgroup" (default)

Description

 • A group metacyclic if it has a cyclic normal subgroup the quotient by which is also cyclic. Every such group $G$ can be generated by two elements $a$ and $b$, with the subgroup $⟨a⟩$ normal in $G$. The group $G$ is then determined by the action of $⟨b⟩$ on $⟨a⟩$. Since $⟨a⟩$ is normal in $G$, it follows that the conjugate ${a}^{b}$ belongs to $⟨a⟩$ so there is a positive integer $k$ for which ${a}^{b}={a}^{-k}$. Thus, a finite metacyclic group $G$ is completely determined by the orders of $a$ and $b$ and the integer $k$.
 • The MetacyclicGroup( m, n, k ) command constructs a metacyclic group with generators $a$ and $b$ as described above, such that ${a}^{b}={a}^{-k}$, and where ${a}^{n}=1$ and ${b}^{m}=1$.
 • Note that the generators $a$ and $b$ need not have orders $n$ and $m$, respectively, but that their orders are necessarily divisors of $n$ and $m$.
 • By default, a permutation group is returned, but you can create a finitely presented group by passing the 'form' = "fpgroup" option.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{MetacyclicGroup}\left(6,8,5\right)$
 $⟨\left({1}{,}{2}{,}{6}{,}{14}{,}{9}{,}{3}\right)\left({4}{,}{7}{,}{15}{,}{25}{,}{19}{,}{10}\right)\left({5}{,}{8}{,}{16}{,}{26}{,}{20}{,}{11}\right)\left({12}{,}{17}{,}{27}{,}{36}{,}{31}{,}{21}\right)\left({13}{,}{18}{,}{28}{,}{37}{,}{32}{,}{22}\right)\left({23}{,}{29}{,}{38}{,}{44}{,}{41}{,}{33}\right)\left({24}{,}{30}{,}{39}{,}{45}{,}{42}{,}{34}\right)\left({35}{,}{40}{,}{46}{,}{48}{,}{47}{,}{43}\right){,}\left({1}{,}{4}{,}{12}{,}{23}{,}{35}{,}{24}{,}{13}{,}{5}\right)\left({2}{,}{7}{,}{17}{,}{29}{,}{40}{,}{30}{,}{18}{,}{8}\right)\left({3}{,}{10}{,}{21}{,}{33}{,}{43}{,}{34}{,}{22}{,}{11}\right)\left({6}{,}{15}{,}{27}{,}{38}{,}{46}{,}{39}{,}{28}{,}{16}\right)\left({9}{,}{19}{,}{31}{,}{41}{,}{47}{,}{42}{,}{32}{,}{20}\right)\left({14}{,}{25}{,}{36}{,}{44}{,}{48}{,}{45}{,}{37}{,}{26}\right)⟩$ (1)
 > $\mathrm{MetacyclicGroup}\left(6,8,5,'\mathrm{form}'="permgroup"\right)$
 $⟨\left({1}{,}{2}{,}{6}{,}{14}{,}{9}{,}{3}\right)\left({4}{,}{7}{,}{15}{,}{25}{,}{19}{,}{10}\right)\left({5}{,}{8}{,}{16}{,}{26}{,}{20}{,}{11}\right)\left({12}{,}{17}{,}{27}{,}{36}{,}{31}{,}{21}\right)\left({13}{,}{18}{,}{28}{,}{37}{,}{32}{,}{22}\right)\left({23}{,}{29}{,}{38}{,}{44}{,}{41}{,}{33}\right)\left({24}{,}{30}{,}{39}{,}{45}{,}{42}{,}{34}\right)\left({35}{,}{40}{,}{46}{,}{48}{,}{47}{,}{43}\right){,}\left({1}{,}{4}{,}{12}{,}{23}{,}{35}{,}{24}{,}{13}{,}{5}\right)\left({2}{,}{7}{,}{17}{,}{29}{,}{40}{,}{30}{,}{18}{,}{8}\right)\left({3}{,}{10}{,}{21}{,}{33}{,}{43}{,}{34}{,}{22}{,}{11}\right)\left({6}{,}{15}{,}{27}{,}{38}{,}{46}{,}{39}{,}{28}{,}{16}\right)\left({9}{,}{19}{,}{31}{,}{41}{,}{47}{,}{42}{,}{32}{,}{20}\right)\left({14}{,}{25}{,}{36}{,}{44}{,}{48}{,}{45}{,}{37}{,}{26}\right)⟩$ (2)
 > $\mathrm{MetacyclicGroup}\left(6,8,5,'\mathrm{form}'="fpgroup"\right)$
 $⟨{}{a}{,}{b}{}{\mid }{}{{a}}^{{6}}{,}{{b}}^{{8}}{,}{{b}}^{{-1}}{}{a}{}{b}{}{{a}}^{{5}}{}⟩$ (3)

In the following example, the first parameter $6$ is a proper multiple of the order of the corresponding generator.

 > $a,b≔\mathrm{op}\left(\mathrm{Generators}\left(\mathrm{MetacyclicGroup}\left(6,8,4\right)\right)\right)$
 ${a}{,}{b}{≔}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{8}{,}{6}\right)\left({5}{,}{9}{,}{7}\right)\left({10}{,}{12}{,}{14}\right)\left({11}{,}{13}{,}{15}\right)\left({16}{,}{20}{,}{18}\right)\left({17}{,}{21}{,}{19}\right)\left({22}{,}{23}{,}{24}\right){,}\left({1}{,}{4}{,}{10}{,}{16}{,}{22}{,}{17}{,}{11}{,}{5}\right)\left({2}{,}{6}{,}{12}{,}{18}{,}{23}{,}{19}{,}{13}{,}{7}\right)\left({3}{,}{8}{,}{14}{,}{20}{,}{24}{,}{21}{,}{15}{,}{9}\right)$ (4)
 > $\mathrm{PermOrder}\left(a\right)$
 ${3}$ (5)
 > $\mathrm{PermOrder}\left(b\right)$
 ${8}$ (6)

Compatibility

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