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Properties

Label 1771440.a
Order \( 2^{4} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 61 \)
Exponent \( 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 61 \)
Nilpotent no
Solvable no
$\card{G^{\mathrm{ab}}}$ \( 1 \)
$\card{Z(G)}$ 2
$\card{\Aut(G)}$ \( 2^{5} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 61 \)
$\card{\mathrm{Out}(G)}$ \( 2^{2} \)
Perm deg. $976$
Trans deg. not computed
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SL(2, 121);
 
Copy content gap:G := SL(2, 121);
 
Copy content sage:G = SL(2, 121)
 
Copy content comment:Define the group as a permutation group
 

Group information

Description:$\SL(2,121)$
Order: \(1771440\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 61 \)
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: \(80520\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 61 \)
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:$\PSL(2,121).C_2^2$, of order \(3542880\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 61 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$, $\PSL(2,121)$
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 10 11 12 15 20 22 24 30 40 60 61 120 122
Elements 1 1 14762 14762 29524 14762 29524 29524 14640 29524 59048 59048 14640 59048 59048 118096 118096 435600 236192 435600 1771440
Conjugacy classes   1 1 1 1 2 1 2 2 2 2 4 4 2 4 4 8 8 30 16 30 125
Divisions 1 1 1 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 1 1 22
Autjugacy classes 1 1 1 1 2 1 1 2 1 2 2 2 1 2 2 4 4 15 8 15 68

Minimal presentations

Permutation degree:$976$
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 60 not computed not computed
Arbitrary not computed not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Groups of Lie type:$\SL(2,121)$, $\SU(2,121)$, $\SpinMinus(4,11)$
Permutation group:Degree $976$ $\langle(1,2,6,16,42,108,265,216,484,787,442,786,754,429,767,590,379,162,378,717,802,965,915,729,388,567,452,207,83,206,469,614,297,122,296,491,545,865,855,794,546,256,418,180,417,580,549,258,105,257,334,138,333,392,168,391,431,771,531,852,609,413,178,412,751,719,533,243,359,151,59,22,8,21,55,141,267,407,175,406,547,736,394,169,393,678,929,974,956,948,749,410,177,70,176,408,745,645,657,605,291,119,46,17,45,115,282,585,892,801,818,568,272,396,458,209,84,32,12,4) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 976 | (1,2,6,16,42,108,265,216,484,787,442,786,754,429,767,590,379,162,378,717,802,965,915,729,388,567,452,207,83,206,469,614,297,122,296,491,545,865,855,794,546,256,418,180,417,580,549,258,105,257,334,138,333,392,168,391,431,771,531,852,609,413,178,412,751,719,533,243,359,151,59,22,8,21,55,141,267,407,175,406,547,736,394,169,393,678,929,974,956,948,749,410,177,70,176,408,745,645,657,605,291,119,46,17,45,115,282,585,892,801,818,568,272,396,458,209,84,32,12,4)(3,9,23,61,155,367,497,803,659,920,790,444,789,884,970,850,971,893,881,748,409,520,292,170,67,25,66,166,387,581,280,114,44,113,276,506,310,128,309,403,188,433,773,670,926,728,385,165,65,24,64,161,377,339,303,125,48,18,47,121,293,607,509,798,964,891,584,281,583,500,224,90,223,498,248,399,673,341,142,340,613,544,254,543,862,799,460,735,895,588,284,587,752,846,526,239,525,843,825,826,815,707,796,952,918,973,889,679,344,502,225,91,116,285,589,415,179,71,27,10)(5,14,36,94,229,512,249,277,577,869,553,261,552,868,561,875,727,384,246,100,38,99,242,287,117,286,591,617,299,123,298,357,150,356,215,338,157,369,278,454,197,453,698,360,447,283,586,800,463,201,462,237,381,163,380,721,603,290,602,902,608,485,783,440,782,514,836,750,411,456,647,322,646,660,599,672,839,879,821,486,217,87,33,86,213,308,629,421,182,72,181,419,443,788,595,837,797,910,687,351,686,475,812,951,890,582,665,335,202,464,556,872,833,503,250,470,255,104,40,15)(7,19,49,127,160,63,159,372,466,204,82,31,81,200,306,126,305,626,562,876,664,700,639,316,638,495,829,919,885,925,955,769,887,575,349,685,633,312,129,311,632,696,810,967,904,604,680,345,320,618,627,559,266,326,450,661,921,936,882,953,753,867,898,592,347,508,691,579,279,578,621,532,570,273,569,416,758,630,601,901,972,886,574,275,573,762,954,913,640,675,860,814,662,845,689,352,688,896,928,677,513,759,480,368,671,725,383,164,382,422,761,933,720,947,923,663,332,137,53,20)(11,29,75,187,430,768,641,593,899,864,958,775,932,682,931,968,894,708,616,842,522,841,805,962,824,718,792,922,883,781,439,191,76,190,435,777,927,676,911,838,959,844,723,916,969,858,538,572,274,571,635,315,424,741,731,389,730,950,935,732,828,493,477,210,476,813,628,307,521,235,96,234,517,648,478,238,524,760,420,668,863,701,830,499,402,173,401,331,136,330,395,230,494,327,654,778,874,669,337,140,54,139,102,39,101,247,539,317,131,50,130,313,634,823,555,734,451,196,79,30)(13,34,88,219,490,705,364,704,715,374,714,944,934,743,404,174,69,26,68,172,398,739,636,709,942,960,785,903,897,809,957,941,819,483,214,482,817,733,744,738,457,536,245,471,511,776,548,301,124,300,619,811,576,888,765,426,185,241,530,370,710,445,742,746,631,558,870,724,692,354,149,58,148,120,220,492,611,294,610,835,510,358,314,637,861,541,253,103,252,212,85,211,184,365,594,448,195,78,194,446,791,655,615,906,711,518,465,203,414,755,853,912,847,840,917,697,505,226,92,35)(28,73,183,423,193,77,192,441,784,914,652,324,651,807,737,540,251,515,231,489,218,488,405,658,329,371,158,62,110,270,565,542,693,937,772,432,481,806,516,233,95,232,400,740,504,487,822,857,537,856,620,907,975,949,726,681,346,145,56,144,295,612,496,222,89,221,152,361,263,107,41,106,259,550,866,940,722,597,438,535,244,534,827,779,436,228,93,227,507,834,943,713,373,712,474,208,473,667,336,666,924,849,528,240,98,37,97,236,523,468,205,467,804,966,930,854,650,428,186,74)(43,111,271,566,880,598,900,703,363,702,747,766,427,376,459,199,80,198,455,260,551,563,877,706,366,154,60,153,362,699,437,780,764,425,763,770,873,557,472,808,831,684,348,683,859,795,449,397,171,386,321,600,289,118,288,596,519,816,643,318,642,461,343,143,342,674,501,479,554,262,350,147,57,146,156,167,390,649,323,133,51,132,319,644,774,820,961,871,656,328,135,52,134,325,653,695,355,694,938,832,905,606,375,716,939,976,963,793,878,564,269,109,268,560,690,353,434,189,264,112)(302,622,908,945,756,527,848,623)(304,624,851,529,757,946,909,625), (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)(22,58,60)(23,62,63)(27,70,72)(29,76,77)(30,78,80)(32,83,85)(34,89,90)(35,91,93)(36,95,96)(40,103,105)(42,109,110)(45,116,117)(46,118,120)(47,122,123)(48,124,126)(49,128,129)(53,136,138)(55,142,143)(59,150,152)(61,156,157)(64,162,163)(65,164,74)(66,167,168)(67,169,171)(69,173,175)(71,178,180)(73,184,185)(75,188,189)(79,137,197)(81,201,202)(82,203,205)(84,208,210)(86,214,215)(87,216,218)(88,194,220)(94,230,231)(97,237,238)(98,239,241)(99,243,244)(100,245,181)(101,248,249)(102,250,251)(104,254,256)(106,260,261)(107,262,264)(108,266,267)(111,272,273)(112,274,275)(113,277,278)(114,279,281)(115,283,284)(119,290,292)(121,294,295)(125,302,304)(127,307,308)(130,314,315)(131,316,318)(132,320,321)(133,322,324)(134,326,247)(135,327,329)(139,335,310)(140,336,338)(141,339,176)(144,344,345)(145,258,282)(146,347,348)(147,349,351)(148,352,223)(149,353,355)(151,358,360)(153,363,364)(154,365,334)(155,368,236)(158,370,161)(159,373,374)(160,375,376)(165,384,386)(166,388,389)(170,395,396)(172,399,400)(174,403,405)(177,409,411)(179,414,416)(182,420,422)(183,424,425)(186,427,429)(187,431,432)(190,436,437)(191,438,440)(192,442,443)(193,444,445)(195,447,449)(196,450,452)(198,456,457)(199,458,460)(200,461,333)(204,385,233)(206,300,470)(207,471,472)(209,259,475)(211,478,479)(212,480,481)(213,417,377)(217,485,487)(219,350,491)(221,493,494)(222,495,497)(224,499,501)(225,407,503)(226,504,506)(227,508,509)(228,510,511)(229,513,514)(232,381,356)(234,518,519)(235,520,522)(240,527,529)(242,531,532)(246,537,538)(252,303,312)(253,361,542)(255,545,387)(257,547,548)(263,555,556)(265,557,558)(268,561,562)(269,563,382)(270,410,469)(271,311,567)(276,575,576)(280,580,582)(285,296,590)(286,592,593)(287,594,595)(288,597,598)(289,599,601)(291,604,606)(293,608,609)(297,613,577)(298,486,615)(299,616,618)(301,620,621)(306,498,627)(309,630,631)(313,635,636)(317,640,641)(319,372,645)(323,648,650)(325,496,634)(328,655,657)(330,421,659)(331,660,661)(332,662,664)(337,668,573)(340,670,671)(341,672,665)(342,675,676)(343,677,678)(346,455,682)(354,691,693)(357,696,406)(359,553,697)(362,700,701)(366,644,707)(369,708,709)(371,398,711)(378,611,718)(379,719,720)(380,722,723)(383,724,726)(390,732,733)(391,651,453)(392,543,439)(393,734,735)(394,552,737)(397,738,401)(402,534,741)(404,742,744)(408,746,747)(412,642,752)(413,753,754)(415,756,757)(418,560,743)(419,759,717)(423,762,584)(426,484,647)(428,585,549)(430,769,770)(433,774,706)(434,775,776)(435,765,778)(441,713,785)(446,792,793)(448,763,794)(451,796,591)(454,766,689)(459,797,798)(462,784,731)(463,570,464)(465,801,802)(466,803,587)(467,736,805)(468,806,807)(473,809,810)(474,574,811)(476,814,695)(477,815,816)(482,653,605)(483,818,820)(488,823,824)(489,619,681)(490,825,626)(492,826,827)(500,831,832)(502,799,607)(505,612,819)(507,751,600)(512,782,632)(515,674,690)(516,837,838)(517,795,565)(521,839,840)(523,539,821)(524,768,581)(525,844,800)(526,845,847)(528,835,850)(533,694,705)(535,853,854)(536,855,699)(540,812,859)(541,860,804)(544,863,864)(546,725,684)(550,856,667)(551,867,571)(554,870,871)(559,716,874)(564,679,879)(566,767,881)(568,882,883)(569,884,761)(572,885,628)(578,889,666)(579,865,877)(583,781,617)(586,893,875)(588,894,813)(589,896,897)(596,873,714)(602,903,892)(603,745,895)(614,749,834)(622,625,740)(623,624,692)(629,730,773)(637,910,760)(638,911,842)(639,912,772)(643,914,702)(646,866,915)(649,869,791)(652,902,916)(654,917,777)(656,886,841)(658,918,919)(663,922,868)(669,880,925)(673,861,876)(680,704,930)(683,927,933)(685,789,934)(686,935,936)(687,846,750)(698,939,940)(703,941,931)(710,852,828)(715,945,946)(721,948,755)(727,888,899)(729,878,862)(739,951,787)(748,952,904)(764,947,858)(771,956,957)(779,926,913)(780,959,953)(783,924,958)(786,944,961)(788,849,833)(790,962,900)(808,920,905)(817,890,963)(822,872,968)(829,908,909)(830,907,965)(836,969,970)(843,928,898)(848,851,954)(857,964,967)(887,972,921)(901,971,942)(906,938,974)(923,929,950)(932,976,943)(955,975,960) >;
 
Copy content gap:G := Group( (1,2,6,16,42,108,265,216,484,787,442,786,754,429,767,590,379,162,378,717,802,965,915,729,388,567,452,207,83,206,469,614,297,122,296,491,545,865,855,794,546,256,418,180,417,580,549,258,105,257,334,138,333,392,168,391,431,771,531,852,609,413,178,412,751,719,533,243,359,151,59,22,8,21,55,141,267,407,175,406,547,736,394,169,393,678,929,974,956,948,749,410,177,70,176,408,745,645,657,605,291,119,46,17,45,115,282,585,892,801,818,568,272,396,458,209,84,32,12,4)(3,9,23,61,155,367,497,803,659,920,790,444,789,884,970,850,971,893,881,748,409,520,292,170,67,25,66,166,387,581,280,114,44,113,276,506,310,128,309,403,188,433,773,670,926,728,385,165,65,24,64,161,377,339,303,125,48,18,47,121,293,607,509,798,964,891,584,281,583,500,224,90,223,498,248,399,673,341,142,340,613,544,254,543,862,799,460,735,895,588,284,587,752,846,526,239,525,843,825,826,815,707,796,952,918,973,889,679,344,502,225,91,116,285,589,415,179,71,27,10)(5,14,36,94,229,512,249,277,577,869,553,261,552,868,561,875,727,384,246,100,38,99,242,287,117,286,591,617,299,123,298,357,150,356,215,338,157,369,278,454,197,453,698,360,447,283,586,800,463,201,462,237,381,163,380,721,603,290,602,902,608,485,783,440,782,514,836,750,411,456,647,322,646,660,599,672,839,879,821,486,217,87,33,86,213,308,629,421,182,72,181,419,443,788,595,837,797,910,687,351,686,475,812,951,890,582,665,335,202,464,556,872,833,503,250,470,255,104,40,15)(7,19,49,127,160,63,159,372,466,204,82,31,81,200,306,126,305,626,562,876,664,700,639,316,638,495,829,919,885,925,955,769,887,575,349,685,633,312,129,311,632,696,810,967,904,604,680,345,320,618,627,559,266,326,450,661,921,936,882,953,753,867,898,592,347,508,691,579,279,578,621,532,570,273,569,416,758,630,601,901,972,886,574,275,573,762,954,913,640,675,860,814,662,845,689,352,688,896,928,677,513,759,480,368,671,725,383,164,382,422,761,933,720,947,923,663,332,137,53,20)(11,29,75,187,430,768,641,593,899,864,958,775,932,682,931,968,894,708,616,842,522,841,805,962,824,718,792,922,883,781,439,191,76,190,435,777,927,676,911,838,959,844,723,916,969,858,538,572,274,571,635,315,424,741,731,389,730,950,935,732,828,493,477,210,476,813,628,307,521,235,96,234,517,648,478,238,524,760,420,668,863,701,830,499,402,173,401,331,136,330,395,230,494,327,654,778,874,669,337,140,54,139,102,39,101,247,539,317,131,50,130,313,634,823,555,734,451,196,79,30)(13,34,88,219,490,705,364,704,715,374,714,944,934,743,404,174,69,26,68,172,398,739,636,709,942,960,785,903,897,809,957,941,819,483,214,482,817,733,744,738,457,536,245,471,511,776,548,301,124,300,619,811,576,888,765,426,185,241,530,370,710,445,742,746,631,558,870,724,692,354,149,58,148,120,220,492,611,294,610,835,510,358,314,637,861,541,253,103,252,212,85,211,184,365,594,448,195,78,194,446,791,655,615,906,711,518,465,203,414,755,853,912,847,840,917,697,505,226,92,35)(28,73,183,423,193,77,192,441,784,914,652,324,651,807,737,540,251,515,231,489,218,488,405,658,329,371,158,62,110,270,565,542,693,937,772,432,481,806,516,233,95,232,400,740,504,487,822,857,537,856,620,907,975,949,726,681,346,145,56,144,295,612,496,222,89,221,152,361,263,107,41,106,259,550,866,940,722,597,438,535,244,534,827,779,436,228,93,227,507,834,943,713,373,712,474,208,473,667,336,666,924,849,528,240,98,37,97,236,523,468,205,467,804,966,930,854,650,428,186,74)(43,111,271,566,880,598,900,703,363,702,747,766,427,376,459,199,80,198,455,260,551,563,877,706,366,154,60,153,362,699,437,780,764,425,763,770,873,557,472,808,831,684,348,683,859,795,449,397,171,386,321,600,289,118,288,596,519,816,643,318,642,461,343,143,342,674,501,479,554,262,350,147,57,146,156,167,390,649,323,133,51,132,319,644,774,820,961,871,656,328,135,52,134,325,653,695,355,694,938,832,905,606,375,716,939,976,963,793,878,564,269,109,268,560,690,353,434,189,264,112)(302,622,908,945,756,527,848,623)(304,624,851,529,757,946,909,625), (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)(22,58,60)(23,62,63)(27,70,72)(29,76,77)(30,78,80)(32,83,85)(34,89,90)(35,91,93)(36,95,96)(40,103,105)(42,109,110)(45,116,117)(46,118,120)(47,122,123)(48,124,126)(49,128,129)(53,136,138)(55,142,143)(59,150,152)(61,156,157)(64,162,163)(65,164,74)(66,167,168)(67,169,171)(69,173,175)(71,178,180)(73,184,185)(75,188,189)(79,137,197)(81,201,202)(82,203,205)(84,208,210)(86,214,215)(87,216,218)(88,194,220)(94,230,231)(97,237,238)(98,239,241)(99,243,244)(100,245,181)(101,248,249)(102,250,251)(104,254,256)(106,260,261)(107,262,264)(108,266,267)(111,272,273)(112,274,275)(113,277,278)(114,279,281)(115,283,284)(119,290,292)(121,294,295)(125,302,304)(127,307,308)(130,314,315)(131,316,318)(132,320,321)(133,322,324)(134,326,247)(135,327,329)(139,335,310)(140,336,338)(141,339,176)(144,344,345)(145,258,282)(146,347,348)(147,349,351)(148,352,223)(149,353,355)(151,358,360)(153,363,364)(154,365,334)(155,368,236)(158,370,161)(159,373,374)(160,375,376)(165,384,386)(166,388,389)(170,395,396)(172,399,400)(174,403,405)(177,409,411)(179,414,416)(182,420,422)(183,424,425)(186,427,429)(187,431,432)(190,436,437)(191,438,440)(192,442,443)(193,444,445)(195,447,449)(196,450,452)(198,456,457)(199,458,460)(200,461,333)(204,385,233)(206,300,470)(207,471,472)(209,259,475)(211,478,479)(212,480,481)(213,417,377)(217,485,487)(219,350,491)(221,493,494)(222,495,497)(224,499,501)(225,407,503)(226,504,506)(227,508,509)(228,510,511)(229,513,514)(232,381,356)(234,518,519)(235,520,522)(240,527,529)(242,531,532)(246,537,538)(252,303,312)(253,361,542)(255,545,387)(257,547,548)(263,555,556)(265,557,558)(268,561,562)(269,563,382)(270,410,469)(271,311,567)(276,575,576)(280,580,582)(285,296,590)(286,592,593)(287,594,595)(288,597,598)(289,599,601)(291,604,606)(293,608,609)(297,613,577)(298,486,615)(299,616,618)(301,620,621)(306,498,627)(309,630,631)(313,635,636)(317,640,641)(319,372,645)(323,648,650)(325,496,634)(328,655,657)(330,421,659)(331,660,661)(332,662,664)(337,668,573)(340,670,671)(341,672,665)(342,675,676)(343,677,678)(346,455,682)(354,691,693)(357,696,406)(359,553,697)(362,700,701)(366,644,707)(369,708,709)(371,398,711)(378,611,718)(379,719,720)(380,722,723)(383,724,726)(390,732,733)(391,651,453)(392,543,439)(393,734,735)(394,552,737)(397,738,401)(402,534,741)(404,742,744)(408,746,747)(412,642,752)(413,753,754)(415,756,757)(418,560,743)(419,759,717)(423,762,584)(426,484,647)(428,585,549)(430,769,770)(433,774,706)(434,775,776)(435,765,778)(441,713,785)(446,792,793)(448,763,794)(451,796,591)(454,766,689)(459,797,798)(462,784,731)(463,570,464)(465,801,802)(466,803,587)(467,736,805)(468,806,807)(473,809,810)(474,574,811)(476,814,695)(477,815,816)(482,653,605)(483,818,820)(488,823,824)(489,619,681)(490,825,626)(492,826,827)(500,831,832)(502,799,607)(505,612,819)(507,751,600)(512,782,632)(515,674,690)(516,837,838)(517,795,565)(521,839,840)(523,539,821)(524,768,581)(525,844,800)(526,845,847)(528,835,850)(533,694,705)(535,853,854)(536,855,699)(540,812,859)(541,860,804)(544,863,864)(546,725,684)(550,856,667)(551,867,571)(554,870,871)(559,716,874)(564,679,879)(566,767,881)(568,882,883)(569,884,761)(572,885,628)(578,889,666)(579,865,877)(583,781,617)(586,893,875)(588,894,813)(589,896,897)(596,873,714)(602,903,892)(603,745,895)(614,749,834)(622,625,740)(623,624,692)(629,730,773)(637,910,760)(638,911,842)(639,912,772)(643,914,702)(646,866,915)(649,869,791)(652,902,916)(654,917,777)(656,886,841)(658,918,919)(663,922,868)(669,880,925)(673,861,876)(680,704,930)(683,927,933)(685,789,934)(686,935,936)(687,846,750)(698,939,940)(703,941,931)(710,852,828)(715,945,946)(721,948,755)(727,888,899)(729,878,862)(739,951,787)(748,952,904)(764,947,858)(771,956,957)(779,926,913)(780,959,953)(783,924,958)(786,944,961)(788,849,833)(790,962,900)(808,920,905)(817,890,963)(822,872,968)(829,908,909)(830,907,965)(836,969,970)(843,928,898)(848,851,954)(857,964,967)(887,972,921)(901,971,942)(906,938,974)(923,929,950)(932,976,943)(955,975,960) );
 
Copy content sage:G = PermutationGroup(['(1,2,6,16,42,108,265,216,484,787,442,786,754,429,767,590,379,162,378,717,802,965,915,729,388,567,452,207,83,206,469,614,297,122,296,491,545,865,855,794,546,256,418,180,417,580,549,258,105,257,334,138,333,392,168,391,431,771,531,852,609,413,178,412,751,719,533,243,359,151,59,22,8,21,55,141,267,407,175,406,547,736,394,169,393,678,929,974,956,948,749,410,177,70,176,408,745,645,657,605,291,119,46,17,45,115,282,585,892,801,818,568,272,396,458,209,84,32,12,4)(3,9,23,61,155,367,497,803,659,920,790,444,789,884,970,850,971,893,881,748,409,520,292,170,67,25,66,166,387,581,280,114,44,113,276,506,310,128,309,403,188,433,773,670,926,728,385,165,65,24,64,161,377,339,303,125,48,18,47,121,293,607,509,798,964,891,584,281,583,500,224,90,223,498,248,399,673,341,142,340,613,544,254,543,862,799,460,735,895,588,284,587,752,846,526,239,525,843,825,826,815,707,796,952,918,973,889,679,344,502,225,91,116,285,589,415,179,71,27,10)(5,14,36,94,229,512,249,277,577,869,553,261,552,868,561,875,727,384,246,100,38,99,242,287,117,286,591,617,299,123,298,357,150,356,215,338,157,369,278,454,197,453,698,360,447,283,586,800,463,201,462,237,381,163,380,721,603,290,602,902,608,485,783,440,782,514,836,750,411,456,647,322,646,660,599,672,839,879,821,486,217,87,33,86,213,308,629,421,182,72,181,419,443,788,595,837,797,910,687,351,686,475,812,951,890,582,665,335,202,464,556,872,833,503,250,470,255,104,40,15)(7,19,49,127,160,63,159,372,466,204,82,31,81,200,306,126,305,626,562,876,664,700,639,316,638,495,829,919,885,925,955,769,887,575,349,685,633,312,129,311,632,696,810,967,904,604,680,345,320,618,627,559,266,326,450,661,921,936,882,953,753,867,898,592,347,508,691,579,279,578,621,532,570,273,569,416,758,630,601,901,972,886,574,275,573,762,954,913,640,675,860,814,662,845,689,352,688,896,928,677,513,759,480,368,671,725,383,164,382,422,761,933,720,947,923,663,332,137,53,20)(11,29,75,187,430,768,641,593,899,864,958,775,932,682,931,968,894,708,616,842,522,841,805,962,824,718,792,922,883,781,439,191,76,190,435,777,927,676,911,838,959,844,723,916,969,858,538,572,274,571,635,315,424,741,731,389,730,950,935,732,828,493,477,210,476,813,628,307,521,235,96,234,517,648,478,238,524,760,420,668,863,701,830,499,402,173,401,331,136,330,395,230,494,327,654,778,874,669,337,140,54,139,102,39,101,247,539,317,131,50,130,313,634,823,555,734,451,196,79,30)(13,34,88,219,490,705,364,704,715,374,714,944,934,743,404,174,69,26,68,172,398,739,636,709,942,960,785,903,897,809,957,941,819,483,214,482,817,733,744,738,457,536,245,471,511,776,548,301,124,300,619,811,576,888,765,426,185,241,530,370,710,445,742,746,631,558,870,724,692,354,149,58,148,120,220,492,611,294,610,835,510,358,314,637,861,541,253,103,252,212,85,211,184,365,594,448,195,78,194,446,791,655,615,906,711,518,465,203,414,755,853,912,847,840,917,697,505,226,92,35)(28,73,183,423,193,77,192,441,784,914,652,324,651,807,737,540,251,515,231,489,218,488,405,658,329,371,158,62,110,270,565,542,693,937,772,432,481,806,516,233,95,232,400,740,504,487,822,857,537,856,620,907,975,949,726,681,346,145,56,144,295,612,496,222,89,221,152,361,263,107,41,106,259,550,866,940,722,597,438,535,244,534,827,779,436,228,93,227,507,834,943,713,373,712,474,208,473,667,336,666,924,849,528,240,98,37,97,236,523,468,205,467,804,966,930,854,650,428,186,74)(43,111,271,566,880,598,900,703,363,702,747,766,427,376,459,199,80,198,455,260,551,563,877,706,366,154,60,153,362,699,437,780,764,425,763,770,873,557,472,808,831,684,348,683,859,795,449,397,171,386,321,600,289,118,288,596,519,816,643,318,642,461,343,143,342,674,501,479,554,262,350,147,57,146,156,167,390,649,323,133,51,132,319,644,774,820,961,871,656,328,135,52,134,325,653,695,355,694,938,832,905,606,375,716,939,976,963,793,878,564,269,109,268,560,690,353,434,189,264,112)(302,622,908,945,756,527,848,623)(304,624,851,529,757,946,909,625)', '(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)(22,58,60)(23,62,63)(27,70,72)(29,76,77)(30,78,80)(32,83,85)(34,89,90)(35,91,93)(36,95,96)(40,103,105)(42,109,110)(45,116,117)(46,118,120)(47,122,123)(48,124,126)(49,128,129)(53,136,138)(55,142,143)(59,150,152)(61,156,157)(64,162,163)(65,164,74)(66,167,168)(67,169,171)(69,173,175)(71,178,180)(73,184,185)(75,188,189)(79,137,197)(81,201,202)(82,203,205)(84,208,210)(86,214,215)(87,216,218)(88,194,220)(94,230,231)(97,237,238)(98,239,241)(99,243,244)(100,245,181)(101,248,249)(102,250,251)(104,254,256)(106,260,261)(107,262,264)(108,266,267)(111,272,273)(112,274,275)(113,277,278)(114,279,281)(115,283,284)(119,290,292)(121,294,295)(125,302,304)(127,307,308)(130,314,315)(131,316,318)(132,320,321)(133,322,324)(134,326,247)(135,327,329)(139,335,310)(140,336,338)(141,339,176)(144,344,345)(145,258,282)(146,347,348)(147,349,351)(148,352,223)(149,353,355)(151,358,360)(153,363,364)(154,365,334)(155,368,236)(158,370,161)(159,373,374)(160,375,376)(165,384,386)(166,388,389)(170,395,396)(172,399,400)(174,403,405)(177,409,411)(179,414,416)(182,420,422)(183,424,425)(186,427,429)(187,431,432)(190,436,437)(191,438,440)(192,442,443)(193,444,445)(195,447,449)(196,450,452)(198,456,457)(199,458,460)(200,461,333)(204,385,233)(206,300,470)(207,471,472)(209,259,475)(211,478,479)(212,480,481)(213,417,377)(217,485,487)(219,350,491)(221,493,494)(222,495,497)(224,499,501)(225,407,503)(226,504,506)(227,508,509)(228,510,511)(229,513,514)(232,381,356)(234,518,519)(235,520,522)(240,527,529)(242,531,532)(246,537,538)(252,303,312)(253,361,542)(255,545,387)(257,547,548)(263,555,556)(265,557,558)(268,561,562)(269,563,382)(270,410,469)(271,311,567)(276,575,576)(280,580,582)(285,296,590)(286,592,593)(287,594,595)(288,597,598)(289,599,601)(291,604,606)(293,608,609)(297,613,577)(298,486,615)(299,616,618)(301,620,621)(306,498,627)(309,630,631)(313,635,636)(317,640,641)(319,372,645)(323,648,650)(325,496,634)(328,655,657)(330,421,659)(331,660,661)(332,662,664)(337,668,573)(340,670,671)(341,672,665)(342,675,676)(343,677,678)(346,455,682)(354,691,693)(357,696,406)(359,553,697)(362,700,701)(366,644,707)(369,708,709)(371,398,711)(378,611,718)(379,719,720)(380,722,723)(383,724,726)(390,732,733)(391,651,453)(392,543,439)(393,734,735)(394,552,737)(397,738,401)(402,534,741)(404,742,744)(408,746,747)(412,642,752)(413,753,754)(415,756,757)(418,560,743)(419,759,717)(423,762,584)(426,484,647)(428,585,549)(430,769,770)(433,774,706)(434,775,776)(435,765,778)(441,713,785)(446,792,793)(448,763,794)(451,796,591)(454,766,689)(459,797,798)(462,784,731)(463,570,464)(465,801,802)(466,803,587)(467,736,805)(468,806,807)(473,809,810)(474,574,811)(476,814,695)(477,815,816)(482,653,605)(483,818,820)(488,823,824)(489,619,681)(490,825,626)(492,826,827)(500,831,832)(502,799,607)(505,612,819)(507,751,600)(512,782,632)(515,674,690)(516,837,838)(517,795,565)(521,839,840)(523,539,821)(524,768,581)(525,844,800)(526,845,847)(528,835,850)(533,694,705)(535,853,854)(536,855,699)(540,812,859)(541,860,804)(544,863,864)(546,725,684)(550,856,667)(551,867,571)(554,870,871)(559,716,874)(564,679,879)(566,767,881)(568,882,883)(569,884,761)(572,885,628)(578,889,666)(579,865,877)(583,781,617)(586,893,875)(588,894,813)(589,896,897)(596,873,714)(602,903,892)(603,745,895)(614,749,834)(622,625,740)(623,624,692)(629,730,773)(637,910,760)(638,911,842)(639,912,772)(643,914,702)(646,866,915)(649,869,791)(652,902,916)(654,917,777)(656,886,841)(658,918,919)(663,922,868)(669,880,925)(673,861,876)(680,704,930)(683,927,933)(685,789,934)(686,935,936)(687,846,750)(698,939,940)(703,941,931)(710,852,828)(715,945,946)(721,948,755)(727,888,899)(729,878,862)(739,951,787)(748,952,904)(764,947,858)(771,956,957)(779,926,913)(780,959,953)(783,924,958)(786,944,961)(788,849,833)(790,962,900)(808,920,905)(817,890,963)(822,872,968)(829,908,909)(830,907,965)(836,969,970)(843,928,898)(848,851,954)(857,964,967)(887,972,921)(901,971,942)(906,938,974)(923,929,950)(932,976,943)(955,975,960)'])
 
Matrix group:$\left\langle \left(\begin{array}{rrrr} 8 & 1 & 8 & 8 \\ 2 & 5 & 3 & 8 \\ 5 & 3 & 6 & 10 \\ 10 & 5 & 9 & 3 \end{array}\right), \left(\begin{array}{rrrr} 6 & 1 & 5 & 9 \\ 3 & 9 & 9 & 5 \\ 10 & 5 & 0 & 10 \\ 7 & 10 & 8 & 3 \end{array}\right), \left(\begin{array}{rrrr} 10 & 0 & 0 & 0 \\ 0 & 10 & 0 & 0 \\ 0 & 0 & 10 & 0 \\ 0 & 0 & 0 & 10 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 4, GF(11) | [[8, 1, 8, 8, 2, 5, 3, 8, 5, 3, 6, 10, 10, 5, 9, 3], [6, 1, 5, 9, 3, 9, 9, 5, 10, 5, 0, 10, 7, 10, 8, 3], [10, 0, 0, 0, 0, 10, 0, 0, 0, 0, 10, 0, 0, 0, 0, 10]] >;
 
Copy content gap:G := Group([[[ Z(11)^3, Z(11)^0, Z(11)^3, Z(11)^3 ], [ Z(11), Z(11)^4, Z(11)^8, Z(11)^3 ], [ Z(11)^4, Z(11)^8, Z(11)^9, Z(11)^5 ], [ Z(11)^5, Z(11)^4, Z(11)^6, Z(11)^8 ]], [[ Z(11)^9, Z(11)^0, Z(11)^4, Z(11)^6 ], [ Z(11)^8, Z(11)^6, Z(11)^6, Z(11)^4 ], [ Z(11)^5, Z(11)^4, 0*Z(11), Z(11)^5 ], [ Z(11)^7, Z(11)^5, Z(11)^3, Z(11)^8 ]], [[ Z(11)^5, 0*Z(11), 0*Z(11), 0*Z(11) ], [ 0*Z(11), Z(11)^5, 0*Z(11), 0*Z(11) ], [ 0*Z(11), 0*Z(11), Z(11)^5, 0*Z(11) ], [ 0*Z(11), 0*Z(11), 0*Z(11), Z(11)^5 ]]]);
 
Copy content sage:MS = MatrixSpace(GF(11), 4, 4) G = MatrixGroup([MS([[8, 1, 8, 8], [2, 5, 3, 8], [5, 3, 6, 10], [10, 5, 9, 3]]), MS([[6, 1, 5, 9], [3, 9, 9, 5], [10, 5, 0, 10], [7, 10, 8, 3]]), MS([[10, 0, 0, 0], [0, 10, 0, 0], [0, 0, 10, 0], [0, 0, 0, 10]])])
 
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 $\SL(2,121)$.

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 $125 \times 125$ character table is not available for this group.

Rational character table

The $22 \times 22$ rational character table is not available for this group.