GroupTheory/ProjectiveSpecialOrthogonalGroup - Maple Help

GroupTheory

 ProjectiveSpecialOrthogonalGroup
 construct a permutation group isomorphic to a projective special orthogonal group

 Calling Sequence ProjectiveSpecialOrthogonalGroup(d, n, q) PSO(d, n, q)

Parameters

 d - 0, 1 or -1 n - a positive integer q - power of a prime number

Description

 • The projective special orthogonal group $PSO\left(d,n,q\right)$ is the quotient of the special orthogonal group $SO\left(d,n,q\right)$ by its center. The value of $d$ must be $0$ for odd $n$, or $1$ or $-1$ for even $n$.
 • The ProjectiveSpecialOrthogonalGroup( d, n, q ) command returns a permutation group isomorphic to the projective special orthogonal group $PSO\left(d,n,q\right)$ .
 • The PSO( d, n, q ) command is provided as an alias.
 • If the argument q is not a prime power (and is non-numeric), then a symbolic group representing $PSO\left(d,n,q\right)$ is returned.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{ProjectiveSpecialOrthogonalGroup}\left(-1,2,7\right)$
 ${G}{≔}{{C}}_{{4}}$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${4}$ (2)
 > $\mathrm{IsCyclic}\left(G\right)$
 ${\mathrm{true}}$ (3)
 > $G≔\mathrm{ProjectiveSpecialOrthogonalGroup}\left(1,2,8\right)$
 ${G}{≔}{{\mathbf{D}}}_{{7}}$ (4)
 > $\mathrm{AreIsomorphic}\left(G,\mathrm{DihedralGroup}\left(7\right)\right)$
 ${\mathrm{true}}$ (5)
 > $G≔\mathrm{PSO}\left(0,3,3\right)$
 ${G}{≔}⟨\left({2}{,}{3}\right)\left({6}{,}{11}\right)\left({7}{,}{12}\right)\left({8}{,}{9}\right){,}\left({1}{,}{2}{,}{4}\right)\left({5}{,}{6}{,}{12}\right)\left({7}{,}{11}{,}{13}\right)\left({8}{,}{9}{,}{10}\right){,}\left({1}{,}{3}\right)\left({5}{,}{7}\right)\left({6}{,}{13}\right)\left({8}{,}{10}\right)⟩$ (6)
 > $\mathrm{AreIsomorphic}\left(G,\mathrm{Symm}\left(4\right)\right)$
 ${\mathrm{true}}$ (7)
 > $G≔\mathrm{PSO}\left(-1,4,9\right)$
 ${G}{≔}⟨\left({2}{,}{15}{,}{41}{,}{19}{,}{51}{,}{3}{,}{17}{,}{46}\right)\left({5}{,}{20}{,}{52}{,}{130}{,}{113}{,}{16}{,}{43}{,}{114}\right)\left({7}{,}{23}{,}{60}{,}{146}{,}{98}{,}{235}{,}{452}{,}{158}\right)\left({9}{,}{26}{,}{69}{,}{167}{,}{323}{,}{545}{,}{572}{,}{378}\right)\left({10}{,}{29}{,}{78}{,}{190}{,}{80}{,}{194}{,}{31}{,}{84}\right)\left({11}{,}{12}{,}{32}{,}{85}{,}{203}{,}{408}{,}{497}{,}{39}\right)\left({13}{,}{35}{,}{94}{,}{227}{,}{287}{,}{252}{,}{284}{,}{508}\right)\left({14}{,}{38}{,}{103}{,}{247}{,}{479}{,}{709}{,}{259}{,}{491}\right)\left({18}{,}{48}{,}{125}{,}{275}{,}{57}{,}{56}{,}{141}{,}{280}\right)\left({21}{,}{55}{,}{139}{,}{127}{,}{119}{,}{44}{,}{116}{,}{22}\right)\left({24}{,}{63}{,}{154}{,}{299}{,}{520}{,}{517}{,}{315}{,}{500}\right)\left({25}{,}{66}{,}{162}{,}{316}{,}{537}{,}{458}{,}{532}{,}{450}\right)\left({27}{,}{72}{,}{175}{,}{345}{,}{562}{,}{743}{,}{782}{,}{392}\right)\left({28}{,}{75}{,}{182}{,}{363}{,}{616}{,}{787}{,}{620}{,}{597}\right)\left({30}{,}{81}{,}{196}{,}{393}{,}{553}{,}{676}{,}{355}{,}{407}\right)\left({33}{,}{88}{,}{212}{,}{264}{,}{499}{,}{511}{,}{724}{,}{431}\right)\left({34}{,}{91}{,}{219}{,}{441}{,}{699}{,}{288}{,}{40}{,}{108}\right)\left({36}{,}{97}{,}{65}{,}{159}{,}{263}{,}{472}{,}{477}{,}{541}\right)\left({37}{,}{100}{,}{239}{,}{298}{,}{296}{,}{294}{,}{516}{,}{464}\right)\left({42}{,}{112}{,}{270}{,}{279}{,}{276}{,}{129}{,}{53}{,}{133}\right)\left({45}{,}{118}{,}{123}{,}{272}{,}{281}{,}{49}{,}{50}{,}{128}\right)\left({47}{,}{122}{,}{137}{,}{134}{,}{277}{,}{120}{,}{126}{,}{115}\right)\left({54}{,}{136}{,}{273}{,}{282}{,}{131}{,}{271}{,}{59}{,}{145}\right)\left({58}{,}{143}{,}{117}{,}{111}{,}{268}{,}{121}{,}{274}{,}{144}\right)\left({61}{,}{149}{,}{241}{,}{322}{,}{544}{,}{319}{,}{422}{,}{152}\right)\left({62}{,}{151}{,}{292}{,}{233}{,}{459}{,}{229}{,}{68}{,}{165}\right)\left({64}{,}{157}{,}{306}{,}{236}{,}{462}{,}{166}{,}{320}{,}{67}\right)\left({71}{,}{172}{,}{337}{,}{573}{,}{649}{,}{586}{,}{77}{,}{188}\right)\left({73}{,}{177}{,}{351}{,}{601}{,}{555}{,}{581}{,}{763}{,}{341}\right)\left({74}{,}{179}{,}{349}{,}{598}{,}{778}{,}{329}{,}{558}{,}{747}\right)\left({76}{,}{185}{,}{370}{,}{626}{,}{653}{,}{748}{,}{780}{,}{818}\right)\left({79}{,}{191}{,}{383}{,}{648}{,}{170}{,}{331}{,}{181}{,}{360}\right)\left({82}{,}{199}{,}{399}{,}{384}{,}{650}{,}{643}{,}{797}{,}{83}\right)\left({86}{,}{206}{,}{208}{,}{417}{,}{687}{,}{418}{,}{101}{,}{242}\right)\left({87}{,}{209}{,}{419}{,}{234}{,}{460}{,}{253}{,}{93}{,}{224}\right)\left({89}{,}{214}{,}{430}{,}{160}\right)\left({90}{,}{216}{,}{435}{,}{95}{,}{230}{,}{109}{,}{262}{,}{495}\right)\left({92}{,}{221}{,}{446}{,}{701}{,}{246}{,}{107}{,}{257}{,}{425}\right)\left({96}{,}{232}{,}{164}{,}{153}{,}{297}{,}{148}{,}{102}{,}{245}\right)\left({99}{,}{213}{,}{416}{,}{470}{,}{522}{,}{494}{,}{521}{,}{218}\right)\left({104}{,}{249}{,}{475}{,}{715}{,}{482}{,}{291}{,}{434}{,}{694}\right)\left({105}{,}{251}{,}{414}{,}{442}{,}{409}{,}{478}{,}{110}{,}{265}\right)\left({106}{,}{254}{,}{484}{,}{718}{,}{258}{,}{490}{,}{698}{,}{321}\right)\left({124}{,}{142}{,}{138}{,}{140}{,}{132}{,}{278}{,}{135}{,}{269}\right)\left({147}{,}{285}{,}{461}{,}{207}{,}{415}{,}{501}{,}{440}{,}{696}\right)\left({150}{,}{289}{,}{309}{,}{530}{,}{313}{,}{506}{,}{722}{,}{713}\right)\left({155}{,}{301}{,}{496}{,}{719}{,}{293}{,}{436}{,}{318}{,}{453}\right)\left({156}{,}{304}{,}{525}{,}{690}{,}{445}{,}{686}{,}{161}{,}{314}\right)\left({163}{,}{317}{,}{539}{,}{474}{,}{714}{,}{483}{,}{240}{,}{411}\right)\left({168}{,}{326}{,}{201}{,}{404}{,}{198}{,}{397}{,}{186}{,}{373}\right)\left({169}{,}{328}{,}{556}{,}{613}{,}{642}{,}{369}{,}{174}{,}{342}\right)\left({171}{,}{334}{,}{567}{,}{625}{,}{552}{,}{739}{,}{336}{,}{571}\right)\left({173}{,}{340}{,}{579}{,}{745}{,}{815}{,}{609}{,}{359}{,}{607}\right)\left({176}{,}{348}{,}{596}{,}{389}{,}{570}{,}{338}{,}{574}{,}{760}\right)\left({178}{,}{354}{,}{580}{,}{757}{,}{402}{,}{674}{,}{765}{,}{791}\right)\left({180}{,}{358}{,}{382}{,}{647}{,}{379}{,}{560}{,}{750}{,}{557}\right)\left({183}{,}{365}{,}{332}{,}{563}{,}{636}{,}{795}{,}{582}{,}{387}\right)\left({184}{,}{367}{,}{551}{,}{627}{,}{547}{,}{734}{,}{189}{,}{380}\right)\left({187}{,}{376}{,}{635}{,}{735}{,}{197}{,}{395}{,}{564}{,}{754}\right)\left({192}{,}{386}{,}{377}{,}{638}{,}{372}{,}{630}{,}{764}{,}{606}\right)\left({193}{,}{388}{,}{347}{,}{594}{,}{755}{,}{675}{,}{587}{,}{610}\right)\left({195}{,}{391}{,}{659}{,}{753}{,}{790}{,}{362}{,}{614}{,}{357}\right)\left({200}{,}{202}{,}{406}{,}{678}{,}{742}{,}{775}{,}{405}{,}{634}\right)\left({204}{,}{410}{,}{524}{,}{711}{,}{688}{,}{256}{,}{486}{,}{444}\right)\left({205}{,}{413}{,}{492}{,}{502}{,}{489}{,}{423}{,}{211}{,}{424}\right)\left({210}{,}{421}{,}{682}{,}{720}{,}{681}{,}{468}{,}{226}{,}{448}\right)\left({215}{,}{432}{,}{510}{,}{488}{,}{691}{,}{451}{,}{536}{,}{693}\right)\left({217}{,}{437}{,}{439}{,}{523}{,}{438}{,}{695}{,}{712}{,}{808}\right)\left({220}{,}{443}{,}{454}{,}{526}{,}{540}{,}{721}{,}{238}{,}{467}\right)\left({222}{,}{4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}{339}{,}{578}{,}{572}{,}{647}{,}{608}{,}{556}{,}{627}\right)\left({176}{,}{350}{,}{568}{,}{758}{,}{759}{,}{611}{,}{784}{,}{785}\right)\left({177}{,}{353}{,}{606}{,}{735}{,}{795}{,}{815}{,}{402}{,}{602}\right)\left({179}{,}{357}{,}{610}{,}{743}{,}{813}{,}{792}{,}{383}{,}{645}\right)\left({182}{,}{342}{,}{584}{,}{612}{,}{586}{,}{323}{,}{547}{,}{736}\right)\left({183}{,}{366}{,}{623}{,}{630}{,}{674}{,}{601}{,}{780}{,}{340}\right)\left({185}{,}{372}{,}{631}{,}{757}{,}{365}{,}{622}{,}{197}{,}{359}\right)\left({191}{,}{385}{,}{604}{,}{782}{,}{786}{,}{565}{,}{755}{,}{659}\right)\left({192}{,}{387}{,}{370}{,}{590}{,}{555}{,}{745}{,}{754}{,}{580}\right)\left({195}{,}{392}{,}{599}{,}{558}{,}{749}{,}{773}{,}{594}{,}{368}\right)\left({199}{,}{401}{,}{673}{,}{756}{,}{326}{,}{553}{,}{742}{,}{730}\right)\left({202}{,}{407}{,}{663}{,}{643}{,}{644}{,}{667}{,}{661}{,}{679}\right)\left({207}{,}{416}{,}{686}{,}{520}{,}{724}{,}{721}{,}{263}{,}{498}\right)\left({217}{,}{438}{,}{437}{,}{523}{,}{712}{,}{808}{,}{728}{,}{439}\right)\left({222}{,}{411}{,}{444}{,}{293}{,}{515}{,}{312}{,}{514}{,}{258}\right)\left({225}{,}{311}{,}{533}{,}{716}{,}{710}{,}{807}{,}{703}{,}{449}\right)\left({228}{,}{453}{,}{704}{,}{250}{,}{483}{,}{534}{,}{698}{,}{512}\right)\left({231}{,}{456}{,}{706}{,}{727}{,}{809}{,}{455}{,}{705}{,}{480}\right)\left({237}{,}{466}{,}{486}{,}{505}{,}{496}{,}{490}{,}{317}{,}{248}\right)\left({249}{,}{482}{,}{289}{,}{430}{,}{503}{,}{309}{,}{433}{,}{693}\right)\left({254}{,}{485}{,}{719}{,}{474}{,}{266}{,}{447}{,}{410}{,}{683}\right)\left({255}{,}{427}{,}{527}{,}{301}{,}{524}{,}{718}{,}{538}{,}{528}\right)\left({299}{,}{521}{,}{499}{,}{415}{,}{314}{,}{435}{,}{689}{,}{541}\right)\left({302}{,}{428}{,}{484}{,}{436}{,}{535}{,}{726}{,}{688}{,}{539}\right)\left({324}{,}{549}{,}{348}{,}{374}{,}{389}{,}{657}{,}{801}{,}{762}\right)\left({328}{,}{557}{,}{367}{,}{378}{,}{364}{,}{620}{,}{737}{,}{812}\right)\left({329}{,}{345}{,}{591}{,}{391}{,}{583}{,}{360}{,}{343}{,}{587}\right)\left({331}{,}{562}{,}{753}{,}{793}{,}{349}{,}{388}{,}{656}{,}{770}\right)\left({335}{,}{570}{,}{741}{,}{371}\right)\left({338}{,}{576}{,}{617}{,}{789}{,}{666}{,}{600}{,}{769}{,}{596}\right)\left({351}{,}{386}{,}{653}{,}{629}{,}{791}{,}{635}{,}{640}{,}{563}\right)\left({354}{,}{607}{,}{764}{,}{581}{,}{376}{,}{637}{,}{738}{,}{748}\right)\left({355}{,}{603}{,}{554}{,}{744}{,}{678}{,}{397}{,}{665}{,}{803}\right)\left({361}{,}{613}{,}{573}{,}{363}{,}{618}{,}{750}{,}{588}{,}{380}\right)\left({373}{,}{399}{,}{668}{,}{577}{,}{575}{,}{634}{,}{752}{,}{561}\right)\left({393}{,}{633}{,}{406}{,}{672}{,}{404}{,}{677}{,}{751}{,}{650}\right)\left({395}{,}{626}{,}{763}{,}{817}{,}{636}{,}{638}{,}{579}{,}{605}\right)\left({396}{,}{546}{,}{733}{,}{776}{,}{592}{,}{774}{,}{664}{,}{800}\right)\left({400}{,}{671}{,}{566}{,}{732}{,}{794}{,}{654}{,}{779}{,}{811}\right)\left({488}{,}{713}{,}{692}{,}{715}\right)\left({509}{,}{723}{,}{513}{,}{697}{,}{708}{,}{700}{,}{806}{,}{729}\right)\left({574}{,}{761}{,}{771}{,}{760}{,}{810}{,}{619}{,}{767}{,}{628}\right)\left({598}{,}{766}{,}{768}{,}{658}{,}{614}{,}{652}{,}{675}{,}{648}\right)\left({641}{,}{746}{,}{740}{,}{804}{,}{819}{,}{680}{,}{796}{,}{805}\right)⟩$ (8)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${265680}$ (9)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{true}}$ (10)
 > $G≔\mathrm{PSO}\left(1,4,9\right)$
 ${G}{≔}⟨\left({2}{,}{15}{,}{39}{,}{19}{,}{46}{,}{3}{,}{17}{,}{44}\right)\left({5}{,}{20}{,}{48}{,}{108}{,}{187}{,}{14}{,}{16}{,}{41}\right)\left({7}{,}{23}{,}{57}{,}{119}{,}{207}{,}{328}{,}{371}{,}{368}\right)\left({9}{,}{26}{,}{66}{,}{145}{,}{257}{,}{383}{,}{375}{,}{172}\right)\left({10}{,}{29}{,}{75}{,}{169}{,}{77}{,}{173}{,}{31}{,}{81}\right)\left({11}{,}{12}{,}{32}{,}{83}{,}{186}{,}{198}{,}{33}{,}{86}\right)\left({13}{,}{35}{,}{89}{,}{92}{,}{192}{,}{91}{,}{116}{,}{18}\right)\left({21}{,}{51}{,}{40}{,}{98}{,}{38}{,}{95}{,}{34}{,}{42}\right)\left({22}{,}{54}{,}{99}{,}{191}{,}{94}{,}{188}{,}{199}{,}{205}\right)\left({24}{,}{60}{,}{128}{,}{228}{,}{149}{,}{266}{,}{379}{,}{132}\right)\left({25}{,}{63}{,}{136}{,}{241}{,}{337}{,}{329}{,}{335}{,}{444}\right)\left({27}{,}{69}{,}{153}{,}{272}{,}{342}{,}{255}{,}{378}{,}{135}\right)\left({28}{,}{72}{,}{67}{,}{148}{,}{264}{,}{154}{,}{275}{,}{406}\right)\left({30}{,}{78}{,}{175}{,}{310}{,}{430}{,}{431}{,}{318}{,}{377}\right)\left({37}{,}{93}{,}{106}{,}{203}{,}{55}{,}{115}{,}{53}{,}{113}\right)\left({43}{,}{101}{,}{200}{,}{52}{,}{112}{,}{195}{,}{97}{,}{197}\right)\left({45}{,}{105}{,}{201}{,}{104}{,}{88}{,}{100}{,}{85}{,}{110}\right)\left({47}{,}{107}{,}{103}{,}{202}{,}{327}{,}{90}{,}{193}{,}{196}\right)\left({49}{,}{109}{,}{204}{,}{114}{,}{118}{,}{102}{,}{96}{,}{84}\right)\left({50}{,}{111}{,}{194}{,}{87}{,}{189}{,}{206}{,}{56}{,}{117}\right)\left({58}{,}{122}{,}{79}{,}{178}{,}{315}{,}{140}{,}{120}{,}{209}\right)\left({59}{,}{125}{,}{65}{,}{142}\right)\left({61}{,}{130}{,}{232}{,}{242}{,}{365}{,}{240}{,}{239}{,}{184}\right)\left({62}{,}{133}{,}{237}{,}{361}{,}{363}{,}{289}{,}{360}{,}{270}\right)\left({64}{,}{139}{,}{248}{,}{370}{,}{312}{,}{314}{,}{82}{,}{183}\right)\left({68}{,}{151}{,}{269}{,}{299}{,}{225}{,}{295}{,}{74}{,}{166}\right)\left({70}{,}{156}{,}{279}{,}{369}{,}{251}{,}{316}{,}{435}{,}{433}\right)\left({71}{,}{157}{,}{282}{,}{210}{,}{333}{,}{283}{,}{286}{,}{412}\right)\left({73}{,}{163}{,}{294}{,}{263}{,}{392}{,}{259}{,}{357}{,}{407}\right)\left({76}{,}{170}{,}{305}{,}{385}{,}{276}{,}{168}{,}{249}{,}{162}\right)\left({80}{,}{180}{,}{319}{,}{437}{,}{349}{,}{364}{,}{323}{,}{426}\right)\left({121}{,}{212}{,}{244}{,}{131}{,}{235}{,}{247}{,}{127}{,}{137}\right)\left({123}{,}{217}{,}{177}{,}{176}{,}{311}{,}{372}{,}{332}{,}{346}\right)\left({124}{,}{220}{,}{185}{,}{134}{,}{238}{,}{356}{,}{443}{,}{354}\right)\left({126}{,}{224}{,}{271}{,}{398}{,}{171}{,}{227}{,}{253}{,}{344}\right)\left({129}{,}{230}{,}{340}{,}{436}{,}{320}{,}{141}{,}{252}{,}{211}\right)\left({138}{,}{245}{,}{218}{,}{341}{,}{330}{,}{214}{,}{144}{,}{208}\right)\left({143}{,}{256}{,}{381}{,}{352}{,}{280}{,}{300}{,}{421}{,}{452}\right)\left({146}{,}{260}{,}{231}{,}{355}{,}{181}{,}{322}{,}{179}{,}{317}\right)\left({147}{,}{262}{,}{390}{,}{165}{,}{298}{,}{293}{,}{152}{,}{150}\right)\left({155}{,}{277}{,}{229}{,}{334}{,}{441}{,}{359}{,}{160}{,}{288}\right)\left({158}{,}{174}{,}{307}{,}{427}{,}{384}{,}{246}{,}{258}{,}{350}\right)\left({159}{,}{285}{,}{274}{,}{403}{,}{362}{,}{450}{,}{395}{,}{397}\right)\left({161}{,}{290}{,}{415}{,}{460}{,}{455}{,}{428}{,}{292}{,}{223}\right)\left({164}{,}{296}{,}{380}{,}{373}{,}{287}{,}{236}{,}{358}{,}{234}\right)\left({167}{,}{302}{,}{284}{,}{388}{,}{281}{,}{409}{,}{446}{,}{442}\right)\left({182}{,}{324}{,}{439}{,}{449}{,}{389}{,}{419}{,}{429}{,}{410}\right)\left({213}{,}{336}{,}{402}{,}{399}{,}{267}{,}{268}{,}{306}{,}{265}\right)\left({215}{,}{338}{,}{445}{,}{347}{,}{221}{,}{345}{,}{447}{,}{339}\right)\left({216}{,}{321}{,}{254}{,}{326}{,}{250}{,}{374}{,}{222}{,}{219}\right)\left({226}{,}{243}{,}{273}{,}{301}{,}{393}{,}{308}{,}{391}{,}{394}\right)\left({233}{,}{325}{,}{440}{,}{459}{,}{454}{,}{303}{,}{425}{,}{366}\right)\left({261}{,}{387}{,}{451}{,}{278}{,}{408}{,}{438}{,}{367}{,}{386}\right)\left({291}{,}{417}{,}{422}{,}{456}{,}{404}{,}{413}{,}{424}{,}{416}\right)\left({297}{,}{348}{,}{400}{,}{458}{,}{304}{,}{420}{,}{457}{,}{331}\right)\left({309}{,}{313}{,}{432}{,}{434}{,}{376}{,}{453}{,}{411}{,}{418}\right)\left({343}{,}{448}{,}{405}{,}{353}{,}{401}{,}{351}{,}{414}{,}{396}\right){,}\left({1}{,}{2}{,}{16}{,}{42}{,}{101}\right)\left({3}{,}{18}{,}{45}{,}{106}{,}{34}\right)\left({4}{,}{5}{,}{21}{,}{52}{,}{110}\right)\left({6}{,}{7}{,}{24}{,}{61}{,}{131}\right)\left({8}{,}{9}{,}{27}{,}{70}{,}{157}\right)\left({10}{,}{30}{,}{79}{,}{75}{,}{132}\right)\left({11}{,}{13}{,}{17}{,}{19}{,}{47}\right)\left({12}{,}{33}{,}{15}{,}{40}{,}{36}\right)\left({14}{,}{37}{,}{94}{,}{92}{,}{188}\right)\left({20}{,}{49}{,}{48}{,}{46}{,}{93}\right)\left({22}{,}{55}{,}{116}{,}{98}{,}{100}\right)\left({23}{,}{58}{,}{123}{,}{218}{,}{342}\right)\left({25}{,}{64}{,}{140}{,}{251}{,}{293}\right)\left({26}{,}{67}{,}{149}{,}{267}{,}{396}\right)\left({28}{,}{73}{,}{164}{,}{297}{,}{212}\right)\left({29}{,}{76}{,}{171}{,}{62}{,}{134}\right)\left({31}{,}{82}{,}{184}{,}{263}{,}{306}\right)\left({32}{,}{84}{,}{113}{,}{50}{,}{54}\right)\left({35}{,}{90}{,}{117}{,}{108}{,}{41}\right)\left({38}{,}{96}{,}{196}{,}{191}{,}{39}\right)\left({43}{,}{103}{,}{104}{,}{197}{,}{192}\right)\left({44}{,}{102}{,}{201}{,}{51}{,}{107}\right)\left({53}{,}{114}{,}{200}{,}{97}{,}{198}\right)\left({56}{,}{87}{,}{190}{,}{95}{,}{85}\right)\left({57}{,}{120}{,}{210}{,}{334}{,}{256}\right)\left({59}{,}{126}{,}{225}{,}{245}{,}{367}\right)\left({60}{,}{129}{,}{208}{,}{331}{,}{128}\right)\left({63}{,}{137}{,}{243}{,}{133}{,}{165}\right)\left({65}{,}{143}{,}{153}{,}{273}{,}{152}\right)\left({66}{,}{146}{,}{261}{,}{388}{,}{450}\right)\left({68}{,}{72}{,}{161}{,}{177}{,}{313}\right)\left({69}{,}{154}{,}{276}{,}{282}{,}{345}\right)\left({71}{,}{159}{,}{286}{,}{413}{,}{365}\right)\left({74}{,}{167}{,}{231}{,}{260}{,}{244}\right)\left({77}{,}{174}{,}{308}{,}{417}{,}{173}\right)\left({78}{,}{176}{,}{312}{,}{264}{,}{340}\right)\left({80}{,}{181}{,}{323}{,}{122}{,}{215}\right)\left({81}{,}{182}{,}{180}{,}{320}{,}{281}\right)\left({83}{,}{86}{,}{187}{,}{193}{,}{204}\right)\left({88}{,}{105}{,}{118}{,}{189}{,}{203}\right)\left({89}{,}{115}{,}{195}{,}{206}{,}{111}\right)\left({91}{,}{194}{,}{99}{,}{186}{,}{327}\right)\left({109}{,}{205}{,}{202}{,}{112}{,}{199}\right)\left({119}{,}{163}{,}{175}{,}{250}{,}{375}\right)\left({121}{,}{213}{,}{309}{,}{428}{,}{385}\right)\left({124}{,}{221}{,}{346}{,}{415}{,}{439}\right)\left({125}{,}{223}{,}{336}{,}{322}{,}{394}\right)\left({127}{,}{227}{,}{311}{,}{406}{,}{425}\right)\left({130}{,}{233}{,}{166}{,}{300}{,}{332}\right)\left({135}{,}{239}{,}{349}{,}{437}{,}{372}\right)\left({136}{,}{242}{,}{366}{,}{419}{,}{414}\right)\left({138}{,}{246}{,}{271}{,}{252}{,}{147}\right)\left({139}{,}{249}{,}{373}{,}{301}{,}{310}\right)\left({141}{,}{253}{,}{228}{,}{325}{,}{411}\right)\left({142}{,}{255}{,}{379}{,}{374}{,}{455}\right)\left({144}{,}{145}{,}{258}{,}{291}{,}{262}\right)\left({148}{,}{220}{,}{344}{,}{433}{,}{335}\right)\left({150}{,}{169}{,}{155}{,}{278}{,}{285}\right)\left({151}{,}{270}{,}{389}{,}{289}{,}{330}\right)\left({156}{,}{280}{,}{294}{,}{420}{,}{324}\right)\left({158}{,}{283}{,}{319}{,}{216}{,}{259}\right)\left({160}{,}{279}{,}{397}{,}{431}{,}{230}\right)\left({162}{,}{292}{,}{403}{,}{222}{,}{348}\right)\left({168}{,}{304}{,}{381}{,}{298}{,}{247}\right)\left({172}{,}{224}{,}{268}{,}{307}{,}{409}\right)\left({178}{,}{316}{,}{226}{,}{299}{,}{424}\right)\left({179}{,}{266}{,}{338}{,}{360}{,}{377}\right)\left({183}{,}{326}{,}{441}{,}{269}{,}{380}\right)\left({185}{,}{219}{,}{343}{,}{378}{,}{209}\right)\left({207}{,}{329}{,}{442}{,}{347}{,}{376}\right)\left({214}{,}{274}{,}{404}{,}{460}{,}{234}\right)\left({217}{,}{314}{,}{434}{,}{370}{,}{427}\right)\left({229}{,}{352}{,}{361}{,}{272}{,}{400}\right)\left({232}{,}{295}{,}{421}{,}{284}{,}{408}\right)\left({235}{,}{357}{,}{390}{,}{328}{,}{410}\right)\left({236}{,}{359}{,}{368}{,}{386}{,}{443}\right)\left({237}{,}{362}{,}{351}{,}{401}{,}{418}\right)\left({238}{,}{363}{,}{451}{,}{305}{,}{426}\right)\left({240}{,}{321}{,}{438}{,}{454}{,}{436}\right)\left({241}{,}{364}{,}{452}{,}{432}{,}{398}\right)\left({248}{,}{371}{,}{393}{,}{459}{,}{423}\right)\left({254}{,}{317}{,}{358}{,}{440}{,}{405}\right)\left({257}{,}{384}{,}{277}{,}{353}{,}{449}\right)\left({265}{,}{356}{,}{315}{,}{391}{,}{448}\right)\left({275}{,}{341}{,}{444}{,}{399}{,}{333}\right)\left({287}{,}{412}{,}{456}{,}{339}{,}{446}\right)\left({288}{,}{392}{,}{337}{,}{395}{,}{453}\right)\left({290}{,}{416}{,}{430}{,}{354}{,}{429}\right)\left({296}{,}{422}{,}{387}{,}{458}{,}{383}\right)\left({303}{,}{402}{,}{447}{,}{407}{,}{369}\right)\left({350}{,}{355}{,}{435}{,}{382}{,}{457}\right){,}\left({1}{,}{3}{,}{19}{,}{44}{,}{46}{,}{17}{,}{15}{,}{39}\right)\left({5}{,}{22}{,}{56}{,}{118}{,}{197}{,}{106}{,}{105}{,}{51}\right)\left({7}{,}{25}{,}{65}{,}{144}{,}{63}{,}{138}{,}{247}{,}{328}\right)\left({8}{,}{10}{,}{31}{,}{75}{,}{81}{,}{29}{,}{77}{,}{173}\right)\left({9}{,}{28}{,}{74}{,}{168}{,}{243}{,}{357}{,}{213}{,}{227}\right)\left({11}{,}{14}{,}{38}{,}{97}{,}{194}{,}{37}{,}{54}{,}{96}\right)\left({12}{,}{34}{,}{88}{,}{114}{,}{94}{,}{195}{,}{20}{,}{50}\right)\left({13}{,}{36}{,}{92}{,}{35}{,}{91}{,}{89}{,}{192}{,}{18}\right)\left({16}{,}{43}{,}{104}{,}{198}{,}{109}{,}{111}{,}{21}{,}{53}\right)\left({23}{,}{59}{,}{127}{,}{57}{,}{121}{,}{214}{,}{337}{,}{444}\right)\left({24}{,}{62}{,}{135}{,}{240}{,}{286}{,}{414}{,}{296}{,}{403}\right)\left({26}{,}{68}{,}{152}{,}{271}{,}{399}{,}{276}{,}{264}{,}{394}\right)\left({27}{,}{71}{,}{160}{,}{289}{,}{228}{,}{351}{,}{130}{,}{234}\right)\left({30}{,}{80}{,}{123}{,}{219}{,}{129}{,}{231}{,}{356}{,}{178}\right)\left({32}{,}{85}{,}{115}{,}{206}{,}{187}{,}{204}{,}{40}{,}{99}\right)\left({33}{,}{87}{,}{191}{,}{45}{,}{52}{,}{42}{,}{102}{,}{55}\right)\left({41}{,}{100}{,}{199}{,}{203}{,}{98}{,}{186}{,}{112}{,}{84}\right)\left({47}{,}{107}{,}{190}{,}{327}{,}{103}{,}{193}{,}{202}{,}{90}\right)\left({48}{,}{83}{,}{113}{,}{200}{,}{205}{,}{95}{,}{189}{,}{201}\right)\left({49}{,}{110}{,}{117}{,}{101}{,}{86}{,}{188}{,}{108}{,}{93}\right)\left({58}{,}{124}{,}{222}{,}{349}{,}{252}{,}{78}{,}{177}{,}{314}\right)\left({60}{,}{69}{,}{155}{,}{159}{,}{287}{,}{365}{,}{363}{,}{333}\right)\left({61}{,}{132}{,}{236}{,}{360}{,}{396}{,}{450}{,}{334}{,}{210}\right)\left({64}{,}{141}{,}{254}{,}{377}{,}{220}{,}{323}{,}{355}{,}{140}\right)\left({66}{,}{147}{,}{263}{,}{393}{,}{406}{,}{224}{,}{225}{,}{336}\right)\left({67}{,}{150}{,}{268}{,}{163}{,}{295}{,}{257}{,}{385}{,}{253}\right)\left({70}{,}{158}{,}{284}{,}{411}{,}{455}{,}{367}{,}{454}{,}{422}\right)\left({72}{,}{162}{,}{293}{,}{383}{,}{126}{,}{226}{,}{269}{,}{392}\right)\left({73}{,}{165}{,}{299}{,}{148}{,}{265}{,}{308}{,}{398}{,}{249}\right)\left({76}{,}{172}{,}{267}{,}{275}{,}{407}{,}{344}{,}{262}{,}{391}\right)\left({79}{,}{179}{,}{318}{,}{230}{,}{354}{,}{139}{,}{250}{,}{176}\right)\left({82}{,}{185}{,}{322}{,}{346}{,}{430}{,}{374}{,}{315}{,}{319}\right)\left({119}{,}{208}{,}{218}{,}{131}{,}{142}{,}{241}{,}{235}{,}{329}\right)\left({120}{,}{211}{,}{311}{,}{146}{,}{180}{,}{321}{,}{312}{,}{431}\right)\left({122}{,}{216}{,}{181}{,}{248}{,}{372}{,}{437}{,}{238}{,}{320}\right)\left({125}{,}{207}{,}{330}{,}{371}{,}{136}{,}{137}{,}{244}{,}{341}\right)\left({128}{,}{229}{,}{353}{,}{157}{,}{270}{,}{397}{,}{255}{,}{380}\right)\left({133}{,}{184}{,}{288}{,}{266}{,}{395}{,}{283}{,}{272}{,}{401}\right)\left({134}{,}{217}{,}{340}{,}{370}{,}{364}{,}{310}{,}{260}{,}{326}\right)\left({143}{,}{256}{,}{382}{,}{280}{,}{381}{,}{421}{,}{352}{,}{300}\right)\left({145}{,}{259}{,}{170}{,}{306}{,}{166}{,}{301}{,}{298}{,}{154}\right)\left({149}{,}{242}{,}{362}{,}{237}{,}{277}{,}{358}{,}{448}{,}{378}\right)\left({151}{,}{171}{,}{294}{,}{375}{,}{273}{,}{402}{,}{390}{,}{305}\right)\left({153}{,}{274}{,}{405}{,}{379}{,}{412}{,}{373}{,}{441}{,}{232}\right)\left({156}{,}{281}{,}{410}{,}{416}{,}{366}{,}{453}{,}{384}{,}{415}\right)\left({161}{,}{291}{,}{418}{,}{261}{,}{389}{,}{442}{,}{427}{,}{459}\right)\left({164}{,}{282}{,}{285}{,}{239}{,}{343}{,}{359}{,}{342}{,}{361}\right)\left({167}{,}{303}{,}{279}{,}{324}{,}{278}{,}{223}{,}{350}{,}{424}\right)\left({174}{,}{309}{,}{429}{,}{316}{,}{413}{,}{460}{,}{446}{,}{408}\right)\left({175}{,}{183}{,}{317}{,}{436}{,}{426}{,}{209}{,}{332}{,}{443}\right)\left({182}{,}{325}{,}{387}{,}{302}{,}{251}{,}{376}{,}{456}{,}{307}\right)\left({212}{,}{335}{,}{245}{,}{368}\right)\left({215}{,}{339}{,}{447}{,}{345}{,}{221}{,}{347}{,}{445}{,}{338}\right)\left({233}{,}{258}{,}{386}{,}{404}{,}{439}{,}{290}{,}{313}{,}{433}\right)\left({246}{,}{369}{,}{438}{,}{432}{,}{440}{,}{388}{,}{292}{,}{419}\right)\left({297}{,}{423}{,}{458}{,}{348}{,}{420}{,}{400}{,}{304}{,}{331}\right)\left({409}{,}{417}{,}{451}{,}{435}{,}{428}{,}{425}{,}{449}{,}{434}\right)⟩$ (11)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${259200}$ (12)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{false}}$ (13)