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Properties

Label 9170703360.a
Order \( 2^{10} \cdot 3^{9} \cdot 5 \cdot 7 \cdot 13 \)
Exponent \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
Nilpotent no
Solvable no
$\card{G^{\mathrm{ab}}}$ \( 1 \)
$\card{Z(G)}$ 2
$\card{\Aut(G)}$ \( 2^{10} \cdot 3^{9} \cdot 5 \cdot 7 \cdot 13 \)
$\card{\mathrm{Out}(G)}$ \( 2 \)
Perm deg. $728$
Trans deg. not computed
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := Sp(6,3);
 
Copy content gap:G := Sp(6,3);
 
Copy content comment:Define the group as a permutation group
 
Copy content sage:G = PermutationGroup(['(1,3,8,18,34,64,115,200,236,138,78,42,22,10,4)(2,5,12,14,6)(7,15,29,54,96,169,286,377,458,304,182,103,58,31,16)(9,19,36,68,123,211,341,506,512,351,221,130,72,38,20)(11,23,44,45,24)(13,26,49,51,27)(17,32,60,107,188,313,468,616,618,472,316,193,111,62,33)(21,39,73,75,40)(25,46,83,147,252,396,550,710,687,560,402,257,150,85,47)(28,52,92,94,53)(30,55,98,81,143,245,385,454,612,602,441,291,174,99,56)(35,65,117,202,327,490,633,613,478,623,498,333,205,119,66)(37,69,124,212,343,308,464,538,379,241,141,240,214,126,70)(41,76,135,231,365,526,598,657,522,361,485,368,233,136,77)(43,79,140,239,376,446,607,523,457,372,531,382,243,142,80)(48,86,152,261,407,369,528,513,352,492,335,410,264,154,87)(50,89,158,160,90)(57,100,176,293,445,606,667,575,474,453,300,450,297,178,101)(59,104,110,184,105)(61,108,190,116,201,325,486,367,232,366,471,470,315,191,109)(63,112,194,278,227,133,74,132,225,358,519,475,318,196,113)(67,120,207,175,121)(71,127,216,218,128)(82,144,247,229,134,228,360,521,421,584,699,543,390,249,145)(84,148,254,255,149)(88,155,266,412,569,653,515,354,514,651,719,579,416,268,156)(91,161,275,276,162)(93,164,279,192,310,466,439,288,171,97,170,287,428,280,165)(95,166,281,189,301,180,102,179,299,238,374,533,432,283,167)(106,185,309,465,590,427,589,463,307,462,614,539,467,311,186)(114,197,320,321,198)(118,187,312,329,203)(122,208,336,337,209)(125,129,219,173,213)(131,222,353,355,223)(137,234,370,260,235)(139,237,373,532,381,438,461,306,183,305,459,380,242,289,172)(146,163,277,392,250)(151,258,403,562,671,698,617,626,480,625,715,660,565,405,259)(153,262,265,409,263)(157,269,303,339,210,338,503,558,401,557,482,323,199,322,270)(159,272,420,317,473,431,282,430,594,641,500,640,585,422,273)(168,284,434,599,573,460,389,248,388,540,679,636,600,436,285)(177,294,447,437,295)(181,302,456,423,274)(195,298,451,386,246)(204,330,493,494,331)(215,344,509,387,345)(217,346,363,230,362,524,644,505,340,504,499,334,359,226,290)(220,348,349)(224,356,516,654,727,728,673,672,702,694,627,481,364,350,357)(244,383,253,397,384)(251,393,546,496,332,495,452,611,693,564,404,563,695,547,394)(256,399,554,556,400)(267,413,574,700,724,664,706,583,604,444,561,675,680,576,414)(271,418,582,469,419)(292,442,603,665,701,708,628,725,609,449,296,448,314,455,443)(319,476,620,681,635,642,501,581,417,580,691,648,517,622,477)(324,483,508,408,484)(326,487,630,631,488)(342,507,646,632,489)(347,510,621,649,511)(371,529,568,619,530)(375,534,711,663,629,601,656,520,491,634,639,497,638,692,535)(378,440,425,588,536)(391,544,435,426,587,424,586,717,705,593,678,553,684,647,545)(395,548,595,713,549)(398,551,686,650,552)(406,566,527,661,567)(411,570,502,643,571)(415,577,655,518,578)(429,591,688,555,592)(433,596,559,668,597)(479,624,674,541,610)(525,658,659)(537,683,703,685,676)(542,652,716,572,637)(605,662,723,670,722)(608,709,704,689,707)(615,721,690,669,720)(645,714,666,682,696)(677,718,726,697,712)', '(1,2)(3,7)(4,9)(5,11)(6,13)(8,17)(10,21)(12,25)(14,18)(15,28)(16,30)(19,35)(20,37)(22,41)(23,43)(26,48)(27,50)(31,57)(32,59)(33,61)(34,63)(36,67)(38,71)(39,56)(40,74)(44,81)(45,82)(47,84)(49,88)(51,91)(53,93)(54,95)(55,97)(58,102)(60,106)(62,110)(64,114)(65,116)(66,118)(68,122)(70,125)(72,129)(73,131)(75,126)(76,134)(78,137)(79,139)(80,141)(83,146)(86,151)(87,153)(89,157)(90,159)(92,163)(96,168)(98,172)(99,173)(100,175)(101,177)(103,181)(104,183)(105,170)(107,187)(108,189)(109,149)(111,192)(112,161)(113,195)(115,199)(117,124)(119,204)(120,206)(121,193)(123,210)(127,215)(128,217)(130,220)(132,224)(133,226)(135,230)(136,232)(138,233)(140,238)(142,242)(143,244)(144,246)(145,248)(147,251)(148,253)(150,256)(152,260)(154,209)(155,265)(156,267)(158,271)(160,274)(162,227)(164,278)(167,282)(174,290)(176,292)(178,296)(179,298)(180,300)(182,303)(184,307)(185,308)(186,310)(188,200)(190,254)(191,314)(194,297)(196,317)(197,319)(198,311)(201,324)(202,326)(203,328)(205,332)(207,334)(208,335)(211,340)(212,342)(214,279)(218,347)(219,346)(221,350)(222,352)(223,354)(228,344)(229,361)(231,364)(234,369)(235,367)(236,371)(237,372)(239,375)(240,377)(241,378)(243,381)(245,373)(247,387)(250,391)(252,395)(255,398)(257,401)(258,313)(259,404)(261,406)(262,408)(263,353)(264,331)(266,411)(268,415)(269,417)(270,376)(272,305)(273,421)(275,424)(276,425)(277,426)(280,427)(281,429)(284,433)(285,435)(286,437)(288,438)(289,440)(293,444)(294,446)(295,432)(299,452)(301,454)(302,455)(304,457)(306,460)(309,449)(312,453)(315,469)(316,471)(318,474)(320,478)(321,479)(322,480)(323,481)(325,485)(327,489)(329,491)(330,492)(333,497)(336,500)(337,501)(338,502)(343,508)(345,488)(348,490)(351,461)(355,409)(357,517)(358,518)(359,520)(360,456)(362,523)(363,525)(365,473)(366,527)(379,537)(380,539)(382,538)(383,531)(384,521)(385,462)(386,535)(388,467)(389,541)(390,542)(392,396)(394,428)(399,553)(400,555)(402,559)(403,561)(405,550)(407,568)(410,569)(412,572)(413,573)(414,575)(416,566)(418,512)(419,466)(420,583)(422,439)(423,524)(430,593)(431,595)(434,598)(441,601)(442,567)(443,599)(445,605)(447,450)(448,608)(451,610)(459,613)(464,533)(465,591)(468,615)(470,563)(472,617)(475,619)(476,536)(477,621)(482,548)(483,628)(484,612)(487,629)(493,635)(494,636)(496,637)(499,571)(504,640)(506,645)(509,647)(510,630)(511,648)(513,552)(514,650)(515,652)(516,603)(519,654)(522,534)(526,660)(528,626)(529,579)(532,676)(543,682)(544,713)(545,574)(547,596)(549,668)(551,674)(554,560)(556,705)(557,589)(564,607)(565,700)(570,699)(576,698)(578,665)(580,683)(582,691)(586,707)(597,678)(602,646)(604,651)(609,704)(611,620)(618,723)(622,673)(623,714)(624,693)(625,687)(627,724)(632,663)(633,672)(634,716)(639,679)(642,677)(643,653)(644,697)(649,710)(655,719)(656,725)(657,702)(659,711)(664,715)(667,690)(684,706)(685,689)(686,709)(692,726)(694,695)(708,728)'])
 

Group information

Description:$\Sp(6,3)$
Order: \(9170703360\)\(\medspace = 2^{10} \cdot 3^{9} \cdot 5 \cdot 7 \cdot 13 \)
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: \(32760\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
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:Group of order \(9170703360\)\(\medspace = 2^{10} \cdot 3^{9} \cdot 5 \cdot 7 \cdot 13 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$, $\PSp(6,3)$
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 7 8 9 10 12 13 14 15 18 20 24 26 28 30 36
Elements 1 14743 5307848 12408552 38211264 111568184 327525120 382112640 382112640 114633792 1302013440 705438720 327525120 305690112 1146337920 458535168 764225280 705438720 655050240 917070336 509483520 9170703360
Conjugacy classes   1 3 7 6 1 35 1 3 8 3 28 2 1 2 20 2 4 2 2 6 4 141
Divisions 1 3 5 6 1 21 1 3 4 3 18 1 1 1 10 2 2 1 1 3 2 90
Autjugacy classes 1 3 5 6 1 21 1 3 4 3 18 2 1 1 10 2 2 2 2 3 2 93

Copy content comment:Compute statistics about the characters of G
 
Copy content magma:// Outputs [<d_1,c_1>, <d_2,c_2>, ...] where c_i is the number of irr. complex chars. of G with degree d_i CharacterDegrees(G);
 
Copy content gap:# Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i CharacterDegrees(G);
 
Copy content sage:# Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i character_degrees = [c[0] for c in G.character_table()] [[n, character_degrees.count(n)] for n in set(character_degrees)]
 
Copy content sage_gap:G.CharacterDegrees()
 

Dimension 1 13 14 26 28 78 91 105 168 182 195 273 364 455 520 546 728 819 910 1092 1170 1260 1365 1456 1820 2106 2184 2340 2457 2520 2730 2835 3120 3276 3640 4095 4368 4536 4914 5265 5460 5824 6552 7280 7371 8190 8736 9477 9828 10206 10920 11648 12285 13104 14040 14560 14742 16380 16640 17472 17920 18954 19656 19683 20412 21840 23296 24570 29120 29484 33280 35840
Irr. complex chars.   1 2 2 0 0 1 2 1 1 3 1 2 2 2 1 4 2 1 2 5 2 2 6 0 5 1 2 0 3 0 7 1 1 2 4 2 3 1 2 1 7 6 2 3 2 3 0 2 2 2 4 6 2 0 1 3 3 1 4 2 4 0 0 1 0 1 0 0 0 0 0 0 141
Irr. rational chars. 1 0 0 1 1 1 0 1 1 2 1 0 1 0 1 1 1 1 1 3 0 0 0 1 2 1 2 1 1 1 6 1 1 0 2 0 2 1 1 1 5 2 1 3 0 2 1 0 1 0 2 4 0 1 1 2 2 2 2 2 0 1 1 1 1 3 2 1 1 1 1 2 90

Minimal presentations

Permutation degree:$728$
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 14 28 28
Arbitrary not computed not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Groups of Lie type:$\Sp(6,3)$
Copy content magma:G := Sp(6,3);
 
Copy content gap:G := Sp(6,3);
 
Copy content sage:MS = MatrixSpace(GF(3), 6, 6) G = MatrixGroup([MS([[1, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 1], [0, 0, 2, 0, 0, 0]]), MS([[2, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 2]])])
 
Permutation group:Degree $728$ $\langle(1,3,8,18,34,64,115,200,236,138,78,42,22,10,4)(2,5,12,14,6)(7,15,29,54,96,169,286,377,458,304,182,103,58,31,16) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 728 | (1,3,8,18,34,64,115,200,236,138,78,42,22,10,4)(2,5,12,14,6)(7,15,29,54,96,169,286,377,458,304,182,103,58,31,16)(9,19,36,68,123,211,341,506,512,351,221,130,72,38,20)(11,23,44,45,24)(13,26,49,51,27)(17,32,60,107,188,313,468,616,618,472,316,193,111,62,33)(21,39,73,75,40)(25,46,83,147,252,396,550,710,687,560,402,257,150,85,47)(28,52,92,94,53)(30,55,98,81,143,245,385,454,612,602,441,291,174,99,56)(35,65,117,202,327,490,633,613,478,623,498,333,205,119,66)(37,69,124,212,343,308,464,538,379,241,141,240,214,126,70)(41,76,135,231,365,526,598,657,522,361,485,368,233,136,77)(43,79,140,239,376,446,607,523,457,372,531,382,243,142,80)(48,86,152,261,407,369,528,513,352,492,335,410,264,154,87)(50,89,158,160,90)(57,100,176,293,445,606,667,575,474,453,300,450,297,178,101)(59,104,110,184,105)(61,108,190,116,201,325,486,367,232,366,471,470,315,191,109)(63,112,194,278,227,133,74,132,225,358,519,475,318,196,113)(67,120,207,175,121)(71,127,216,218,128)(82,144,247,229,134,228,360,521,421,584,699,543,390,249,145)(84,148,254,255,149)(88,155,266,412,569,653,515,354,514,651,719,579,416,268,156)(91,161,275,276,162)(93,164,279,192,310,466,439,288,171,97,170,287,428,280,165)(95,166,281,189,301,180,102,179,299,238,374,533,432,283,167)(106,185,309,465,590,427,589,463,307,462,614,539,467,311,186)(114,197,320,321,198)(118,187,312,329,203)(122,208,336,337,209)(125,129,219,173,213)(131,222,353,355,223)(137,234,370,260,235)(139,237,373,532,381,438,461,306,183,305,459,380,242,289,172)(146,163,277,392,250)(151,258,403,562,671,698,617,626,480,625,715,660,565,405,259)(153,262,265,409,263)(157,269,303,339,210,338,503,558,401,557,482,323,199,322,270)(159,272,420,317,473,431,282,430,594,641,500,640,585,422,273)(168,284,434,599,573,460,389,248,388,540,679,636,600,436,285)(177,294,447,437,295)(181,302,456,423,274)(195,298,451,386,246)(204,330,493,494,331)(215,344,509,387,345)(217,346,363,230,362,524,644,505,340,504,499,334,359,226,290)(220,348,349)(224,356,516,654,727,728,673,672,702,694,627,481,364,350,357)(244,383,253,397,384)(251,393,546,496,332,495,452,611,693,564,404,563,695,547,394)(256,399,554,556,400)(267,413,574,700,724,664,706,583,604,444,561,675,680,576,414)(271,418,582,469,419)(292,442,603,665,701,708,628,725,609,449,296,448,314,455,443)(319,476,620,681,635,642,501,581,417,580,691,648,517,622,477)(324,483,508,408,484)(326,487,630,631,488)(342,507,646,632,489)(347,510,621,649,511)(371,529,568,619,530)(375,534,711,663,629,601,656,520,491,634,639,497,638,692,535)(378,440,425,588,536)(391,544,435,426,587,424,586,717,705,593,678,553,684,647,545)(395,548,595,713,549)(398,551,686,650,552)(406,566,527,661,567)(411,570,502,643,571)(415,577,655,518,578)(429,591,688,555,592)(433,596,559,668,597)(479,624,674,541,610)(525,658,659)(537,683,703,685,676)(542,652,716,572,637)(605,662,723,670,722)(608,709,704,689,707)(615,721,690,669,720)(645,714,666,682,696)(677,718,726,697,712), (1,2)(3,7)(4,9)(5,11)(6,13)(8,17)(10,21)(12,25)(14,18)(15,28)(16,30)(19,35)(20,37)(22,41)(23,43)(26,48)(27,50)(31,57)(32,59)(33,61)(34,63)(36,67)(38,71)(39,56)(40,74)(44,81)(45,82)(47,84)(49,88)(51,91)(53,93)(54,95)(55,97)(58,102)(60,106)(62,110)(64,114)(65,116)(66,118)(68,122)(70,125)(72,129)(73,131)(75,126)(76,134)(78,137)(79,139)(80,141)(83,146)(86,151)(87,153)(89,157)(90,159)(92,163)(96,168)(98,172)(99,173)(100,175)(101,177)(103,181)(104,183)(105,170)(107,187)(108,189)(109,149)(111,192)(112,161)(113,195)(115,199)(117,124)(119,204)(120,206)(121,193)(123,210)(127,215)(128,217)(130,220)(132,224)(133,226)(135,230)(136,232)(138,233)(140,238)(142,242)(143,244)(144,246)(145,248)(147,251)(148,253)(150,256)(152,260)(154,209)(155,265)(156,267)(158,271)(160,274)(162,227)(164,278)(167,282)(174,290)(176,292)(178,296)(179,298)(180,300)(182,303)(184,307)(185,308)(186,310)(188,200)(190,254)(191,314)(194,297)(196,317)(197,319)(198,311)(201,324)(202,326)(203,328)(205,332)(207,334)(208,335)(211,340)(212,342)(214,279)(218,347)(219,346)(221,350)(222,352)(223,354)(228,344)(229,361)(231,364)(234,369)(235,367)(236,371)(237,372)(239,375)(240,377)(241,378)(243,381)(245,373)(247,387)(250,391)(252,395)(255,398)(257,401)(258,313)(259,404)(261,406)(262,408)(263,353)(264,331)(266,411)(268,415)(269,417)(270,376)(272,305)(273,421)(275,424)(276,425)(277,426)(280,427)(281,429)(284,433)(285,435)(286,437)(288,438)(289,440)(293,444)(294,446)(295,432)(299,452)(301,454)(302,455)(304,457)(306,460)(309,449)(312,453)(315,469)(316,471)(318,474)(320,478)(321,479)(322,480)(323,481)(325,485)(327,489)(329,491)(330,492)(333,497)(336,500)(337,501)(338,502)(343,508)(345,488)(348,490)(351,461)(355,409)(357,517)(358,518)(359,520)(360,456)(362,523)(363,525)(365,473)(366,527)(379,537)(380,539)(382,538)(383,531)(384,521)(385,462)(386,535)(388,467)(389,541)(390,542)(392,396)(394,428)(399,553)(400,555)(402,559)(403,561)(405,550)(407,568)(410,569)(412,572)(413,573)(414,575)(416,566)(418,512)(419,466)(420,583)(422,439)(423,524)(430,593)(431,595)(434,598)(441,601)(442,567)(443,599)(445,605)(447,450)(448,608)(451,610)(459,613)(464,533)(465,591)(468,615)(470,563)(472,617)(475,619)(476,536)(477,621)(482,548)(483,628)(484,612)(487,629)(493,635)(494,636)(496,637)(499,571)(504,640)(506,645)(509,647)(510,630)(511,648)(513,552)(514,650)(515,652)(516,603)(519,654)(522,534)(526,660)(528,626)(529,579)(532,676)(543,682)(544,713)(545,574)(547,596)(549,668)(551,674)(554,560)(556,705)(557,589)(564,607)(565,700)(570,699)(576,698)(578,665)(580,683)(582,691)(586,707)(597,678)(602,646)(604,651)(609,704)(611,620)(618,723)(622,673)(623,714)(624,693)(625,687)(627,724)(632,663)(633,672)(634,716)(639,679)(642,677)(643,653)(644,697)(649,710)(655,719)(656,725)(657,702)(659,711)(664,715)(667,690)(684,706)(685,689)(686,709)(692,726)(694,695)(708,728) >;
 
Copy content gap:G := Group( (1,3,8,18,34,64,115,200,236,138,78,42,22,10,4)(2,5,12,14,6)(7,15,29,54,96,169,286,377,458,304,182,103,58,31,16)(9,19,36,68,123,211,341,506,512,351,221,130,72,38,20)(11,23,44,45,24)(13,26,49,51,27)(17,32,60,107,188,313,468,616,618,472,316,193,111,62,33)(21,39,73,75,40)(25,46,83,147,252,396,550,710,687,560,402,257,150,85,47)(28,52,92,94,53)(30,55,98,81,143,245,385,454,612,602,441,291,174,99,56)(35,65,117,202,327,490,633,613,478,623,498,333,205,119,66)(37,69,124,212,343,308,464,538,379,241,141,240,214,126,70)(41,76,135,231,365,526,598,657,522,361,485,368,233,136,77)(43,79,140,239,376,446,607,523,457,372,531,382,243,142,80)(48,86,152,261,407,369,528,513,352,492,335,410,264,154,87)(50,89,158,160,90)(57,100,176,293,445,606,667,575,474,453,300,450,297,178,101)(59,104,110,184,105)(61,108,190,116,201,325,486,367,232,366,471,470,315,191,109)(63,112,194,278,227,133,74,132,225,358,519,475,318,196,113)(67,120,207,175,121)(71,127,216,218,128)(82,144,247,229,134,228,360,521,421,584,699,543,390,249,145)(84,148,254,255,149)(88,155,266,412,569,653,515,354,514,651,719,579,416,268,156)(91,161,275,276,162)(93,164,279,192,310,466,439,288,171,97,170,287,428,280,165)(95,166,281,189,301,180,102,179,299,238,374,533,432,283,167)(106,185,309,465,590,427,589,463,307,462,614,539,467,311,186)(114,197,320,321,198)(118,187,312,329,203)(122,208,336,337,209)(125,129,219,173,213)(131,222,353,355,223)(137,234,370,260,235)(139,237,373,532,381,438,461,306,183,305,459,380,242,289,172)(146,163,277,392,250)(151,258,403,562,671,698,617,626,480,625,715,660,565,405,259)(153,262,265,409,263)(157,269,303,339,210,338,503,558,401,557,482,323,199,322,270)(159,272,420,317,473,431,282,430,594,641,500,640,585,422,273)(168,284,434,599,573,460,389,248,388,540,679,636,600,436,285)(177,294,447,437,295)(181,302,456,423,274)(195,298,451,386,246)(204,330,493,494,331)(215,344,509,387,345)(217,346,363,230,362,524,644,505,340,504,499,334,359,226,290)(220,348,349)(224,356,516,654,727,728,673,672,702,694,627,481,364,350,357)(244,383,253,397,384)(251,393,546,496,332,495,452,611,693,564,404,563,695,547,394)(256,399,554,556,400)(267,413,574,700,724,664,706,583,604,444,561,675,680,576,414)(271,418,582,469,419)(292,442,603,665,701,708,628,725,609,449,296,448,314,455,443)(319,476,620,681,635,642,501,581,417,580,691,648,517,622,477)(324,483,508,408,484)(326,487,630,631,488)(342,507,646,632,489)(347,510,621,649,511)(371,529,568,619,530)(375,534,711,663,629,601,656,520,491,634,639,497,638,692,535)(378,440,425,588,536)(391,544,435,426,587,424,586,717,705,593,678,553,684,647,545)(395,548,595,713,549)(398,551,686,650,552)(406,566,527,661,567)(411,570,502,643,571)(415,577,655,518,578)(429,591,688,555,592)(433,596,559,668,597)(479,624,674,541,610)(525,658,659)(537,683,703,685,676)(542,652,716,572,637)(605,662,723,670,722)(608,709,704,689,707)(615,721,690,669,720)(645,714,666,682,696)(677,718,726,697,712), (1,2)(3,7)(4,9)(5,11)(6,13)(8,17)(10,21)(12,25)(14,18)(15,28)(16,30)(19,35)(20,37)(22,41)(23,43)(26,48)(27,50)(31,57)(32,59)(33,61)(34,63)(36,67)(38,71)(39,56)(40,74)(44,81)(45,82)(47,84)(49,88)(51,91)(53,93)(54,95)(55,97)(58,102)(60,106)(62,110)(64,114)(65,116)(66,118)(68,122)(70,125)(72,129)(73,131)(75,126)(76,134)(78,137)(79,139)(80,141)(83,146)(86,151)(87,153)(89,157)(90,159)(92,163)(96,168)(98,172)(99,173)(100,175)(101,177)(103,181)(104,183)(105,170)(107,187)(108,189)(109,149)(111,192)(112,161)(113,195)(115,199)(117,124)(119,204)(120,206)(121,193)(123,210)(127,215)(128,217)(130,220)(132,224)(133,226)(135,230)(136,232)(138,233)(140,238)(142,242)(143,244)(144,246)(145,248)(147,251)(148,253)(150,256)(152,260)(154,209)(155,265)(156,267)(158,271)(160,274)(162,227)(164,278)(167,282)(174,290)(176,292)(178,296)(179,298)(180,300)(182,303)(184,307)(185,308)(186,310)(188,200)(190,254)(191,314)(194,297)(196,317)(197,319)(198,311)(201,324)(202,326)(203,328)(205,332)(207,334)(208,335)(211,340)(212,342)(214,279)(218,347)(219,346)(221,350)(222,352)(223,354)(228,344)(229,361)(231,364)(234,369)(235,367)(236,371)(237,372)(239,375)(240,377)(241,378)(243,381)(245,373)(247,387)(250,391)(252,395)(255,398)(257,401)(258,313)(259,404)(261,406)(262,408)(263,353)(264,331)(266,411)(268,415)(269,417)(270,376)(272,305)(273,421)(275,424)(276,425)(277,426)(280,427)(281,429)(284,433)(285,435)(286,437)(288,438)(289,440)(293,444)(294,446)(295,432)(299,452)(301,454)(302,455)(304,457)(306,460)(309,449)(312,453)(315,469)(316,471)(318,474)(320,478)(321,479)(322,480)(323,481)(325,485)(327,489)(329,491)(330,492)(333,497)(336,500)(337,501)(338,502)(343,508)(345,488)(348,490)(351,461)(355,409)(357,517)(358,518)(359,520)(360,456)(362,523)(363,525)(365,473)(366,527)(379,537)(380,539)(382,538)(383,531)(384,521)(385,462)(386,535)(388,467)(389,541)(390,542)(392,396)(394,428)(399,553)(400,555)(402,559)(403,561)(405,550)(407,568)(410,569)(412,572)(413,573)(414,575)(416,566)(418,512)(419,466)(420,583)(422,439)(423,524)(430,593)(431,595)(434,598)(441,601)(442,567)(443,599)(445,605)(447,450)(448,608)(451,610)(459,613)(464,533)(465,591)(468,615)(470,563)(472,617)(475,619)(476,536)(477,621)(482,548)(483,628)(484,612)(487,629)(493,635)(494,636)(496,637)(499,571)(504,640)(506,645)(509,647)(510,630)(511,648)(513,552)(514,650)(515,652)(516,603)(519,654)(522,534)(526,660)(528,626)(529,579)(532,676)(543,682)(544,713)(545,574)(547,596)(549,668)(551,674)(554,560)(556,705)(557,589)(564,607)(565,700)(570,699)(576,698)(578,665)(580,683)(582,691)(586,707)(597,678)(602,646)(604,651)(609,704)(611,620)(618,723)(622,673)(623,714)(624,693)(625,687)(627,724)(632,663)(633,672)(634,716)(639,679)(642,677)(643,653)(644,697)(649,710)(655,719)(656,725)(657,702)(659,711)(664,715)(667,690)(684,706)(685,689)(686,709)(692,726)(694,695)(708,728) );
 
Copy content sage:G = PermutationGroup(['(1,3,8,18,34,64,115,200,236,138,78,42,22,10,4)(2,5,12,14,6)(7,15,29,54,96,169,286,377,458,304,182,103,58,31,16)(9,19,36,68,123,211,341,506,512,351,221,130,72,38,20)(11,23,44,45,24)(13,26,49,51,27)(17,32,60,107,188,313,468,616,618,472,316,193,111,62,33)(21,39,73,75,40)(25,46,83,147,252,396,550,710,687,560,402,257,150,85,47)(28,52,92,94,53)(30,55,98,81,143,245,385,454,612,602,441,291,174,99,56)(35,65,117,202,327,490,633,613,478,623,498,333,205,119,66)(37,69,124,212,343,308,464,538,379,241,141,240,214,126,70)(41,76,135,231,365,526,598,657,522,361,485,368,233,136,77)(43,79,140,239,376,446,607,523,457,372,531,382,243,142,80)(48,86,152,261,407,369,528,513,352,492,335,410,264,154,87)(50,89,158,160,90)(57,100,176,293,445,606,667,575,474,453,300,450,297,178,101)(59,104,110,184,105)(61,108,190,116,201,325,486,367,232,366,471,470,315,191,109)(63,112,194,278,227,133,74,132,225,358,519,475,318,196,113)(67,120,207,175,121)(71,127,216,218,128)(82,144,247,229,134,228,360,521,421,584,699,543,390,249,145)(84,148,254,255,149)(88,155,266,412,569,653,515,354,514,651,719,579,416,268,156)(91,161,275,276,162)(93,164,279,192,310,466,439,288,171,97,170,287,428,280,165)(95,166,281,189,301,180,102,179,299,238,374,533,432,283,167)(106,185,309,465,590,427,589,463,307,462,614,539,467,311,186)(114,197,320,321,198)(118,187,312,329,203)(122,208,336,337,209)(125,129,219,173,213)(131,222,353,355,223)(137,234,370,260,235)(139,237,373,532,381,438,461,306,183,305,459,380,242,289,172)(146,163,277,392,250)(151,258,403,562,671,698,617,626,480,625,715,660,565,405,259)(153,262,265,409,263)(157,269,303,339,210,338,503,558,401,557,482,323,199,322,270)(159,272,420,317,473,431,282,430,594,641,500,640,585,422,273)(168,284,434,599,573,460,389,248,388,540,679,636,600,436,285)(177,294,447,437,295)(181,302,456,423,274)(195,298,451,386,246)(204,330,493,494,331)(215,344,509,387,345)(217,346,363,230,362,524,644,505,340,504,499,334,359,226,290)(220,348,349)(224,356,516,654,727,728,673,672,702,694,627,481,364,350,357)(244,383,253,397,384)(251,393,546,496,332,495,452,611,693,564,404,563,695,547,394)(256,399,554,556,400)(267,413,574,700,724,664,706,583,604,444,561,675,680,576,414)(271,418,582,469,419)(292,442,603,665,701,708,628,725,609,449,296,448,314,455,443)(319,476,620,681,635,642,501,581,417,580,691,648,517,622,477)(324,483,508,408,484)(326,487,630,631,488)(342,507,646,632,489)(347,510,621,649,511)(371,529,568,619,530)(375,534,711,663,629,601,656,520,491,634,639,497,638,692,535)(378,440,425,588,536)(391,544,435,426,587,424,586,717,705,593,678,553,684,647,545)(395,548,595,713,549)(398,551,686,650,552)(406,566,527,661,567)(411,570,502,643,571)(415,577,655,518,578)(429,591,688,555,592)(433,596,559,668,597)(479,624,674,541,610)(525,658,659)(537,683,703,685,676)(542,652,716,572,637)(605,662,723,670,722)(608,709,704,689,707)(615,721,690,669,720)(645,714,666,682,696)(677,718,726,697,712)', '(1,2)(3,7)(4,9)(5,11)(6,13)(8,17)(10,21)(12,25)(14,18)(15,28)(16,30)(19,35)(20,37)(22,41)(23,43)(26,48)(27,50)(31,57)(32,59)(33,61)(34,63)(36,67)(38,71)(39,56)(40,74)(44,81)(45,82)(47,84)(49,88)(51,91)(53,93)(54,95)(55,97)(58,102)(60,106)(62,110)(64,114)(65,116)(66,118)(68,122)(70,125)(72,129)(73,131)(75,126)(76,134)(78,137)(79,139)(80,141)(83,146)(86,151)(87,153)(89,157)(90,159)(92,163)(96,168)(98,172)(99,173)(100,175)(101,177)(103,181)(104,183)(105,170)(107,187)(108,189)(109,149)(111,192)(112,161)(113,195)(115,199)(117,124)(119,204)(120,206)(121,193)(123,210)(127,215)(128,217)(130,220)(132,224)(133,226)(135,230)(136,232)(138,233)(140,238)(142,242)(143,244)(144,246)(145,248)(147,251)(148,253)(150,256)(152,260)(154,209)(155,265)(156,267)(158,271)(160,274)(162,227)(164,278)(167,282)(174,290)(176,292)(178,296)(179,298)(180,300)(182,303)(184,307)(185,308)(186,310)(188,200)(190,254)(191,314)(194,297)(196,317)(197,319)(198,311)(201,324)(202,326)(203,328)(205,332)(207,334)(208,335)(211,340)(212,342)(214,279)(218,347)(219,346)(221,350)(222,352)(223,354)(228,344)(229,361)(231,364)(234,369)(235,367)(236,371)(237,372)(239,375)(240,377)(241,378)(243,381)(245,373)(247,387)(250,391)(252,395)(255,398)(257,401)(258,313)(259,404)(261,406)(262,408)(263,353)(264,331)(266,411)(268,415)(269,417)(270,376)(272,305)(273,421)(275,424)(276,425)(277,426)(280,427)(281,429)(284,433)(285,435)(286,437)(288,438)(289,440)(293,444)(294,446)(295,432)(299,452)(301,454)(302,455)(304,457)(306,460)(309,449)(312,453)(315,469)(316,471)(318,474)(320,478)(321,479)(322,480)(323,481)(325,485)(327,489)(329,491)(330,492)(333,497)(336,500)(337,501)(338,502)(343,508)(345,488)(348,490)(351,461)(355,409)(357,517)(358,518)(359,520)(360,456)(362,523)(363,525)(365,473)(366,527)(379,537)(380,539)(382,538)(383,531)(384,521)(385,462)(386,535)(388,467)(389,541)(390,542)(392,396)(394,428)(399,553)(400,555)(402,559)(403,561)(405,550)(407,568)(410,569)(412,572)(413,573)(414,575)(416,566)(418,512)(419,466)(420,583)(422,439)(423,524)(430,593)(431,595)(434,598)(441,601)(442,567)(443,599)(445,605)(447,450)(448,608)(451,610)(459,613)(464,533)(465,591)(468,615)(470,563)(472,617)(475,619)(476,536)(477,621)(482,548)(483,628)(484,612)(487,629)(493,635)(494,636)(496,637)(499,571)(504,640)(506,645)(509,647)(510,630)(511,648)(513,552)(514,650)(515,652)(516,603)(519,654)(522,534)(526,660)(528,626)(529,579)(532,676)(543,682)(544,713)(545,574)(547,596)(549,668)(551,674)(554,560)(556,705)(557,589)(564,607)(565,700)(570,699)(576,698)(578,665)(580,683)(582,691)(586,707)(597,678)(602,646)(604,651)(609,704)(611,620)(618,723)(622,673)(623,714)(624,693)(625,687)(627,724)(632,663)(633,672)(634,716)(639,679)(642,677)(643,653)(644,697)(649,710)(655,719)(656,725)(657,702)(659,711)(664,715)(667,690)(684,706)(685,689)(686,709)(692,726)(694,695)(708,728)'])
 
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 $\Sp(6,3)$.

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

See the $141 \times 141$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $90 \times 90$ rational character table.