 RandomSmallGroup - Maple Help

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GroupTheory

 RandomSmallGroup
 return a random group from the database of small groups Calling Sequence RandomSmallGroup( idopt, ordopt, formopt ) Parameters

 idopt - (optional) option of the form id = true (or just id) or id = false ordopt - (optional) option of the form maxorder = n, for a positive integer n formopt - (optional) option of the form form = "permgroup" (default), form = "fpgroup" or form = "id" Description

 • The RandomSmallGroup() command returns a randomly selected group from the database of small groups, as a permutation group.
 • The id option controls how the group is selected. If the option id = true (or just id) is passed, then a randomly selected order in the range 1 .. 511 is first selected, and then, within the groups of that order, a random group is returned. If the id = false option is passed, then a truly (pseudo-)randomly selected group is returned from the database of small groups.  Note that most groups in the small groups database have order equal to $256$, so this is usually not what is wanted, which is why the default option is id = true.
 • The form option determines the form of what is returned. By default, a permutation group is returned. To have a finitely presented group returned, use the form = "fpgroup" option. Sometimes, only a valid small group ID is required, in which case, use the form = "id" option. Examples

 ${"SEED ="}{,}{66302275552}$ (1)
 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{RandomSmallGroup}\left(\right):$
 > $\mathrm{RandomSmallGroup}\left('\mathrm{id}'=\mathrm{false}\right)$
 ${\mathrm{< a permutation group on 256 letters with 8 generators >}}$ (2)
 > $G≔\mathrm{RandomSmallGroup}\left('\mathrm{id}','\mathrm{maxorder}'=200\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right)\left({3}{,}{8}\right)\left({4}{,}{7}\right)\left({5}{,}{10}\right)\left({6}{,}{9}\right)\left({11}{,}{22}\right)\left({12}{,}{24}\right)\left({13}{,}{23}\right)\left({14}{,}{19}\right)\left({15}{,}{21}\right)\left({16}{,}{20}\right)\left({17}{,}{26}\right)\left({18}{,}{25}\right)\left({27}{,}{42}\right)\left({28}{,}{41}\right)\left({29}{,}{44}\right)\left({30}{,}{43}\right)\left({31}{,}{38}\right)\left({32}{,}{37}\right)\left({33}{,}{40}\right)\left({34}{,}{39}\right)\left({35}{,}{46}\right)\left({36}{,}{45}\right)\left({47}{,}{60}\right)\left({48}{,}{59}\right)\left({49}{,}{62}\right)\left({50}{,}{61}\right)\left({51}{,}{56}\right)\left({52}{,}{55}\right)\left({53}{,}{58}\right)\left({54}{,}{57}\right)\left({63}{,}{70}\right)\left({64}{,}{69}\right)\left({65}{,}{68}\right)\left({66}{,}{67}\right){,}\left({1}{,}{3}{,}{11}{,}{14}{,}{4}\right)\left({2}{,}{7}{,}{19}{,}{22}{,}{8}\right)\left({5}{,}{12}{,}{27}{,}{31}{,}{15}\right)\left({6}{,}{13}{,}{28}{,}{32}{,}{16}\right)\left({9}{,}{20}{,}{37}{,}{41}{,}{23}\right)\left({10}{,}{21}{,}{38}{,}{42}{,}{24}\right)\left({17}{,}{29}{,}{47}{,}{51}{,}{33}\right)\left({18}{,}{30}{,}{48}{,}{52}{,}{34}\right)\left({25}{,}{39}{,}{55}{,}{59}{,}{43}\right)\left({26}{,}{40}{,}{56}{,}{60}{,}{44}\right)\left({35}{,}{49}{,}{63}{,}{65}{,}{53}\right)\left({36}{,}{50}{,}{64}{,}{66}{,}{54}\right)\left({45}{,}{57}{,}{67}{,}{69}{,}{61}\right)\left({46}{,}{58}{,}{68}{,}{70}{,}{62}\right){,}\left({1}{,}{5}{,}{17}{,}{35}{,}{36}{,}{18}{,}{6}\right)\left({2}{,}{9}{,}{25}{,}{45}{,}{46}{,}{26}{,}{10}\right)\left({3}{,}{12}{,}{29}{,}{49}{,}{50}{,}{30}{,}{13}\right)\left({4}{,}{15}{,}{33}{,}{53}{,}{54}{,}{34}{,}{16}\right)\left({7}{,}{20}{,}{39}{,}{57}{,}{58}{,}{40}{,}{21}\right)\left({8}{,}{23}{,}{43}{,}{61}{,}{62}{,}{44}{,}{24}\right)\left({11}{,}{27}{,}{47}{,}{63}{,}{64}{,}{48}{,}{28}\right)\left({14}{,}{31}{,}{51}{,}{65}{,}{66}{,}{52}{,}{32}\right)\left({19}{,}{37}{,}{55}{,}{67}{,}{68}{,}{56}{,}{38}\right)\left({22}{,}{41}{,}{59}{,}{69}{,}{70}{,}{60}{,}{42}\right)⟩$ (3)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${70}$ (4)
 > $G≔\mathrm{RandomSmallGroup}\left('\mathrm{form}'="fpgroup"\right)$
 ${G}{≔}⟨{}{\mathrm{a1}}{}{\mid }{}{{\mathrm{a1}}}^{{484}}{}⟩$ (5)
 > $G≔\mathrm{RandomSmallGroup}\left('\mathrm{maxorder}'=100,'\mathrm{form}'="fpgroup"\right)$
 ${G}{≔}⟨{}{\mathrm{_a}}{,}{\mathrm{_b}}{}{\mid }{}{{\mathrm{_b}}}^{{2}}{,}{\mathrm{_a}}{}{\mathrm{_b}}{}{\mathrm{_a}}{}{\mathrm{_b}}{,}{{\mathrm{_a}}}^{{38}}{}⟩$ (6)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${76}$ (7)
 > $\mathrm{RandomSmallGroup}\left('\mathrm{form}'="id"\right)$
 $\left[{72}{,}{5}\right]$ (8)
 > $\mathrm{id}≔\mathrm{RandomSmallGroup}\left('\mathrm{form}'='\mathrm{id}','\mathrm{maxorder}'=150\right)$
 ${\mathrm{id}}{≔}\left[{27}{,}{1}\right]$ (9)
 > $\mathrm{evalb}\left({\mathrm{id}}_{1}\le 150\right)$
 ${\mathrm{true}}$ (10) Compatibility

 • The GroupTheory[RandomSmallGroup] command was introduced in Maple 2017.