LMFDB - Abstract group 204073344.ou: $C_3^8.(C_6^3:S_3.S

comment:Define the group as a permutation group

magma:G := PermutationGroup< 36 | (1,23,11,34,27,14)(2,24,10,35,25,13)(3,22,12,36,26,15)(4,31,20,7,29,17,5,33,21,9,30,16,6,32,19,8,28,18), (1,9,34,4,3,7,36,6,2,8,35,5)(10,33,15,30)(11,31,13,29)(12,32,14,28)(16,23,18,24)(17,22)(19,26)(20,27,21,25) >;

gap:G := Group( (1,23,11,34,27,14)(2,24,10,35,25,13)(3,22,12,36,26,15)(4,31,20,7,29,17,5,33,21,9,30,16,6,32,19,8,28,18), (1,9,34,4,3,7,36,6,2,8,35,5)(10,33,15,30)(11,31,13,29)(12,32,14,28)(16,23,18,24)(17,22)(19,26)(20,27,21,25) );

sage:G = PermutationGroup(['(1,23,11,34,27,14)(2,24,10,35,25,13)(3,22,12,36,26,15)(4,31,20,7,29,17,5,33,21,9,30,16,6,32,19,8,28,18)', '(1,9,34,4,3,7,36,6,2,8,35,5)(10,33,15,30)(11,31,13,29)(12,32,14,28)(16,23,18,24)(17,22)(19,26)(20,27,21,25)'])

sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(30639763228382971989236756756496165680078338279050011444754898106788777211275258520417650388402908700672798374443656475403266250230375065201557851005826190483221416364958583864742272533951314920673284459460409031829589375779485707753774412716997321062894512180952847439232136429540052023068332359604735464539159740504186225192726812447729762533380398127741077461871232632196935013767938980443052162362556337945898743155869134838404028700749701607761742102913812950345408490419260202653890972169134963475276110266804750456067591594283568652711810314827268363164694480770677380673805209238652815506535660989766543869425300271837178781189616208910369597516834871831049620766997055792344412210442379141987828665289409302145575564317571228716942438375671856504451824272571039022057360435845133723951161553613090045167173542404005152681899519067156518088861044893677081165448325623466521513892284865696326243213353678721984303207799252008548900187823575620690893552908561781198978398162605340307544069103678262777159040497770980273205982488006229175494071807,204073344)'); a = G.1; b = G.3; c = G.5; d = G.7; e = G.9; f = G.11; g = G.13; h = G.14; i = G.15; j = G.16; k = G.17; l = G.18; m = G.19; n = G.20;

Group information

Description:	$C_3^8.(C_6^3:S_3.S_4)$
Order:	$204073344$$\medspace = 2^{7} \cdot 3^{13} $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order()
Exponent:	$36$$\medspace = 2^{2} \cdot 3^{2} $	comment:Exponent of the group magma:Exponent(G); gap:Exponent(G); sage:G.exponent() sage_gap:G.Exponent()
Automorphism group:	Group of order $1586874322944$$\medspace = 2^{12} \cdot 3^{18} $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage_gap:G.AutomorphismGroup()
Composition factors:	$C_2$ x 7, $C_3$ x 13	comment:Composition factors of the group magma:CompositionFactors(G); gap:CompositionSeries(G); sage:G.composition_series() sage_gap:G.CompositionSeries()
Derived length:	$4$	comment:Derived length of the group magma:DerivedLength(G); gap:DerivedLength(G); sage_gap:G.DerivedLength()

This group is nonabelian and solvable. Whether it is monomial or rational has not been computed.

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))

Statistics about orders of elements in this group have not been computed.

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Show commands: Gap / Magma / SageMath

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k, l, m, n \mid c^{6}=d^{6}=e^{6}= \!\cdots\! \rangle}$ < a, b, c, d, e, f, g, h, i, j, k, l, m, n \| c^6,d^6,e^6,f^6,g^3,h^3,i^3,j^3,k^3,l^3,m^3,n^3,[c,g],[c,n],[d,n],[g,h],[g,i],[g,j],[g,k],[g,l],[g,m],[g,n],[h,i],[h,j],[h,k],[h,l],[h,m],[h,n],[i,j],[i,k],[i,l],[i,m],[i,n],[j,k],[j,l],[j,m],[j,n],[k,l],[k,m],[k,n],[l,m],[l,n],[m,n], g^2i^2j^2klmn^2a^-4, ghij^2kb^-6, b^5c^3d^3e^2f^4g^2h^2ijk^2lm^2na^-1b^-1a, cde^3fh^2jk^2lma^-1c^-1a, c^2d^5e^4fghklmn^2a^-1d^-1a, c^4d^2e^5fg^2hj^2kl^2na^-1e^-1a, c^2e^2fi^2jkl^2na^-1f^-1a, c^2d^4e^2i^2mn^2a^-1g^-1a, c^2d^2f^4ghikl^2ma^-1h^-1a, d^4e^4f^4gh^2jlna^-1i^-1a, d^4e^4f^4gik^2lm^2n^2a^-1j^-1a, c^2e^4f^2gi^2jkl^2m^2n^2a^-1k^-1a, d^4e^4f^4hk^2lm^2n^2a^-1l^-1a, c^4e^2f^4gh^2j^2k^2a^-1m^-1a, c^4d^4f^2ghk^2a^-1n^-1a, c^3de^2f^4g^2h^2ilm^2n^2b^-1c^-1b, cd^4e^4f^2gi^2k^2l^2b^-1d^-1b, c^2d^4e^5fi^2jlmn^2b^-1e^-1b, d^4ef^2gi^2k^2lm^2b^-1f^-1b, c^4d^4e^4f^4h^2ijknb^-1g^-1b, c^4d^4f^2g^2ij^2km^2b^-1h^-1b, e^2f^4g^2h^2j^2k^2m^2n^2b^-1i^-1b, e^2f^4h^2jmn^2b^-1j^-1b, c^4d^4e^2h^2i^2jn^2b^-1k^-1b, e^2f^4g^2hjkn^2b^-1l^-1b, c^2d^2e^4g^2h^2ij^2k^2l^2m^2nb^-1m^-1b, c^2d^2f^4ghi^2jknb^-1n^-1b, dg^2hi^2c^-1d^-1c, eij^2k^2l^2mnc^-1e^-1c, fg^2h^2ijkmnc^-1f^-1c, ijk^2lm^2n^2c^-1h^-1c, hijk^2c^-1i^-1c, h^2kc^-1j^-1c, ij^2k^2lm^2n^2c^-1k^-1c, h^2j^2klc^-1l^-1c, hjk^2mc^-1m^-1c, el^2m^2nd^-1e^-1d, fghl^2mnd^-1f^-1d, g^2i^2lm^2n^2d^-1g^-1d, gid^-1h^-1d, ghil^2mnd^-1i^-1d, gh^2ijd^-1j^-1d, g^2ikl^2mnd^-1k^-1d, gh^2ild^-1l^-1d, hiln^2d^-1m^-1d, fgh^2i^2jle^-1f^-1e, gh^2i^2ke^-1g^-1e, g^2i^2jk^2lmn^2e^-1h^-1e, g^2h^2jkm^2e^-1i^-1e, g^2h^2ikm^2e^-1j^-1e, gj^2k^2ln^2e^-1k^-1e, g^2hj^2k^2lm^2e^-1l^-1e, i^2jme^-1m^-1e, hik^2ne^-1n^-1e, g^2h^2iklm^2n^2f^-1g^-1f, gh^2jkl^2f^-1h^-1f, ghijlmnf^-1i^-1f, ghj^2kl^2f^-1j^-1f, g^2hi^2j^2mnf^-1k^-1f, ghklf^-1l^-1f, h^2i^2jk^2n^2f^-1m^-1f, jl^2nf^-1n^-1f >
comment:Define the group with the given generators and relations magma:G := PCGroup([20, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 40, 3422062101, 7641765842, 1930104862, 162, 3683886083, 2156418583, 46033983, 3630631204, 9900409224, 4737706244, 2471107264, 284, 9134663045, 1153342105, 3176763885, 2746851905, 23650932486, 8994394106, 726627766, 3398596806, 4174046, 552848206, 406, 23422110727, 3655388187, 8155136687, 51920707, 15750979208, 16222680028, 7546130688, 1361151068, 846113128, 20298708, 160614488, 28558048, 528, 29694768009, 1922572829, 3913593649, 1068986469, 263263329, 19154752330, 12022369950, 3220224530, 2648133910, 867097530, 837991550, 161861170, 2337890, 12969850, 650, 32752120331, 23214781471, 9815921331, 6096713831, 1122647131, 31184811, 41414343372, 8087072, 4671881812, 2618854752, 342295332, 992332, 9630092, 11406138253, 16085744673, 3493129973, 4917459193, 2276346333, 604933, 55994573, 1009893, 22999507214, 4749580834, 14129121654, 3275715674, 78148894, 221292134, 16999374, 5758414, 62186065935, 5543976995, 12781808695, 2778704715, 36080735, 2972295, 17925295, 1200215, 68162911696, 14128680996, 11699196056, 122230756, 2776032096, 256599496, 59168336, 6105936, 69880942097, 7983774757, 12562750137, 5686057517, 179159137, 20139977, 24909297, 708697, 2644325778, 10265253158, 12895456618, 637861998, 453574178, 379948458, 10588498, 10458578, 2069721619, 30139776039, 7208904059, 7482273679, 33998579, 6480219]); a,b,c,d,e,f,g,h,i,j,k,l,m,n := Explode([G.1, G.3, G.5, G.7, G.9, G.11, G.13, G.14, G.15, G.16, G.17, G.18, G.19, G.20]); AssignNames(~G, ["a", "a2", "b", "b2", "c", "c2", "d", "d2", "e", "e2", "f", "f2", "g", "h", "i", "j", "k", "l", "m", "n"]); gap:G := PcGroupCode(30639763228382971989236756756496165680078338279050011444754898106788777211275258520417650388402908700672798374443656475403266250230375065201557851005826190483221416364958583864742272533951314920673284459460409031829589375779485707753774412716997321062894512180952847439232136429540052023068332359604735464539159740504186225192726812447729762533380398127741077461871232632196935013767938980443052162362556337945898743155869134838404028700749701607761742102913812950345408490419260202653890972169134963475276110266804750456067591594283568652711810314827268363164694480770677380673805209238652815506535660989766543869425300271837178781189616208910369597516834871831049620766997055792344412210442379141987828665289409302145575564317571228716942438375671856504451824272571039022057360435845133723951161553613090045167173542404005152681899519067156518088861044893677081165448325623466521513892284865696326243213353678721984303207799252008548900187823575620690893552908561781198978398162605340307544069103678262777159040497770980273205982488006229175494071807,204073344); a := G.1; b := G.3; c := G.5; d := G.7; e := G.9; f := G.11; g := G.13; h := G.14; i := G.15; j := G.16; k := G.17; l := G.18; m := G.19; n := G.20; sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(30639763228382971989236756756496165680078338279050011444754898106788777211275258520417650388402908700672798374443656475403266250230375065201557851005826190483221416364958583864742272533951314920673284459460409031829589375779485707753774412716997321062894512180952847439232136429540052023068332359604735464539159740504186225192726812447729762533380398127741077461871232632196935013767938980443052162362556337945898743155869134838404028700749701607761742102913812950345408490419260202653890972169134963475276110266804750456067591594283568652711810314827268363164694480770677380673805209238652815506535660989766543869425300271837178781189616208910369597516834871831049620766997055792344412210442379141987828665289409302145575564317571228716942438375671856504451824272571039022057360435845133723951161553613090045167173542404005152681899519067156518088861044893677081165448325623466521513892284865696326243213353678721984303207799252008548900187823575620690893552908561781198978398162605340307544069103678262777159040497770980273205982488006229175494071807,204073344)'); a = G.1; b = G.3; c = G.5; d = G.7; e = G.9; f = G.11; g = G.13; h = G.14; i = G.15; j = G.16; k = G.17; l = G.18; m = G.19; n = G.20; sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(30639763228382971989236756756496165680078338279050011444754898106788777211275258520417650388402908700672798374443656475403266250230375065201557851005826190483221416364958583864742272533951314920673284459460409031829589375779485707753774412716997321062894512180952847439232136429540052023068332359604735464539159740504186225192726812447729762533380398127741077461871232632196935013767938980443052162362556337945898743155869134838404028700749701607761742102913812950345408490419260202653890972169134963475276110266804750456067591594283568652711810314827268363164694480770677380673805209238652815506535660989766543869425300271837178781189616208910369597516834871831049620766997055792344412210442379141987828665289409302145575564317571228716942438375671856504451824272571039022057360435845133723951161553613090045167173542404005152681899519067156518088861044893677081165448325623466521513892284865696326243213353678721984303207799252008548900187823575620690893552908561781198978398162605340307544069103678262777159040497770980273205982488006229175494071807,204073344)'); a = G.1; b = G.3; c = G.5; d = G.7; e = G.9; f = G.11; g = G.13; h = G.14; i = G.15; j = G.16; k = G.17; l = G.18; m = G.19; n = G.20;
Permutation group:	Degree $36$ $\langle(1,23,11,34,27,14)(2,24,10,35,25,13)(3,22,12,36,26,15)(4,31,20,7,29,17,5,33,21,9,30,16,6,32,19,8,28,18) \!\cdots\! \rangle$ (1,23,11,34,27,14)(2,24,10,35,25,13)(3,22,12,36,26,15)(4,31,20,7,29,17,5,33,21,9,30,16,6,32,19,8,28,18), (1,9,34,4,3,7,36,6,2,8,35,5)(10,33,15,30)(11,31,13,29)(12,32,14,28)(16,23,18,24)(17,22)(19,26)(20,27,21,25)
comment:Define the group as a permutation group magma:G := PermutationGroup< 36 \| (1,23,11,34,27,14)(2,24,10,35,25,13)(3,22,12,36,26,15)(4,31,20,7,29,17,5,33,21,9,30,16,6,32,19,8,28,18), (1,9,34,4,3,7,36,6,2,8,35,5)(10,33,15,30)(11,31,13,29)(12,32,14,28)(16,23,18,24)(17,22)(19,26)(20,27,21,25) >; gap:G := Group( (1,23,11,34,27,14)(2,24,10,35,25,13)(3,22,12,36,26,15)(4,31,20,7,29,17,5,33,21,9,30,16,6,32,19,8,28,18), (1,9,34,4,3,7,36,6,2,8,35,5)(10,33,15,30)(11,31,13,29)(12,32,14,28)(16,23,18,24)(17,22)(19,26)(20,27,21,25) ); sage:G = PermutationGroup(['(1,23,11,34,27,14)(2,24,10,35,25,13)(3,22,12,36,26,15)(4,31,20,7,29,17,5,33,21,9,30,16,6,32,19,8,28,18)', '(1,9,34,4,3,7,36,6,2,8,35,5)(10,33,15,30)(11,31,13,29)(12,32,14,28)(16,23,18,24)(17,22)(19,26)(20,27,21,25)'])
Transitive group:	36T82964				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_3^{12}.C_2^6)$ . $S_3$	$(C_3^{12}.C_2^4)$ . $S_4$ (3)	$C_3^{12}$ . $(C_2^4.S_4)$	$(C_3^{12}.C_2^5)$ . $D_6$	all 41

Elements of the group are displayed as permutations of degree 36.

Homology

Abelianization:	$C_{2} \times C_{4} $	comment:The abelianization of the group magma:quo< G \| CommutatorSubgroup(G) >; gap:FactorGroup(G, DerivedSubgroup(G)); sage:G.quotient(G.commutator())
Schur multiplier:	$C_{6}^{4}$	comment:The Schur multiplier of the group gap:AbelianInvariantsMultiplier(G); sage:G.homology(2) sage_gap:G.AbelianInvariantsMultiplier()
Commutator length:	not computed	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 70 normal subgroups (35 characteristic).

Characteristic subgroups are shown in this color. Normal (but not 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:	not computed	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 $C_3^4$	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()

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 10 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

The character tables for this group have not been computed.

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	204073344.ou
Order	\( 2^{7} \cdot 3^{13} \)
Exponent	\( 2^{2} \cdot 3^{2} \)
Nilpotent	no
Solvable	yes
$\card{G^{\mathrm{ab}}}$	\( 2^{3} \)
$\card{Z(G)}$	1
$\card{\Aut(G)}$	\( 2^{12} \cdot 3^{18} \)
$\card{\mathrm{Out}(G)}$	\( 2^{5} \cdot 3^{5} \)
Perm deg.	not computed
Trans deg.	$36$
Rank	$2$

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