LMFDB - Abstract group 4585351680.b: $\PSp(6,3)$

Show commands: Gap / Magma / SageMath

magma:G := PSp(6,3);

gap:G := PSp(6,3);

sage:G = PSp(6,3)

comment:Define the group as a permutation group

Group information

Description:	$\PSp(6,3)$
Order:	$4585351680$$\medspace = 2^{9} \cdot 3^{9} \cdot 5 \cdot 7 \cdot 13 $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order()
Exponent:	$32760$$\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 $	comment:Exponent of the group magma:Exponent(G); gap:Exponent(G); sage:G.exponent() sage_gap:G.Exponent()
Automorphism group:	Group of order $9170703360$$\medspace = 2^{10} \cdot 3^{9} \cdot 5 \cdot 7 \cdot 13 $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage_gap:G.AutomorphismGroup()
Composition factors:	$\PSp(6,3)$	comment:Composition factors of the group magma:CompositionFactors(G); gap:CompositionSeries(G); sage:G.composition_series() sage_gap:G.CompositionSeries()
Derived length:	$0$	comment:Derived length of the group magma:DerivedLength(G); gap:DerivedLength(G); sage_gap:G.DerivedLength()

This group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).

comment:Determine if the group G is abelian

magma:IsAbelian(G);

gap:IsAbelian(G);

sage:G.is_abelian()

sage_gap:G.IsAbelian()

comment:Determine if the group G is cyclic

magma:IsCyclic(G);

gap:IsCyclic(G);

sage:G.is_cyclic()

sage_gap:G.IsCyclic()

comment:Determine if the group G is nilpotent

magma:IsNilpotent(G);

gap:IsNilpotentGroup(G);

sage:G.is_nilpotent()

sage_gap:G.IsNilpotentGroup()

comment:Determine if the group G is solvable

magma:IsSolvable(G);

gap:IsSolvableGroup(G);

sage:G.is_solvable()

sage_gap:G.IsSolvableGroup()

comment:Determine if the group G is supersolvable

gap:IsSupersolvableGroup(G);

sage:G.is_supersolvable()

sage_gap:G.IsSupersolvableGroup()

comment:Determine if the group G is simple

magma:IsSimple(G);

gap:IsSimpleGroup(G);

sage_gap:G.IsSimpleGroup()

Group statistics

comment:Compute statistics for the group G

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;

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");

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	30	36
Elements	1	196911	5307848	6014736	38211264	191115288	327525120	191056320	382112640	38211264	513021600	705438720	327525120	305690112	382112640	229267584	382112640	305690112	254741760	4585351680
Conjugacy classes	1	2	7	3	1	16	1	2	8	1	14	2	1	2	6	1	2	2	2	74
Divisions	1	2	5	3	1	10	1	2	4	1	9	1	1	1	3	1	1	1	1	49
Autjugacy classes	1	2	5	3	1	10	1	2	4	1	9	2	1	1	3	1	1	1	1	50

comment:Compute statistics about the characters of G

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);

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);

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)]

sage_gap:G.CharacterDegrees()

Dimension	1	13	26	78	91	105	168	182	195	273	455	546	728	819	910	1092	1170	1365	1456	1820	2106	2184	2340	2457	2730	2835	3640	4095	4368	4536	4914	5265	5460	5824	6552	7280	7371	8190	9477	10920	11648	12285	13104	14560	14742	16380	16640	17472	17920	18954	19683	21840	23296	24570	29120	35840
Irr. complex chars.	1	2	0	1	2	1	1	0	1	2	2	2	2	1	0	0	2	6	0	3	1	2	0	3	1	1	2	2	0	1	0	1	4	3	2	0	2	1	2	2	3	2	0	3	0	1	2	1	2	0	1	0	0	0	0	0	74
Irr. rational chars.	1	0	1	1	0	1	1	1	1	0	0	1	0	1	1	1	0	0	1	1	1	0	1	1	4	1	1	0	1	1	1	1	2	1	0	1	0	2	0	1	2	0	1	1	1	1	2	1	0	1	1	1	1	1	1	1	49

Minimal presentations

Permutation degree:	$364$
Transitive degree:	$364$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	13	26	26
Arbitrary	not computed	not computed	not computed

Constructions

Show commands: Gap / Magma / SageMath

Groups of Lie type:	$\PSp(6,3)$, $\PSigmaSp(6,3)$
magma:G := PSp(6,3); gap:G := PSp(6,3); sage:G = PSp(6,3) magma:G := PSigmaSp(6,3);
Permutation group:	Degree $364$ $\langle(1,17,143,138,267,69,72,197,272,278,82,253,6,3,2)(4,5,19,223,222)(7,289,303,39,53,68,101,269,183,349,171,188,189,182,50) \!\cdots\! \rangle$ (1,17,143,138,267,69,72,197,272,278,82,253,6,3,2)(4,5,19,223,222)(7,289,303,39,53,68,101,269,183,349,171,188,189,182,50)(8,310,314,100,216,59,274,287,227,174,74,324,67,350,170)(9,275,342,245,44,276,246,331,265,61,347,113,130,48,306)(10,177,163,37,120,234,355,137,221,293,346,273,220,268,353)(11,316,256,134,202)(12,363,87,231,70,299,191,277,46,112,239,99,111,213,326)(13,83,357,215,71)(14,255,136,38,203,152,107,47,122,257,26,145,281,290,126)(15,325,335,356,58,73,327,129,271,364,81,266,201,36,51)(16,43,65,248,282)(18,218,318,184,233,27,55,150,291,86,187,154,94,260,251)(20,166,180,270,62,280,258,146,309,108,24,133,307,261,322)(21,117,41,323,304,232,132,135,205,292,42,124,103,35,344)(22,175,321,259,161,96,288,179,181,224,301,198,89,332,296)(23,317,359,142,153)(25,63,200,195,127)(28,109,308,264,110,351,337,167,240,140,176,119,343,219,345)(29,169,199,148,116)(30,283,77,330,361,186,84,165,158,336,80,263,229,33,34)(31,294,340,92,312)(32,90,147,212,247)(40,206,54,217,141)(45,131,159,60,193)(49,56,328,279,285)(52,156,286,157,320)(57,121,297,243,85,105,115,95,194,155,226,302,250,249,300)(64,228,334,102,98)(75,97,78,211,237,235,236,118,360,352,114,354,204,341,139)(76,208,149,173,254)(79,358,225,125,214)(88,164,196,311,284)(91,338,207,190,93)(104,362,295,151,230)(106,315,252,333,262)(123,178,192,144,348)(128,238,329,185,210)(160,305,242,168,209)(162,319,244)(172,298,313,339,241), (7,188)(8,311)(9,358)(11,353)(12,100)(13,21)(14,104)(15,272)(16,174)(17,300)(19,302)(20,235)(22,132)(23,263)(24,203)(25,344)(26,173)(27,255)(28,339)(29,195)(30,244)(31,193)(32,357)(33,303)(34,258)(35,56)(36,123)(38,333)(39,150)(40,341)(41,322)(42,190)(44,329)(45,148)(46,305)(47,261)(48,262)(49,170)(50,285)(51,157)(52,266)(53,168)(54,121)(55,138)(57,337)(58,144)(59,65)(60,342)(61,228)(62,233)(63,240)(64,98)(66,141)(67,277)(68,321)(69,197)(70,296)(71,325)(72,270)(73,218)(74,172)(75,345)(76,364)(77,247)(78,212)(79,152)(82,113)(83,147)(84,217)(85,119)(86,103)(87,346)(88,92)(89,354)(91,165)(93,265)(94,158)(95,273)(96,180)(97,227)(99,177)(101,236)(102,245)(105,281)(106,306)(109,309)(110,133)(114,336)(115,242)(116,291)(117,146)(118,295)(120,160)(122,241)(125,130)(126,140)(127,229)(128,331)(131,149)(136,155)(137,269)(139,186)(142,292)(143,275)(145,268)(151,169)(156,256)(159,347)(161,167)(162,237)(163,318)(166,280)(171,350)(175,199)(176,294)(178,184)(179,239)(181,201)(182,310)(183,189)(185,205)(187,221)(192,288)(194,206)(196,232)(198,207)(200,231)(202,213)(204,238)(208,290)(209,243)(210,260)(211,282)(214,264)(215,323)(216,313)(219,312)(220,324)(222,249)(226,253)(230,293)(234,314)(248,283)(254,279)(257,361)(259,278)(267,335)(271,340)(274,330)(276,338)(284,349)(286,327)(289,299)(297,363)(298,334)(304,362)(315,351)(317,360)(320,356)(326,343)
comment:Define the group as a permutation group magma:G := PermutationGroup< 364 \| (1,17,143,138,267,69,72,197,272,278,82,253,6,3,2)(4,5,19,223,222)(7,289,303,39,53,68,101,269,183,349,171,188,189,182,50)(8,310,314,100,216,59,274,287,227,174,74,324,67,350,170)(9,275,342,245,44,276,246,331,265,61,347,113,130,48,306)(10,177,163,37,120,234,355,137,221,293,346,273,220,268,353)(11,316,256,134,202)(12,363,87,231,70,299,191,277,46,112,239,99,111,213,326)(13,83,357,215,71)(14,255,136,38,203,152,107,47,122,257,26,145,281,290,126)(15,325,335,356,58,73,327,129,271,364,81,266,201,36,51)(16,43,65,248,282)(18,218,318,184,233,27,55,150,291,86,187,154,94,260,251)(20,166,180,270,62,280,258,146,309,108,24,133,307,261,322)(21,117,41,323,304,232,132,135,205,292,42,124,103,35,344)(22,175,321,259,161,96,288,179,181,224,301,198,89,332,296)(23,317,359,142,153)(25,63,200,195,127)(28,109,308,264,110,351,337,167,240,140,176,119,343,219,345)(29,169,199,148,116)(30,283,77,330,361,186,84,165,158,336,80,263,229,33,34)(31,294,340,92,312)(32,90,147,212,247)(40,206,54,217,141)(45,131,159,60,193)(49,56,328,279,285)(52,156,286,157,320)(57,121,297,243,85,105,115,95,194,155,226,302,250,249,300)(64,228,334,102,98)(75,97,78,211,237,235,236,118,360,352,114,354,204,341,139)(76,208,149,173,254)(79,358,225,125,214)(88,164,196,311,284)(91,338,207,190,93)(104,362,295,151,230)(106,315,252,333,262)(123,178,192,144,348)(128,238,329,185,210)(160,305,242,168,209)(162,319,244)(172,298,313,339,241), (7,188)(8,311)(9,358)(11,353)(12,100)(13,21)(14,104)(15,272)(16,174)(17,300)(19,302)(20,235)(22,132)(23,263)(24,203)(25,344)(26,173)(27,255)(28,339)(29,195)(30,244)(31,193)(32,357)(33,303)(34,258)(35,56)(36,123)(38,333)(39,150)(40,341)(41,322)(42,190)(44,329)(45,148)(46,305)(47,261)(48,262)(49,170)(50,285)(51,157)(52,266)(53,168)(54,121)(55,138)(57,337)(58,144)(59,65)(60,342)(61,228)(62,233)(63,240)(64,98)(66,141)(67,277)(68,321)(69,197)(70,296)(71,325)(72,270)(73,218)(74,172)(75,345)(76,364)(77,247)(78,212)(79,152)(82,113)(83,147)(84,217)(85,119)(86,103)(87,346)(88,92)(89,354)(91,165)(93,265)(94,158)(95,273)(96,180)(97,227)(99,177)(101,236)(102,245)(105,281)(106,306)(109,309)(110,133)(114,336)(115,242)(116,291)(117,146)(118,295)(120,160)(122,241)(125,130)(126,140)(127,229)(128,331)(131,149)(136,155)(137,269)(139,186)(142,292)(143,275)(145,268)(151,169)(156,256)(159,347)(161,167)(162,237)(163,318)(166,280)(171,350)(175,199)(176,294)(178,184)(179,239)(181,201)(182,310)(183,189)(185,205)(187,221)(192,288)(194,206)(196,232)(198,207)(200,231)(202,213)(204,238)(208,290)(209,243)(210,260)(211,282)(214,264)(215,323)(216,313)(219,312)(220,324)(222,249)(226,253)(230,293)(234,314)(248,283)(254,279)(257,361)(259,278)(267,335)(271,340)(274,330)(276,338)(284,349)(286,327)(289,299)(297,363)(298,334)(304,362)(315,351)(317,360)(320,356)(326,343) >; gap:G := Group( (1,17,143,138,267,69,72,197,272,278,82,253,6,3,2)(4,5,19,223,222)(7,289,303,39,53,68,101,269,183,349,171,188,189,182,50)(8,310,314,100,216,59,274,287,227,174,74,324,67,350,170)(9,275,342,245,44,276,246,331,265,61,347,113,130,48,306)(10,177,163,37,120,234,355,137,221,293,346,273,220,268,353)(11,316,256,134,202)(12,363,87,231,70,299,191,277,46,112,239,99,111,213,326)(13,83,357,215,71)(14,255,136,38,203,152,107,47,122,257,26,145,281,290,126)(15,325,335,356,58,73,327,129,271,364,81,266,201,36,51)(16,43,65,248,282)(18,218,318,184,233,27,55,150,291,86,187,154,94,260,251)(20,166,180,270,62,280,258,146,309,108,24,133,307,261,322)(21,117,41,323,304,232,132,135,205,292,42,124,103,35,344)(22,175,321,259,161,96,288,179,181,224,301,198,89,332,296)(23,317,359,142,153)(25,63,200,195,127)(28,109,308,264,110,351,337,167,240,140,176,119,343,219,345)(29,169,199,148,116)(30,283,77,330,361,186,84,165,158,336,80,263,229,33,34)(31,294,340,92,312)(32,90,147,212,247)(40,206,54,217,141)(45,131,159,60,193)(49,56,328,279,285)(52,156,286,157,320)(57,121,297,243,85,105,115,95,194,155,226,302,250,249,300)(64,228,334,102,98)(75,97,78,211,237,235,236,118,360,352,114,354,204,341,139)(76,208,149,173,254)(79,358,225,125,214)(88,164,196,311,284)(91,338,207,190,93)(104,362,295,151,230)(106,315,252,333,262)(123,178,192,144,348)(128,238,329,185,210)(160,305,242,168,209)(162,319,244)(172,298,313,339,241), (7,188)(8,311)(9,358)(11,353)(12,100)(13,21)(14,104)(15,272)(16,174)(17,300)(19,302)(20,235)(22,132)(23,263)(24,203)(25,344)(26,173)(27,255)(28,339)(29,195)(30,244)(31,193)(32,357)(33,303)(34,258)(35,56)(36,123)(38,333)(39,150)(40,341)(41,322)(42,190)(44,329)(45,148)(46,305)(47,261)(48,262)(49,170)(50,285)(51,157)(52,266)(53,168)(54,121)(55,138)(57,337)(58,144)(59,65)(60,342)(61,228)(62,233)(63,240)(64,98)(66,141)(67,277)(68,321)(69,197)(70,296)(71,325)(72,270)(73,218)(74,172)(75,345)(76,364)(77,247)(78,212)(79,152)(82,113)(83,147)(84,217)(85,119)(86,103)(87,346)(88,92)(89,354)(91,165)(93,265)(94,158)(95,273)(96,180)(97,227)(99,177)(101,236)(102,245)(105,281)(106,306)(109,309)(110,133)(114,336)(115,242)(116,291)(117,146)(118,295)(120,160)(122,241)(125,130)(126,140)(127,229)(128,331)(131,149)(136,155)(137,269)(139,186)(142,292)(143,275)(145,268)(151,169)(156,256)(159,347)(161,167)(162,237)(163,318)(166,280)(171,350)(175,199)(176,294)(178,184)(179,239)(181,201)(182,310)(183,189)(185,205)(187,221)(192,288)(194,206)(196,232)(198,207)(200,231)(202,213)(204,238)(208,290)(209,243)(210,260)(211,282)(214,264)(215,323)(216,313)(219,312)(220,324)(222,249)(226,253)(230,293)(234,314)(248,283)(254,279)(257,361)(259,278)(267,335)(271,340)(274,330)(276,338)(284,349)(286,327)(289,299)(297,363)(298,334)(304,362)(315,351)(317,360)(320,356)(326,343) ); sage:G = PermutationGroup(['(1,17,143,138,267,69,72,197,272,278,82,253,6,3,2)(4,5,19,223,222)(7,289,303,39,53,68,101,269,183,349,171,188,189,182,50)(8,310,314,100,216,59,274,287,227,174,74,324,67,350,170)(9,275,342,245,44,276,246,331,265,61,347,113,130,48,306)(10,177,163,37,120,234,355,137,221,293,346,273,220,268,353)(11,316,256,134,202)(12,363,87,231,70,299,191,277,46,112,239,99,111,213,326)(13,83,357,215,71)(14,255,136,38,203,152,107,47,122,257,26,145,281,290,126)(15,325,335,356,58,73,327,129,271,364,81,266,201,36,51)(16,43,65,248,282)(18,218,318,184,233,27,55,150,291,86,187,154,94,260,251)(20,166,180,270,62,280,258,146,309,108,24,133,307,261,322)(21,117,41,323,304,232,132,135,205,292,42,124,103,35,344)(22,175,321,259,161,96,288,179,181,224,301,198,89,332,296)(23,317,359,142,153)(25,63,200,195,127)(28,109,308,264,110,351,337,167,240,140,176,119,343,219,345)(29,169,199,148,116)(30,283,77,330,361,186,84,165,158,336,80,263,229,33,34)(31,294,340,92,312)(32,90,147,212,247)(40,206,54,217,141)(45,131,159,60,193)(49,56,328,279,285)(52,156,286,157,320)(57,121,297,243,85,105,115,95,194,155,226,302,250,249,300)(64,228,334,102,98)(75,97,78,211,237,235,236,118,360,352,114,354,204,341,139)(76,208,149,173,254)(79,358,225,125,214)(88,164,196,311,284)(91,338,207,190,93)(104,362,295,151,230)(106,315,252,333,262)(123,178,192,144,348)(128,238,329,185,210)(160,305,242,168,209)(162,319,244)(172,298,313,339,241)', '(7,188)(8,311)(9,358)(11,353)(12,100)(13,21)(14,104)(15,272)(16,174)(17,300)(19,302)(20,235)(22,132)(23,263)(24,203)(25,344)(26,173)(27,255)(28,339)(29,195)(30,244)(31,193)(32,357)(33,303)(34,258)(35,56)(36,123)(38,333)(39,150)(40,341)(41,322)(42,190)(44,329)(45,148)(46,305)(47,261)(48,262)(49,170)(50,285)(51,157)(52,266)(53,168)(54,121)(55,138)(57,337)(58,144)(59,65)(60,342)(61,228)(62,233)(63,240)(64,98)(66,141)(67,277)(68,321)(69,197)(70,296)(71,325)(72,270)(73,218)(74,172)(75,345)(76,364)(77,247)(78,212)(79,152)(82,113)(83,147)(84,217)(85,119)(86,103)(87,346)(88,92)(89,354)(91,165)(93,265)(94,158)(95,273)(96,180)(97,227)(99,177)(101,236)(102,245)(105,281)(106,306)(109,309)(110,133)(114,336)(115,242)(116,291)(117,146)(118,295)(120,160)(122,241)(125,130)(126,140)(127,229)(128,331)(131,149)(136,155)(137,269)(139,186)(142,292)(143,275)(145,268)(151,169)(156,256)(159,347)(161,167)(162,237)(163,318)(166,280)(171,350)(175,199)(176,294)(178,184)(179,239)(181,201)(182,310)(183,189)(185,205)(187,221)(192,288)(194,206)(196,232)(198,207)(200,231)(202,213)(204,238)(208,290)(209,243)(210,260)(211,282)(214,264)(215,323)(216,313)(219,312)(220,324)(222,249)(226,253)(230,293)(234,314)(248,283)(254,279)(257,361)(259,278)(267,335)(271,340)(274,330)(276,338)(284,349)(286,327)(289,299)(297,363)(298,334)(304,362)(315,351)(317,360)(320,356)(326,343)'])
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\Sp(6,3)$.

Homology

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

Subgroups

comment:List of subgroups of the group

magma:Subgroups(G);

gap:AllSubgroups(G);

sage:G.subgroups()

sage_gap:G.AllSubgroups()

There are 2 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$	comment:Center of the group magma:Center(G); gap:Center(G); sage:G.center() sage_gap:G.Center()
Commutator:	a subgroup isomorphic to $\PSp(6,3)$	comment:Commutator subgroup of the group G magma:CommutatorSubgroup(G); gap:DerivedSubgroup(G); sage:G.commutator() sage_gap:G.DerivedSubgroup()
Frattini:	a subgroup isomorphic to $C_1$	comment:Frattini subgroup of the group G magma:FrattiniSubgroup(G); gap:FrattiniSubgroup(G); sage:G.frattini_subgroup() sage_gap:G.FrattiniSubgroup()
Fitting:	not computed	comment:Fitting subgroup of the group G magma:FittingSubgroup(G); gap:FittingSubgroup(G); sage:G.fitting_subgroup() sage_gap:G.FittingSubgroup()
Radical:	not computed	comment:Radical of the group G magma:Radical(G); gap:SolvableRadical(G); sage_gap:G.SolvableRadical()
Socle:	not computed	comment:Socle of the group G magma:Socle(G); gap:Socle(G); sage:G.socle() sage_gap:G.Socle()
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_8^2:C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^6:\He_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$

Subgroup diagram and profile

Series

Derived series

not computed

comment:Derived series of the group GF

magma:DerivedSeries(G);

gap:DerivedSeriesOfGroup(G);

sage:G.derived_series()

sage_gap:G.DerivedSeriesOfGroup()

Chief series

not computed

comment:Chief series of the group G

magma:ChiefSeries(G);

gap:ChiefSeries(G);

sage_gap:G.ChiefSeries()

Lower central series

not computed

comment:The lower central series of the group G

magma:LowerCentralSeries(G);

gap:LowerCentralSeriesOfGroup(G);

sage:G.lower_central_series()

sage_gap:G.LowerCentralSeriesOfGroup()

Upper central series

not computed

comment:The upper central series of the group G

magma:UpperCentralSeries(G);

gap:UpperCentralSeriesOfGroup(G);

sage:G.upper_central_series()

sage_gap:G.UpperCentralSeriesOfGroup()

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

comment:Character table

magma:CharacterTable(G); // Output not guaranteed to exactly match the LMFDB table

gap:CharacterTable(G); # Output not guaranteed to exactly match the LMFDB table

sage:G.character_table() # Output not guaranteed to exactly match the LMFDB table

sage_gap:G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table

Complex character table

See the $74 \times 74$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $49 \times 49$ rational character table.

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Label	4585351680.b
Order	\( 2^{9} \cdot 3^{9} \cdot 5 \cdot 7 \cdot 13 \)
Exponent	\( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)

Simple	yes
$\card{G^{\mathrm{ab}}}$	\( 1 \)
$\card{Z(G)}$	1
$\card{\Aut(G)}$	\( 2^{10} \cdot 3^{9} \cdot 5 \cdot 7 \cdot 13 \)
$\card{\mathrm{Out}(G)}$	\( 2 \)
Perm deg.	$364$
Trans deg.	$364$
Rank	$2$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Group information

Group statistics

Minimal presentations

Minimal degrees of faithful linear representations

Constructions

Homology

Subgroups

Special subgroups

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups (quotient in parentheses)

Normal subgroups up to automorphism (quotient in parentheses)

Series

Supergroups

Character theory

Complex character table

Rational character table

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.