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Properties

Label 704880.a
Order \( 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 89 \)
Exponent \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 89 \)
Nilpotent no
Solvable no
$\card{G^{\mathrm{ab}}}$ \( 1 \)
$\card{Z(G)}$ 2
$\card{\Aut(G)}$ \( 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 89 \)
$\card{\mathrm{Out}(G)}$ \( 2 \)
Perm deg. $720$
Trans deg. not computed
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SL(2, 89);
 
Copy content gap:G := SL(2, 89);
 
Copy content sage:G = SL(2, 89)
 
Copy content comment:Define the group as a permutation group
 

Group information

Description:$\SL(2,89)$
Order: \(704880\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 89 \)
Copy content comment:Order of the group
 
Copy content magma:Order(G);
 
Copy content gap:Order(G);
 
Copy content sage:G.order()
 
Copy content sage_gap:G.Order()
 
Exponent: \(352440\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 89 \)
Copy content comment:Exponent of the group
 
Copy content magma:Exponent(G);
 
Copy content gap:Exponent(G);
 
Copy content sage:G.exponent()
 
Copy content sage_gap:G.Exponent()
 
Automorphism group:$\PGL(2,89)$, of order \(704880\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 89 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$, $\PSL(2,89)$
Copy content comment:Composition factors of the group
 
Copy content magma:CompositionFactors(G);
 
Copy content gap:CompositionSeries(G);
 
Copy content sage:G.composition_series()
 
Copy content sage_gap:G.CompositionSeries()
 
Derived length:$0$
Copy content comment:Derived length of the group
 
Copy content magma:DerivedLength(G);
 
Copy content gap:DerivedLength(G);
 
Copy content sage_gap:G.DerivedLength()
 

This group is nonabelian and quasisimple (hence nonsolvable and perfect). Whether it is almost simple has not been computed.

Copy content comment:Determine if the group G is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content gap:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content sage_gap:G.IsAbelian()
 
Copy content comment:Determine if the group G is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content gap:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content sage_gap:G.IsCyclic()
 
Copy content comment:Determine if the group G is nilpotent
 
Copy content magma:IsNilpotent(G);
 
Copy content gap:IsNilpotentGroup(G);
 
Copy content sage:G.is_nilpotent()
 
Copy content sage_gap:G.IsNilpotentGroup()
 
Copy content comment:Determine if the group G is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content gap:IsSolvableGroup(G);
 
Copy content sage:G.is_solvable()
 
Copy content sage_gap:G.IsSolvableGroup()
 
Copy content comment:Determine if the group G is supersolvable
 
Copy content gap:IsSupersolvableGroup(G);
 
Copy content sage:G.is_supersolvable()
 
Copy content sage_gap:G.IsSupersolvableGroup()
 
Copy content comment:Determine if the group G is simple
 
Copy content magma:IsSimple(G);
 
Copy content gap:IsSimpleGroup(G);
 
Copy content sage_gap:G.IsSimpleGroup()
 

Group statistics

Copy content comment:Compute statistics for the group G
 
Copy content magma:// Magma code to output the first two rows of the group statistics table element_orders := [Order(g) : g in G]; orders := Set(element_orders); printf "Orders: %o\n", orders; printf "Elements: %o %o\n", [#[x : x in element_orders | x eq n] : n in orders], Order(G); cc_orders := [cc[1] : cc in ConjugacyClasses(G)]; printf "Conjugacy classes: %o %o\n", [#[x : x in cc_orders | x eq n] : n in orders], #cc_orders;
 
Copy content gap:# Gap code to output the first two rows of the group statistics table element_orders := List(Elements(G), g -> Order(g)); orders := Set(element_orders); Print("Orders: ", orders, "\n"); element_counts := List(orders, n -> Length(Filtered(element_orders, x -> x = n))); Print("Elements: ", element_counts, " ", Size(G), "\n"); cc_orders := List(ConjugacyClasses(G), cc -> Order(Representative(cc))); cc_counts := List(orders, n -> Length(Filtered(cc_orders, x -> x = n))); Print("Conjugacy classes: ", cc_counts, " ", Length(ConjugacyClasses(G)), "\n");
 
Copy content sage:# Sage code to output the first two rows of the group statistics table element_orders = [g.order() for g in G] orders = sorted(list(set(element_orders))) print("Orders:", orders) print("Elements:", [element_orders.count(n) for n in orders], G.order()) cc_orders = [cc[0].order() for cc in G.conjugacy_classes()] print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))
 

Order 1 2 3 4 5 6 8 9 10 11 15 18 22 30 44 45 88 89 90 178
Elements 1 1 7832 8010 15664 7832 16020 23496 15664 40050 31328 23496 40050 31328 80100 93984 160200 7920 93984 7920 704880
Conjugacy classes   1 1 1 1 2 1 2 3 2 5 4 3 5 4 10 12 20 2 12 2 93
Divisions 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 20
Autjugacy classes 1 1 1 1 2 1 2 3 2 5 4 3 5 4 10 12 20 1 12 1 91

Minimal presentations

Permutation degree:$720$
Transitive degree:not computed
Rank: $2$
Inequivalent generating pairs: not computed

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 44 not computed not computed
Arbitrary not computed not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Groups of Lie type:$\SL(2,89)$, $\SU(2,89)$
Permutation group:Degree $720$ $\langle(1,2,6,16,42,107,197,382,403,281,127,280,302,325,234,103,40,15,5,14,36,93,214,409,617,443,352,178,145,57,144,301,521,680,586,372,439,330,208,89,207,396,606,491,666,604,394,590,652,609,398,608,700,632,683,581,365,427,630,612,400,611,708,649,466,648,686,535,312,471,622,701,582,366,185,77,184,363,311,534,435,232,102,123,83,32,12,4) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 720 | (1,2,6,16,42,107,197,382,403,281,127,280,302,325,234,103,40,15,5,14,36,93,214,409,617,443,352,178,145,57,144,301,521,680,586,372,439,330,208,89,207,396,606,491,666,604,394,590,652,609,398,608,700,632,683,581,365,427,630,612,400,611,708,649,466,648,686,535,312,471,622,701,582,366,185,77,184,363,311,534,435,232,102,123,83,32,12,4)(3,9,23,60,149,220,417,242,106,41,105,238,445,355,180,354,397,249,459,432,303,254,464,635,625,452,644,461,252,231,101,39,100,227,426,628,655,478,544,661,520,300,517,657,715,693,645,712,703,584,369,505,672,675,631,428,482,486,272,381,196,155,62,154,320,292,134,52,133,289,203,390,542,322,296,514,326,159,64,24,63,116,259,348,173,70,27,10)(7,19,49,126,277,493,667,695,562,344,171,69,170,94,143,56,142,256,465,572,577,358,576,501,284,500,670,594,414,523,488,273,124,48,18,47,120,266,477,568,699,716,658,634,707,684,527,306,146,58,22,8,21,55,140,298,162,331,484,347,172,346,457,247,109,246,456,337,524,304,368,187,78,30,11,29,74,88,206,340,167,115,258,418,221,136,53,20)(13,34,87,122,271,479,267,285,130,50,129,282,189,79,176,71,175,315,151,314,472,490,274,462,384,198,84,82,195,378,595,692,551,333,163,332,550,422,460,250,287,503,494,492,276,125,275,168,68,26,67,165,336,553,532,685,682,522,569,516,677,717,719,681,575,357,181,356,573,359,578,607,664,602,580,579,468,650,593,377,193,81,31,80,190,211,91,35)(17,45,114,257,186,367,364,469,629,688,539,615,405,408,226,99,38,98,223,202,86,33,85,76,128,278,495,669,552,334,313,150,177,157,156,265,119,251,111,43,110,248,454,605,395,415,218,95,217,412,545,324,158,323,543,496,598,674,512,673,592,375,192,295,513,647,663,489,662,526,525,305,476,654,713,720,718,714,706,596,705,690,589,374,191,263,118,46)(25,65,161,328,506,603,393,205,392,339,166,135,141,270,121,269,453,243,283,499,668,555,338,554,380,386,199,385,564,502,329,279,321,473,262,362,316,536,451,241,450,619,710,694,561,343,434,627,424,225,423,402,376,360,268,480,345,559,341,558,618,531,309,148,59,147,307,529,665,698,614,404,533,310,436,528,433,463,455,556,659,481,642,541,319,335,164,66)(28,72,137,294,510,431,230,245,108,201,389,599,679,519,299,518,678,563,588,509,574,601,391,204,97,37,96,219,416,621,620,507,290,407,388,200,387,571,497,624,567,351,566,697,656,676,515,297,139,54,138,264,474,651,546,549,327,160,174,349,255,113,44,112,253,210,90,209,401,613,704,702,583,626,420,222,419,623,711,709,633,458,508,291,498,353,179,73)(51,131,286,430,228,429,557,487,383,597,641,444,640,600,689,540,318,153,61,152,224,216,411,570,511,653,475,421,467,413,410,215,229,293,483,660,485,470,399,610,538,317,537,687,587,373,308,530,547,361,183,75,182,261,117,260,213,92,212,406,616,643,646,671,591,696,565,350,425,244,379,194,371,188,370,585,548,691,560,342,169,237,104,236,442,504,288,132)(233,437,636,638,447,239,446,438)(235,440,449,240,448,639,637,441), (1,3,5)(2,7,8)(4,11,13)(6,17,18)(9,24,25)(10,26,28)(12,31,33)(14,37,38)(15,39,41)(16,43,44)(19,50,51)(20,52,54)(21,56,57)(22,58,59)(23,61,62)(27,69,71)(29,75,76)(30,77,79)(32,82,84)(34,88,89)(35,90,92)(36,94,95)(40,102,104)(42,108,109)(45,115,116)(46,117,119)(47,121,122)(48,123,125)(49,127,128)(53,135,137)(55,141,140)(60,150,151)(63,156,157)(64,158,160)(65,114,162)(66,163,73)(67,166,167)(68,143,169)(70,172,174)(72,177,178)(74,180,181)(78,186,188)(80,191,155)(81,192,194)(83,196,197)(85,199,200)(86,201,203)(87,204,205)(91,146,134)(93,215,216)(96,220,221)(97,222,173)(98,224,139)(99,225,175)(100,228,229)(101,230,202)(103,233,235)(105,239,240)(106,241,243)(107,132,244)(110,249,250)(111,251,252)(112,130,254)(113,147,256)(118,262,264)(120,267,268)(124,272,274)(126,278,279)(129,283,284)(131,248,287)(133,290,291)(136,293,219)(138,295,296)(142,299,300)(144,302,303)(145,304,305)(148,308,310)(149,311,312)(152,316,317)(153,236,319)(154,321,322)(159,325,238)(161,329,330)(164,334,327)(165,337,338)(168,340,341)(170,343,218)(171,344,345)(176,350,351)(179,333,214)(182,358,359)(183,360,362)(184,364,314)(185,365,320)(187,332,369)(189,372,368)(190,373,263)(193,376,378)(195,380,357)(198,383,246)(206,394,395)(207,397,398)(208,399,400)(209,402,275)(210,403,384)(211,404,405)(212,388,407)(213,255,408)(217,413,414)(223,421,422)(226,297,425)(227,427,428)(231,432,433)(232,434,436)(234,288,439)(237,443,444)(242,424,452)(245,454,455)(247,381,458)(253,462,463)(257,466,331)(258,467,468)(259,389,469)(260,470,471)(261,323,472)(265,475,476)(266,478,415)(269,481,482)(270,328,483)(271,484,485)(273,487,489)(276,491,488)(277,464,494)(280,496,497)(281,406,361)(282,386,498)(285,501,307)(286,502,499)(289,505,506)(292,509,336)(294,511,512)(298,516,517)(301,522,523)(306,526,528)(309,524,532)(313,342,315)(318,539,349)(324,544,546)(326,547,548)(335,529,370)(339,556,557)(346,563,555)(347,366,562)(348,564,355)(352,568,549)(353,569,570)(354,571,572)(356,473,574)(363,579,580)(367,583,393)(371,551,477)(374,588,590)(375,591,510)(377,417,594)(379,552,596)(382,453,535)(385,598,503)(387,530,465)(390,456,495)(391,600,492)(392,602,550)(396,607,578)(401,507,520)(409,608,618)(410,619,554)(411,620,621)(412,416,423)(418,599,622)(419,624,625)(420,626,565)(426,629,519)(429,632,540)(430,457,633)(431,634,635)(435,630,537)(437,441,631)(438,440,538)(442,638,639)(445,558,536)(446,449,642)(447,448,612)(450,643,610)(451,636,637)(459,645,646)(460,647,606)(461,576,525)(474,577,652)(479,656,657)(480,515,658)(486,661,533)(490,664,665)(493,668,667)(500,669,582)(504,671,659)(508,581,615)(513,651,644)(514,675,611)(518,616,543)(521,545,681)(527,683,662)(531,672,541)(534,670,682)(542,589,690)(553,587,684)(559,627,560)(561,693,628)(566,650,686)(567,685,698)(573,601,700)(575,597,688)(584,699,704)(585,696,648)(586,617,703)(592,649,613)(593,697,694)(595,680,692)(603,677,673)(604,707,695)(605,641,705)(609,687,653)(614,709,663)(623,689,711)(640,678,674)(654,679,710)(655,706,714)(660,717,713)(666,715,708)(676,718,701)(691,702,716)(712,720,719) >;
 
Copy content gap:G := Group( (1,2,6,16,42,107,197,382,403,281,127,280,302,325,234,103,40,15,5,14,36,93,214,409,617,443,352,178,145,57,144,301,521,680,586,372,439,330,208,89,207,396,606,491,666,604,394,590,652,609,398,608,700,632,683,581,365,427,630,612,400,611,708,649,466,648,686,535,312,471,622,701,582,366,185,77,184,363,311,534,435,232,102,123,83,32,12,4)(3,9,23,60,149,220,417,242,106,41,105,238,445,355,180,354,397,249,459,432,303,254,464,635,625,452,644,461,252,231,101,39,100,227,426,628,655,478,544,661,520,300,517,657,715,693,645,712,703,584,369,505,672,675,631,428,482,486,272,381,196,155,62,154,320,292,134,52,133,289,203,390,542,322,296,514,326,159,64,24,63,116,259,348,173,70,27,10)(7,19,49,126,277,493,667,695,562,344,171,69,170,94,143,56,142,256,465,572,577,358,576,501,284,500,670,594,414,523,488,273,124,48,18,47,120,266,477,568,699,716,658,634,707,684,527,306,146,58,22,8,21,55,140,298,162,331,484,347,172,346,457,247,109,246,456,337,524,304,368,187,78,30,11,29,74,88,206,340,167,115,258,418,221,136,53,20)(13,34,87,122,271,479,267,285,130,50,129,282,189,79,176,71,175,315,151,314,472,490,274,462,384,198,84,82,195,378,595,692,551,333,163,332,550,422,460,250,287,503,494,492,276,125,275,168,68,26,67,165,336,553,532,685,682,522,569,516,677,717,719,681,575,357,181,356,573,359,578,607,664,602,580,579,468,650,593,377,193,81,31,80,190,211,91,35)(17,45,114,257,186,367,364,469,629,688,539,615,405,408,226,99,38,98,223,202,86,33,85,76,128,278,495,669,552,334,313,150,177,157,156,265,119,251,111,43,110,248,454,605,395,415,218,95,217,412,545,324,158,323,543,496,598,674,512,673,592,375,192,295,513,647,663,489,662,526,525,305,476,654,713,720,718,714,706,596,705,690,589,374,191,263,118,46)(25,65,161,328,506,603,393,205,392,339,166,135,141,270,121,269,453,243,283,499,668,555,338,554,380,386,199,385,564,502,329,279,321,473,262,362,316,536,451,241,450,619,710,694,561,343,434,627,424,225,423,402,376,360,268,480,345,559,341,558,618,531,309,148,59,147,307,529,665,698,614,404,533,310,436,528,433,463,455,556,659,481,642,541,319,335,164,66)(28,72,137,294,510,431,230,245,108,201,389,599,679,519,299,518,678,563,588,509,574,601,391,204,97,37,96,219,416,621,620,507,290,407,388,200,387,571,497,624,567,351,566,697,656,676,515,297,139,54,138,264,474,651,546,549,327,160,174,349,255,113,44,112,253,210,90,209,401,613,704,702,583,626,420,222,419,623,711,709,633,458,508,291,498,353,179,73)(51,131,286,430,228,429,557,487,383,597,641,444,640,600,689,540,318,153,61,152,224,216,411,570,511,653,475,421,467,413,410,215,229,293,483,660,485,470,399,610,538,317,537,687,587,373,308,530,547,361,183,75,182,261,117,260,213,92,212,406,616,643,646,671,591,696,565,350,425,244,379,194,371,188,370,585,548,691,560,342,169,237,104,236,442,504,288,132)(233,437,636,638,447,239,446,438)(235,440,449,240,448,639,637,441), (1,3,5)(2,7,8)(4,11,13)(6,17,18)(9,24,25)(10,26,28)(12,31,33)(14,37,38)(15,39,41)(16,43,44)(19,50,51)(20,52,54)(21,56,57)(22,58,59)(23,61,62)(27,69,71)(29,75,76)(30,77,79)(32,82,84)(34,88,89)(35,90,92)(36,94,95)(40,102,104)(42,108,109)(45,115,116)(46,117,119)(47,121,122)(48,123,125)(49,127,128)(53,135,137)(55,141,140)(60,150,151)(63,156,157)(64,158,160)(65,114,162)(66,163,73)(67,166,167)(68,143,169)(70,172,174)(72,177,178)(74,180,181)(78,186,188)(80,191,155)(81,192,194)(83,196,197)(85,199,200)(86,201,203)(87,204,205)(91,146,134)(93,215,216)(96,220,221)(97,222,173)(98,224,139)(99,225,175)(100,228,229)(101,230,202)(103,233,235)(105,239,240)(106,241,243)(107,132,244)(110,249,250)(111,251,252)(112,130,254)(113,147,256)(118,262,264)(120,267,268)(124,272,274)(126,278,279)(129,283,284)(131,248,287)(133,290,291)(136,293,219)(138,295,296)(142,299,300)(144,302,303)(145,304,305)(148,308,310)(149,311,312)(152,316,317)(153,236,319)(154,321,322)(159,325,238)(161,329,330)(164,334,327)(165,337,338)(168,340,341)(170,343,218)(171,344,345)(176,350,351)(179,333,214)(182,358,359)(183,360,362)(184,364,314)(185,365,320)(187,332,369)(189,372,368)(190,373,263)(193,376,378)(195,380,357)(198,383,246)(206,394,395)(207,397,398)(208,399,400)(209,402,275)(210,403,384)(211,404,405)(212,388,407)(213,255,408)(217,413,414)(223,421,422)(226,297,425)(227,427,428)(231,432,433)(232,434,436)(234,288,439)(237,443,444)(242,424,452)(245,454,455)(247,381,458)(253,462,463)(257,466,331)(258,467,468)(259,389,469)(260,470,471)(261,323,472)(265,475,476)(266,478,415)(269,481,482)(270,328,483)(271,484,485)(273,487,489)(276,491,488)(277,464,494)(280,496,497)(281,406,361)(282,386,498)(285,501,307)(286,502,499)(289,505,506)(292,509,336)(294,511,512)(298,516,517)(301,522,523)(306,526,528)(309,524,532)(313,342,315)(318,539,349)(324,544,546)(326,547,548)(335,529,370)(339,556,557)(346,563,555)(347,366,562)(348,564,355)(352,568,549)(353,569,570)(354,571,572)(356,473,574)(363,579,580)(367,583,393)(371,551,477)(374,588,590)(375,591,510)(377,417,594)(379,552,596)(382,453,535)(385,598,503)(387,530,465)(390,456,495)(391,600,492)(392,602,550)(396,607,578)(401,507,520)(409,608,618)(410,619,554)(411,620,621)(412,416,423)(418,599,622)(419,624,625)(420,626,565)(426,629,519)(429,632,540)(430,457,633)(431,634,635)(435,630,537)(437,441,631)(438,440,538)(442,638,639)(445,558,536)(446,449,642)(447,448,612)(450,643,610)(451,636,637)(459,645,646)(460,647,606)(461,576,525)(474,577,652)(479,656,657)(480,515,658)(486,661,533)(490,664,665)(493,668,667)(500,669,582)(504,671,659)(508,581,615)(513,651,644)(514,675,611)(518,616,543)(521,545,681)(527,683,662)(531,672,541)(534,670,682)(542,589,690)(553,587,684)(559,627,560)(561,693,628)(566,650,686)(567,685,698)(573,601,700)(575,597,688)(584,699,704)(585,696,648)(586,617,703)(592,649,613)(593,697,694)(595,680,692)(603,677,673)(604,707,695)(605,641,705)(609,687,653)(614,709,663)(623,689,711)(640,678,674)(654,679,710)(655,706,714)(660,717,713)(666,715,708)(676,718,701)(691,702,716)(712,720,719) );
 
Copy content sage:G = PermutationGroup(['(1,2,6,16,42,107,197,382,403,281,127,280,302,325,234,103,40,15,5,14,36,93,214,409,617,443,352,178,145,57,144,301,521,680,586,372,439,330,208,89,207,396,606,491,666,604,394,590,652,609,398,608,700,632,683,581,365,427,630,612,400,611,708,649,466,648,686,535,312,471,622,701,582,366,185,77,184,363,311,534,435,232,102,123,83,32,12,4)(3,9,23,60,149,220,417,242,106,41,105,238,445,355,180,354,397,249,459,432,303,254,464,635,625,452,644,461,252,231,101,39,100,227,426,628,655,478,544,661,520,300,517,657,715,693,645,712,703,584,369,505,672,675,631,428,482,486,272,381,196,155,62,154,320,292,134,52,133,289,203,390,542,322,296,514,326,159,64,24,63,116,259,348,173,70,27,10)(7,19,49,126,277,493,667,695,562,344,171,69,170,94,143,56,142,256,465,572,577,358,576,501,284,500,670,594,414,523,488,273,124,48,18,47,120,266,477,568,699,716,658,634,707,684,527,306,146,58,22,8,21,55,140,298,162,331,484,347,172,346,457,247,109,246,456,337,524,304,368,187,78,30,11,29,74,88,206,340,167,115,258,418,221,136,53,20)(13,34,87,122,271,479,267,285,130,50,129,282,189,79,176,71,175,315,151,314,472,490,274,462,384,198,84,82,195,378,595,692,551,333,163,332,550,422,460,250,287,503,494,492,276,125,275,168,68,26,67,165,336,553,532,685,682,522,569,516,677,717,719,681,575,357,181,356,573,359,578,607,664,602,580,579,468,650,593,377,193,81,31,80,190,211,91,35)(17,45,114,257,186,367,364,469,629,688,539,615,405,408,226,99,38,98,223,202,86,33,85,76,128,278,495,669,552,334,313,150,177,157,156,265,119,251,111,43,110,248,454,605,395,415,218,95,217,412,545,324,158,323,543,496,598,674,512,673,592,375,192,295,513,647,663,489,662,526,525,305,476,654,713,720,718,714,706,596,705,690,589,374,191,263,118,46)(25,65,161,328,506,603,393,205,392,339,166,135,141,270,121,269,453,243,283,499,668,555,338,554,380,386,199,385,564,502,329,279,321,473,262,362,316,536,451,241,450,619,710,694,561,343,434,627,424,225,423,402,376,360,268,480,345,559,341,558,618,531,309,148,59,147,307,529,665,698,614,404,533,310,436,528,433,463,455,556,659,481,642,541,319,335,164,66)(28,72,137,294,510,431,230,245,108,201,389,599,679,519,299,518,678,563,588,509,574,601,391,204,97,37,96,219,416,621,620,507,290,407,388,200,387,571,497,624,567,351,566,697,656,676,515,297,139,54,138,264,474,651,546,549,327,160,174,349,255,113,44,112,253,210,90,209,401,613,704,702,583,626,420,222,419,623,711,709,633,458,508,291,498,353,179,73)(51,131,286,430,228,429,557,487,383,597,641,444,640,600,689,540,318,153,61,152,224,216,411,570,511,653,475,421,467,413,410,215,229,293,483,660,485,470,399,610,538,317,537,687,587,373,308,530,547,361,183,75,182,261,117,260,213,92,212,406,616,643,646,671,591,696,565,350,425,244,379,194,371,188,370,585,548,691,560,342,169,237,104,236,442,504,288,132)(233,437,636,638,447,239,446,438)(235,440,449,240,448,639,637,441)', '(1,3,5)(2,7,8)(4,11,13)(6,17,18)(9,24,25)(10,26,28)(12,31,33)(14,37,38)(15,39,41)(16,43,44)(19,50,51)(20,52,54)(21,56,57)(22,58,59)(23,61,62)(27,69,71)(29,75,76)(30,77,79)(32,82,84)(34,88,89)(35,90,92)(36,94,95)(40,102,104)(42,108,109)(45,115,116)(46,117,119)(47,121,122)(48,123,125)(49,127,128)(53,135,137)(55,141,140)(60,150,151)(63,156,157)(64,158,160)(65,114,162)(66,163,73)(67,166,167)(68,143,169)(70,172,174)(72,177,178)(74,180,181)(78,186,188)(80,191,155)(81,192,194)(83,196,197)(85,199,200)(86,201,203)(87,204,205)(91,146,134)(93,215,216)(96,220,221)(97,222,173)(98,224,139)(99,225,175)(100,228,229)(101,230,202)(103,233,235)(105,239,240)(106,241,243)(107,132,244)(110,249,250)(111,251,252)(112,130,254)(113,147,256)(118,262,264)(120,267,268)(124,272,274)(126,278,279)(129,283,284)(131,248,287)(133,290,291)(136,293,219)(138,295,296)(142,299,300)(144,302,303)(145,304,305)(148,308,310)(149,311,312)(152,316,317)(153,236,319)(154,321,322)(159,325,238)(161,329,330)(164,334,327)(165,337,338)(168,340,341)(170,343,218)(171,344,345)(176,350,351)(179,333,214)(182,358,359)(183,360,362)(184,364,314)(185,365,320)(187,332,369)(189,372,368)(190,373,263)(193,376,378)(195,380,357)(198,383,246)(206,394,395)(207,397,398)(208,399,400)(209,402,275)(210,403,384)(211,404,405)(212,388,407)(213,255,408)(217,413,414)(223,421,422)(226,297,425)(227,427,428)(231,432,433)(232,434,436)(234,288,439)(237,443,444)(242,424,452)(245,454,455)(247,381,458)(253,462,463)(257,466,331)(258,467,468)(259,389,469)(260,470,471)(261,323,472)(265,475,476)(266,478,415)(269,481,482)(270,328,483)(271,484,485)(273,487,489)(276,491,488)(277,464,494)(280,496,497)(281,406,361)(282,386,498)(285,501,307)(286,502,499)(289,505,506)(292,509,336)(294,511,512)(298,516,517)(301,522,523)(306,526,528)(309,524,532)(313,342,315)(318,539,349)(324,544,546)(326,547,548)(335,529,370)(339,556,557)(346,563,555)(347,366,562)(348,564,355)(352,568,549)(353,569,570)(354,571,572)(356,473,574)(363,579,580)(367,583,393)(371,551,477)(374,588,590)(375,591,510)(377,417,594)(379,552,596)(382,453,535)(385,598,503)(387,530,465)(390,456,495)(391,600,492)(392,602,550)(396,607,578)(401,507,520)(409,608,618)(410,619,554)(411,620,621)(412,416,423)(418,599,622)(419,624,625)(420,626,565)(426,629,519)(429,632,540)(430,457,633)(431,634,635)(435,630,537)(437,441,631)(438,440,538)(442,638,639)(445,558,536)(446,449,642)(447,448,612)(450,643,610)(451,636,637)(459,645,646)(460,647,606)(461,576,525)(474,577,652)(479,656,657)(480,515,658)(486,661,533)(490,664,665)(493,668,667)(500,669,582)(504,671,659)(508,581,615)(513,651,644)(514,675,611)(518,616,543)(521,545,681)(527,683,662)(531,672,541)(534,670,682)(542,589,690)(553,587,684)(559,627,560)(561,693,628)(566,650,686)(567,685,698)(573,601,700)(575,597,688)(584,699,704)(585,696,648)(586,617,703)(592,649,613)(593,697,694)(595,680,692)(603,677,673)(604,707,695)(605,641,705)(609,687,653)(614,709,663)(623,689,711)(640,678,674)(654,679,710)(655,706,714)(660,717,713)(666,715,708)(676,718,701)(691,702,716)(712,720,719)'])
 
Matrix group:$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{89})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 2, GF(89) | [[1, 1, 0, 1], [1, 0, 1, 1]] >;
 
Copy content gap:G := Group([[[ Z(89)^0, Z(89)^0 ], [ 0*Z(89), Z(89)^0 ]], [[ Z(89)^0, 0*Z(89) ], [ Z(89)^0, Z(89)^0 ]]]);
 
Copy content sage:MS = MatrixSpace(GF(89), 2, 2) G = MatrixGroup([MS([[1, 1], [0, 1]]), MS([[1, 0], [1, 1]])])
 
Direct product: not computed
Semidirect product: not computed
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\SL(2,89)$.

Homology

Abelianization: $C_1 $
Copy content comment:The abelianization of the group
 
Copy content magma:quo< G | CommutatorSubgroup(G) >;
 
Copy content gap:FactorGroup(G, DerivedSubgroup(G));
 
Copy content sage:G.quotient(G.commutator())
 
Schur multiplier: $C_1$
Copy content comment:The Schur multiplier of the group
 
Copy content gap:AbelianInvariantsMultiplier(G);
 
Copy content sage:G.homology(2)
 
Copy content sage_gap:G.AbelianInvariantsMultiplier()
 
Commutator length: $1$
Copy content comment:The commutator length of the group
 
Copy content gap:CommutatorLength(G);
 
Copy content sage_gap:G.CommutatorLength()
 

Subgroups

Copy content comment:List of subgroups of the group
 
Copy content magma:Subgroups(G);
 
Copy content gap:AllSubgroups(G);
 
Copy content sage:G.subgroups()
 
Copy content sage_gap:G.AllSubgroups()
 

Subgroup data has not been computed.

Character theory

Copy content comment:Character table
 
Copy content magma:CharacterTable(G); // Output not guaranteed to exactly match the LMFDB table
 
Copy content gap:CharacterTable(G); # Output not guaranteed to exactly match the LMFDB table
 
Copy content sage:G.character_table() # Output not guaranteed to exactly match the LMFDB table
 
Copy content sage_gap:G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table
 

Complex character table

The $93 \times 93$ character table is not available for this group.

Rational character table

The $20 \times 20$ rational character table is not available for this group.