RandomSmallGroup - Maple Help

Home : Support : Online Help : Mathematics : Group Theory : RandomSmallGroup

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

 ${"SEED ="}{,}{59308905600}$ (1)
 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{RandomSmallGroup}\left(\right)$
 $⟨\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}{,}{62}\right)\left({48}{,}{61}\right)\left({49}{,}{64}\right)\left({50}{,}{63}\right)\left({51}{,}{58}\right)\left({52}{,}{57}\right)\left({53}{,}{60}\right)\left({54}{,}{59}\right)\left({55}{,}{66}\right)\left({56}{,}{65}\right)\left({67}{,}{82}\right)\left({68}{,}{81}\right)\left({69}{,}{84}\right)\left({70}{,}{83}\right)\left({71}{,}{78}\right)\left({72}{,}{77}\right)\left({73}{,}{80}\right)\left({74}{,}{79}\right)\left({75}{,}{86}\right)\left({76}{,}{85}\right)\left({87}{,}{102}\right)\left({88}{,}{101}\right)\left({89}{,}{104}\right)\left({90}{,}{103}\right)\left({91}{,}{98}\right)\left({92}{,}{97}\right)\left({93}{,}{100}\right)\left({94}{,}{99}\right)\left({95}{,}{106}\right)\left({96}{,}{105}\right)\left({107}{,}{122}\right)\left({108}{,}{121}\right)\left({109}{,}{124}\right)\left({110}{,}{123}\right)\left({111}{,}{118}\right)\left({112}{,}{117}\right)\left({113}{,}{120}\right)\left({114}{,}{119}\right)\left({115}{,}{126}\right)\left({116}{,}{125}\right)\left({127}{,}{142}\right)\left({128}{,}{141}\right)\left({129}{,}{144}\right)\left({130}{,}{143}\right)\left({131}{,}{138}\right)\left({132}{,}{137}\right)\left({133}{,}{140}\right)\left({134}{,}{139}\right)\left({135}{,}{146}\right)\left({136}{,}{145}\right)\left({147}{,}{162}\right)\left({148}{,}{161}\right)\left({149}{,}{164}\right)\left({150}{,}{163}\right)\left({151}{,}{158}\right)\left({152}{,}{157}\right)\left({153}{,}{160}\right)\left({154}{,}{159}\right)\left({155}{,}{166}\right)\left({156}{,}{165}\right)\left({167}{,}{180}\right)\left({168}{,}{179}\right)\left({169}{,}{182}\right)\left({170}{,}{181}\right)\left({171}{,}{176}\right)\left({172}{,}{175}\right)\left({173}{,}{178}\right)\left({174}{,}{177}\right)\left({183}{,}{190}\right)\left({184}{,}{189}\right)\left({185}{,}{188}\right)\left({186}{,}{187}\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}{,}{57}{,}{61}{,}{43}\right)\left({26}{,}{40}{,}{58}{,}{62}{,}{44}\right)\left({35}{,}{49}{,}{67}{,}{71}{,}{53}\right)\left({36}{,}{50}{,}{68}{,}{72}{,}{54}\right)\left({45}{,}{59}{,}{77}{,}{81}{,}{63}\right)\left({46}{,}{60}{,}{78}{,}{82}{,}{64}\right)\left({55}{,}{69}{,}{87}{,}{91}{,}{73}\right)\left({56}{,}{70}{,}{88}{,}{92}{,}{74}\right)\left({65}{,}{79}{,}{97}{,}{101}{,}{83}\right)\left({66}{,}{80}{,}{98}{,}{102}{,}{84}\right)\left({75}{,}{89}{,}{107}{,}{111}{,}{93}\right)\left({76}{,}{90}{,}{108}{,}{112}{,}{94}\right)\left({85}{,}{99}{,}{117}{,}{121}{,}{103}\right)\left({86}{,}{100}{,}{118}{,}{122}{,}{104}\right)\left({95}{,}{109}{,}{127}{,}{131}{,}{113}\right)\left({96}{,}{110}{,}{128}{,}{132}{,}{114}\right)\left({105}{,}{119}{,}{137}{,}{141}{,}{123}\right)\left({106}{,}{120}{,}{138}{,}{142}{,}{124}\right)\left({115}{,}{129}{,}{147}{,}{151}{,}{133}\right)\left({116}{,}{130}{,}{148}{,}{152}{,}{134}\right)\left({125}{,}{139}{,}{157}{,}{161}{,}{143}\right)\left({126}{,}{140}{,}{158}{,}{162}{,}{144}\right)\left({135}{,}{149}{,}{167}{,}{171}{,}{153}\right)\left({136}{,}{150}{,}{168}{,}{172}{,}{154}\right)\left({145}{,}{159}{,}{175}{,}{179}{,}{163}\right)\left({146}{,}{160}{,}{176}{,}{180}{,}{164}\right)\left({155}{,}{169}{,}{183}{,}{185}{,}{173}\right)\left({156}{,}{170}{,}{184}{,}{186}{,}{174}\right)\left({165}{,}{177}{,}{187}{,}{189}{,}{181}\right)\left({166}{,}{178}{,}{188}{,}{190}{,}{182}\right){,}\left({1}{,}{5}{,}{17}{,}{35}{,}{55}{,}{75}{,}{95}{,}{115}{,}{135}{,}{155}{,}{156}{,}{136}{,}{116}{,}{96}{,}{76}{,}{56}{,}{36}{,}{18}{,}{6}\right)\left({2}{,}{9}{,}{25}{,}{45}{,}{65}{,}{85}{,}{105}{,}{125}{,}{145}{,}{165}{,}{166}{,}{146}{,}{126}{,}{106}{,}{86}{,}{66}{,}{46}{,}{26}{,}{10}\right)\left({3}{,}{12}{,}{29}{,}{49}{,}{69}{,}{89}{,}{109}{,}{129}{,}{149}{,}{169}{,}{170}{,}{150}{,}{130}{,}{110}{,}{90}{,}{70}{,}{50}{,}{30}{,}{13}\right)\left({4}{,}{15}{,}{33}{,}{53}{,}{73}{,}{93}{,}{113}{,}{133}{,}{153}{,}{173}{,}{174}{,}{154}{,}{134}{,}{114}{,}{94}{,}{74}{,}{54}{,}{34}{,}{16}\right)\left({7}{,}{20}{,}{39}{,}{59}{,}{79}{,}{99}{,}{119}{,}{139}{,}{159}{,}{177}{,}{178}{,}{160}{,}{140}{,}{120}{,}{100}{,}{80}{,}{60}{,}{40}{,}{21}\right)\left({8}{,}{23}{,}{43}{,}{63}{,}{83}{,}{103}{,}{123}{,}{143}{,}{163}{,}{181}{,}{182}{,}{164}{,}{144}{,}{124}{,}{104}{,}{84}{,}{64}{,}{44}{,}{24}\right)\left({11}{,}{27}{,}{47}{,}{67}{,}{87}{,}{107}{,}{127}{,}{147}{,}{167}{,}{183}{,}{184}{,}{168}{,}{148}{,}{128}{,}{108}{,}{88}{,}{68}{,}{48}{,}{28}\right)\left({14}{,}{31}{,}{51}{,}{71}{,}{91}{,}{111}{,}{131}{,}{151}{,}{171}{,}{185}{,}{186}{,}{172}{,}{152}{,}{132}{,}{112}{,}{92}{,}{72}{,}{52}{,}{32}\right)\left({19}{,}{37}{,}{57}{,}{77}{,}{97}{,}{117}{,}{137}{,}{157}{,}{175}{,}{187}{,}{188}{,}{176}{,}{158}{,}{138}{,}{118}{,}{98}{,}{78}{,}{58}{,}{38}\right)\left({22}{,}{41}{,}{61}{,}{81}{,}{101}{,}{121}{,}{141}{,}{161}{,}{179}{,}{189}{,}{190}{,}{180}{,}{162}{,}{142}{,}{122}{,}{102}{,}{82}{,}{62}{,}{42}\right)⟩$ (2)
 > $\mathrm{RandomSmallGroup}\left('\mathrm{id}'=\mathrm{false}\right)$
 ${\mathrm{⟨a permutation group on 256 letters with 8 generators⟩}}$ (3)
 > $G≔\mathrm{RandomSmallGroup}\left('\mathrm{id}','\mathrm{maxorder}'=200\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{4}{,}{6}{,}{8}{,}{10}{,}{12}{,}{14}{,}{16}{,}{18}{,}{20}{,}{22}{,}{24}{,}{26}{,}{28}{,}{30}{,}{32}{,}{34}{,}{36}{,}{38}{,}{40}{,}{42}{,}{44}{,}{46}{,}{48}{,}{50}{,}{52}{,}{54}{,}{56}{,}{58}{,}{60}{,}{62}{,}{64}{,}{66}{,}{68}{,}{70}{,}{72}{,}{74}{,}{76}{,}{78}{,}{80}{,}{82}{,}{84}{,}{86}{,}{88}{,}{90}{,}{92}{,}{94}{,}{96}{,}{98}{,}{100}{,}{102}{,}{104}{,}{106}{,}{108}{,}{110}{,}{112}{,}{114}{,}{116}{,}{118}{,}{120}{,}{122}{,}{124}{,}{126}{,}{128}{,}{130}{,}{132}{,}{134}{,}{136}{,}{138}{,}{140}{,}{142}{,}{144}{,}{146}{,}{148}{,}{150}{,}{152}{,}{154}{,}{156}{,}{158}{,}{160}{,}{162}{,}{164}{,}{166}{,}{168}{,}{170}{,}{172}{,}{173}{,}{171}{,}{169}{,}{167}{,}{165}{,}{163}{,}{161}{,}{159}{,}{157}{,}{155}{,}{153}{,}{151}{,}{149}{,}{147}{,}{145}{,}{143}{,}{141}{,}{139}{,}{137}{,}{135}{,}{133}{,}{131}{,}{129}{,}{127}{,}{125}{,}{123}{,}{121}{,}{119}{,}{117}{,}{115}{,}{113}{,}{111}{,}{109}{,}{107}{,}{105}{,}{103}{,}{101}{,}{99}{,}{97}{,}{95}{,}{93}{,}{91}{,}{89}{,}{87}{,}{85}{,}{83}{,}{81}{,}{79}{,}{77}{,}{75}{,}{73}{,}{71}{,}{69}{,}{67}{,}{65}{,}{63}{,}{61}{,}{59}{,}{57}{,}{55}{,}{53}{,}{51}{,}{49}{,}{47}{,}{45}{,}{43}{,}{41}{,}{39}{,}{37}{,}{35}{,}{33}{,}{31}{,}{29}{,}{27}{,}{25}{,}{23}{,}{21}{,}{19}{,}{17}{,}{15}{,}{13}{,}{11}{,}{9}{,}{7}{,}{5}{,}{3}\right)⟩$ (4)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${173}$ (5)
 > $G≔\mathrm{RandomSmallGroup}\left('\mathrm{form}'="fpgroup"\right)$
 ${G}{≔}⟨{}{\mathrm{a1}}{,}{\mathrm{a2}}{,}{\mathrm{a3}}{,}{\mathrm{a4}}{}{\mid }{}{{\mathrm{a1}}}^{{2}}{,}{{\mathrm{a2}}}^{{2}}{,}{{\mathrm{a3}}}^{{2}}{,}{{\mathrm{a1}}}^{{-1}}{}{{\mathrm{a2}}}^{{-1}}{}{\mathrm{a1}}{}{\mathrm{a2}}{,}{{\mathrm{a1}}}^{{-1}}{}{{\mathrm{a3}}}^{{-1}}{}{\mathrm{a1}}{}{\mathrm{a3}}{,}{{\mathrm{a1}}}^{{-1}}{}{{\mathrm{a4}}}^{{-1}}{}{\mathrm{a1}}{}{\mathrm{a4}}{,}{{\mathrm{a2}}}^{{-1}}{}{{\mathrm{a1}}}^{{-1}}{}{\mathrm{a2}}{}{\mathrm{a1}}{,}{{\mathrm{a2}}}^{{-1}}{}{{\mathrm{a3}}}^{{-1}}{}{\mathrm{a2}}{}{\mathrm{a3}}{,}{{\mathrm{a2}}}^{{-1}}{}{{\mathrm{a4}}}^{{-1}}{}{\mathrm{a2}}{}{\mathrm{a4}}{,}{{\mathrm{a3}}}^{{-1}}{}{{\mathrm{a1}}}^{{-1}}{}{\mathrm{a3}}{}{\mathrm{a1}}{,}{{\mathrm{a3}}}^{{-1}}{}{{\mathrm{a2}}}^{{-1}}{}{\mathrm{a3}}{}{\mathrm{a2}}{,}{{\mathrm{a3}}}^{{-1}}{}{{\mathrm{a4}}}^{{-1}}{}{\mathrm{a3}}{}{\mathrm{a4}}{,}{{\mathrm{a4}}}^{{-1}}{}{{\mathrm{a1}}}^{{-1}}{}{\mathrm{a4}}{}{\mathrm{a1}}{,}{{\mathrm{a4}}}^{{-1}}{}{{\mathrm{a2}}}^{{-1}}{}{\mathrm{a4}}{}{\mathrm{a2}}{,}{{\mathrm{a4}}}^{{-1}}{}{{\mathrm{a3}}}^{{-1}}{}{\mathrm{a4}}{}{\mathrm{a3}}{,}{{\mathrm{a4}}}^{{37}}{}⟩$ (6)
 > $G≔\mathrm{RandomSmallGroup}\left('\mathrm{maxorder}'=100,'\mathrm{form}'="fpgroup"\right)$
 ${G}{≔}⟨{}{\mathrm{_a}}{,}{\mathrm{_b}}{}{\mid }{}{{\mathrm{_b}}}^{{3}}{,}{{\mathrm{_a}}}^{{-1}}{}{{\mathrm{_b}}}^{{-1}}{}{\mathrm{_a}}{}{\mathrm{_b}}{,}{{\mathrm{_a}}}^{{9}}{}⟩$ (7)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${27}$ (8)
 > $\mathrm{RandomSmallGroup}\left('\mathrm{form}'="id"\right)$
 $\left[{153}{,}{2}\right]$ (9)
 > $\mathrm{id}≔\mathrm{RandomSmallGroup}\left('\mathrm{form}'='\mathrm{id}','\mathrm{maxorder}'=150\right)$
 ${\mathrm{id}}{≔}\left[{97}{,}{1}\right]$ (10)
 > $\mathrm{evalb}\left(\mathrm{id}\left[1\right]\le 150\right)$
 ${\mathrm{true}}$ (11)

Compatibility

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