↑

Properties

Label 314928.nv
Order \( 2^{4} \cdot 3^{9} \)
Exponent \( 2^{2} \cdot 3^{2} \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{3} \)
$\card{Z(G)}$ \( 1 \)
$\card{\Aut(G)}$ \( 2^{5} \cdot 3^{11} \)
$\card{\mathrm{Out}(G)}$ \( 2 \cdot 3^{2} \)
Perm deg. $24$
Trans deg. $36$
Rank $3$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Construction of abstract group
 
Copy content magma:G := PermutationGroup< 24 | (2,4)(3,7)(5,11,9,17,15,13)(6,12,10,14,16,18)(20,22)(21,23), (1,3,7)(2,6,14,4,10,12,8,16,18)(11,17,13)(19,21,20,23,22,24), (1,2,5,10,17,12,3,8,15,6,13,14,7,4,9,16,11,18)(19,20,22)(23,24) >;
 
Copy content gap:G := Group( (2,4)(3,7)(5,11,9,17,15,13)(6,12,10,14,16,18)(20,22)(21,23), (1,3,7)(2,6,14,4,10,12,8,16,18)(11,17,13)(19,21,20,23,22,24), (1,2,5,10,17,12,3,8,15,6,13,14,7,4,9,16,11,18)(19,20,22)(23,24) );
 
Copy content sage:G = PermutationGroup(['(2,4)(3,7)(5,11,9,17,15,13)(6,12,10,14,16,18)(20,22)(21,23)', '(1,3,7)(2,6,14,4,10,12,8,16,18)(11,17,13)(19,21,20,23,22,24)', '(1,2,5,10,17,12,3,8,15,6,13,14,7,4,9,16,11,18)(19,20,22)(23,24)'])
 
Copy content sage_gap:G = gap.new('Group( (2,4)(3,7)(5,11,9,17,15,13)(6,12,10,14,16,18)(20,22)(21,23), (1,3,7)(2,6,14,4,10,12,8,16,18)(11,17,13)(19,21,20,23,22,24), (1,2,5,10,17,12,3,8,15,6,13,14,7,4,9,16,11,18)(19,20,22)(23,24) )')
 
Copy content oscar:G = @permutation_group(24, (2,4)(3,7)(5,11,9,17,15,13)(6,12,10,14,16,18)(20,22)(21,23), (1,3,7)(2,6,14,4,10,12,8,16,18)(11,17,13)(19,21,20,23,22,24), (1,2,5,10,17,12,3,8,15,6,13,14,7,4,9,16,11,18)(19,20,22)(23,24))
 

Group information

Description:$C_3^7.D_6:D_6$
Order: \(314928\)\(\medspace = 2^{4} \cdot 3^{9} \)
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()
 
Copy content oscar:order(G)
 
Exponent: \(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
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()
 
Copy content oscar:exponent(G)
 
Automorphism group:$C_3^6.C_3^3.C_6^2.C_2^3$, of order \(5668704\)\(\medspace = 2^{5} \cdot 3^{11} \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage:libgap(G).AutomorphismGroup()
 
Copy content sage_gap:G.AutomorphismGroup()
 
Copy content oscar:automorphism_group(G)
 
Composition factors:$C_2$ x 4, $C_3$ x 9
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()
 
Copy content oscar:composition_series(G)
 
Derived length:$3$
Copy content comment:Derived length of the group
 
Copy content magma:DerivedLength(G);
 
Copy content gap:DerivedLength(G);
 
Copy content sage:libgap(G).DerivedLength()
 
Copy content sage_gap:G.DerivedLength()
 
Copy content oscar:derived_length(G)
 

This group is nonabelian and solvable. Whether it is monomial 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 oscar:is_abelian(G)
 
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 oscar:is_cyclic(G)
 
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 oscar:is_nilpotent(G)
 
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 oscar:is_solvable(G)
 
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 oscar:is_supersolvable(G)
 
Copy content comment:Determine if the group G is simple
 
Copy content magma:IsSimple(G);
 
Copy content gap:IsSimpleGroup(G);
 
Copy content sage:G.is_simple()
 
Copy content sage_gap:G.IsSimpleGroup()
 
Copy content oscar:is_simple(G)
 

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))
 
Copy content sage_gap:# Sage code (using the GAP interface) to output the first two rows of the group statistics table element_orders = [g.Order() for g in G.Elements()] orders = sorted(list(set(element_orders))) print("Orders:", orders) print("Elements:", [element_orders.count(n) for n in orders], G.Order()) cc_orders = [cc.Representative().Order() for cc in G.ConjugacyClasses()] print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))
 
Copy content oscar:# Oscar code to output the first two rows of the group statistics table element_orders = [order(g) for g in elements(G)] orders = sort(unique(element_orders)) println("Orders: ", orders) element_counts = [count(==(n), element_orders) for n in orders] println("Elements: ", element_counts, " ", order(G)) ccs = conjugacy_classes(G) cc_orders = [order(representative(cc)) for cc in ccs] cc_counts = [count(==(n), cc_orders) for n in orders] println("Conjugacy classes: ", cc_counts, " ", length(ccs))
 

Order 1 2 3 4 6 9 12 18 36
Elements 1 4551 6560 1944 128856 13122 33048 83106 43740 314928
Conjugacy classes   1 7 74 2 149 55 10 75 5 378
Divisions 1 7 62 2 112 47 8 63 4 306
Autjugacy classes 1 7 44 2 94 33 8 43 5 237

Minimal presentations

Permutation degree:$24$
Transitive degree:$36$
Rank: $3$
Inequivalent generating triples: not computed

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 24 not computed not computed
Arbitrary not computed not computed not computed

Constructions

Show commands: Gap / Magma / Oscar / SageMath


Presentation: ${\langle a, b, c, d, e, f, g, h, i \mid c^{6}=f^{3}=i^{9}=[e,f]=[e,i]=[f,g]= \!\cdots\! \rangle}$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([13, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3382704, 4594149, 66, 11906234, 1968436, 8770323, 5402920, 3456209, 146, 5999764, 7376477, 276150, 20189525, 3705174, 5069875, 687536, 11991, 226, 18214566, 11688787, 6984464, 28437, 404644, 132762, 5181703, 9824276, 6735489, 1832110, 920459, 156072, 152029, 50642, 303272, 6916125, 25319, 4285, 36791049, 13079062, 2187155, 2574048, 861961, 554654, 45717, 78490, 7146, 11963962, 10007735, 2630664, 1570189, 81572, 28389, 38711, 2701, 2714, 42164939, 13927704, 5144293, 2145362, 1615599, 533596, 151721, 11828, 2013, 622, 35481900, 17740969]); a,b,c,d,e,f,g,h,i := Explode([G.1, G.2, G.4, G.6, G.8, G.9, G.10, G.11, G.12]); AssignNames(~G, ["a", "b", "b2", "c", "c2", "d", "d2", "e", "f", "g", "h", "i", "i3"]);
 
Copy content gap:G := PcGroupCode(24538067914315831980504839417288933641760499969173279440472023756586213279685606194822613855276881525987115782085512097765754151126848019876977298197516884263537513734998253998080335581393343660017123703690640809418159325789572754199733869587608133545802656583597301300722949225290524805792099424253281481525185921394783994948369543642353662618297755323600087176566065468449188120645985663,314928); a := G.1; b := G.2; c := G.4; d := G.6; e := G.8; f := G.9; g := G.10; h := G.11; i := G.12;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(24538067914315831980504839417288933641760499969173279440472023756586213279685606194822613855276881525987115782085512097765754151126848019876977298197516884263537513734998253998080335581393343660017123703690640809418159325789572754199733869587608133545802656583597301300722949225290524805792099424253281481525185921394783994948369543642353662618297755323600087176566065468449188120645985663,314928)'); a = G.1; b = G.2; c = G.4; d = G.6; e = G.8; f = G.9; g = G.10; h = G.11; i = G.12;
 
Copy content sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(24538067914315831980504839417288933641760499969173279440472023756586213279685606194822613855276881525987115782085512097765754151126848019876977298197516884263537513734998253998080335581393343660017123703690640809418159325789572754199733869587608133545802656583597301300722949225290524805792099424253281481525185921394783994948369543642353662618297755323600087176566065468449188120645985663,314928)'); a = G.1; b = G.2; c = G.4; d = G.6; e = G.8; f = G.9; g = G.10; h = G.11; i = G.12;
 
Permutation group:Degree $24$ $\langle(2,4)(3,7)(5,11,9,17,15,13)(6,12,10,14,16,18)(20,22)(21,23), (1,3,7)(2,6,14,4,10,12,8,16,18) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 24 | (2,4)(3,7)(5,11,9,17,15,13)(6,12,10,14,16,18)(20,22)(21,23), (1,3,7)(2,6,14,4,10,12,8,16,18)(11,17,13)(19,21,20,23,22,24), (1,2,5,10,17,12,3,8,15,6,13,14,7,4,9,16,11,18)(19,20,22)(23,24) >;
 
Copy content gap:G := Group( (2,4)(3,7)(5,11,9,17,15,13)(6,12,10,14,16,18)(20,22)(21,23), (1,3,7)(2,6,14,4,10,12,8,16,18)(11,17,13)(19,21,20,23,22,24), (1,2,5,10,17,12,3,8,15,6,13,14,7,4,9,16,11,18)(19,20,22)(23,24) );
 
Copy content sage:G = PermutationGroup(['(2,4)(3,7)(5,11,9,17,15,13)(6,12,10,14,16,18)(20,22)(21,23)', '(1,3,7)(2,6,14,4,10,12,8,16,18)(11,17,13)(19,21,20,23,22,24)', '(1,2,5,10,17,12,3,8,15,6,13,14,7,4,9,16,11,18)(19,20,22)(23,24)'])
 
Copy content sage_gap:G = gap.new('Group( (2,4)(3,7)(5,11,9,17,15,13)(6,12,10,14,16,18)(20,22)(21,23), (1,3,7)(2,6,14,4,10,12,8,16,18)(11,17,13)(19,21,20,23,22,24), (1,2,5,10,17,12,3,8,15,6,13,14,7,4,9,16,11,18)(19,20,22)(23,24) )')
 
Copy content oscar:G = @permutation_group(24, (2,4)(3,7)(5,11,9,17,15,13)(6,12,10,14,16,18)(20,22)(21,23), (1,3,7)(2,6,14,4,10,12,8,16,18)(11,17,13)(19,21,20,23,22,24), (1,2,5,10,17,12,3,8,15,6,13,14,7,4,9,16,11,18)(19,20,22)(23,24))
 
Transitive group: 36T27696 more information
Copy content magma:G := TransitiveGroup(36, 27696);
 
Copy content gap:G := TransitiveGroup(36, 27696);
 
Copy content sage:G = TransitiveGroup(36, 27696)
 
Copy content sage_gap:G = libgap.TransitiveGroup(36, 27696)
 
Copy content oscar:G = transitive_group(36, 27696)
 
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: not computed
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Possibly split product: $(C_3^7.D_6)$ . $D_6$ $(C_3^7.D_6)$ . $D_6$ $C_3^7$ . $(D_6:D_6)$ $(C_3^6.S_3^3)$ . $C_2$ all 92

Elements of the group are displayed as permutations of degree 24.

Homology

Abelianization: $C_{2}^{3} $
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())
 
Copy content sage_gap:G.FactorGroup(G.DerivedSubgroup())
 
Copy content oscar:quo(G, derived_subgroup(G)[1])
 
Schur multiplier: $C_{2}^{3}$
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()
 
Copy content oscar:subgroups(G)
 

There are 118 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_1$ $G/Z \simeq$ $C_3^7.D_6:D_6$
Copy content comment:Center of the group
 
Copy content magma:Center(G);
 
Copy content gap:Center(G);
 
Copy content sage:G.center()
 
Copy content sage_gap:G.Center()
 
Copy content oscar:center(G)
 
Commutator: $G' \simeq$ $C_3^7.C_3.C_6$ $G/G' \simeq$ $C_2^3$
Copy content comment:Commutator subgroup of the group G
 
Copy content magma:CommutatorSubgroup(G);
 
Copy content gap:DerivedSubgroup(G);
 
Copy content sage:G.commutator()
 
Copy content sage_gap:G.DerivedSubgroup()
 
Copy content oscar:derived_subgroup(G)
 
Frattini: $\Phi \simeq$ $C_3^4$ $G/\Phi \simeq$ $C_3\wr C_4:D_6$
Copy content comment:Frattini subgroup of the group G
 
Copy content magma:FrattiniSubgroup(G);
 
Copy content gap:FrattiniSubgroup(G);
 
Copy content sage:G.frattini_subgroup()
 
Copy content sage_gap:G.FrattiniSubgroup()
 
Copy content oscar:frattini_subgroup(G)
 
Fitting: $\operatorname{Fit} \simeq$ $C_3^7.C_3^2$ $G/\operatorname{Fit} \simeq$ $C_2\times D_4$
Copy content comment:Fitting subgroup of the group G
 
Copy content magma:FittingSubgroup(G);
 
Copy content gap:FittingSubgroup(G);
 
Copy content sage:G.fitting_subgroup()
 
Copy content sage_gap:G.FittingSubgroup()
 
Copy content oscar:fitting_subgroup(G)
 
Radical: $R \simeq$ $C_3^7.D_6:D_6$ $G/R \simeq$ $C_1$
Copy content comment:Radical of the group G
 
Copy content magma:Radical(G);
 
Copy content gap:SolvableRadical(G);
 
Copy content sage_gap:G.SolvableRadical()
 
Copy content oscar:solvable_radical(G)
 
Socle: $\operatorname{soc} \simeq$ $C_3^4$ $G/\operatorname{soc} \simeq$ $(C_3\times C_6^2):S_3^2$
Copy content comment:Socle of the group G
 
Copy content magma:Socle(G);
 
Copy content gap:Socle(G);
 
Copy content sage:G.socle()
 
Copy content sage_gap:G.Socle()
 
Copy content oscar:socle(G)
 
2-Sylow subgroup: $P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3^7.C_3^2$

Subgroup diagram and profile

Series

Derived series $C_3^7.D_6:D_6$ $\rhd$ $C_3^7.C_3.C_6$ $\rhd$ $C_3^6$ $\rhd$ $C_1$
Copy content comment:Derived series of the group G
 
Copy content magma:DerivedSeries(G);
 
Copy content gap:DerivedSeriesOfGroup(G);
 
Copy content sage:G.derived_series()
 
Copy content sage_gap:G.DerivedSeriesOfGroup()
 
Copy content oscar:derived_series(G)
 
Chief series $C_3^7.D_6:D_6$ $\rhd$ $C_3^7.C_6^2.C_2$ $\rhd$ $C_3^7.C_6^2$ $\rhd$ $C_3^7.C_3.C_6$ $\rhd$ $C_3^7.C_3^2$ $\rhd$ $C_3^2\times C_3^4.C_3^2$ $\rhd$ $C_3^2\times C_3^4.C_3$ $\rhd$ $C_3^6$ $\rhd$ $C_3^5$ $\rhd$ $C_3^3$ $\rhd$ $C_3^2$ $\rhd$ $C_3$ $\rhd$ $C_1$
Copy content comment:Chief series of the group G
 
Copy content magma:ChiefSeries(G);
 
Copy content gap:ChiefSeries(G);
 
Copy content sage:libgap(G).ChiefSeries()
 
Copy content sage_gap:G.ChiefSeries()
 
Copy content oscar:chief_series(G)
 
Lower central series $C_3^7.D_6:D_6$ $\rhd$ $C_3^7.C_3.C_6$ $\rhd$ $C_3^7.C_3^2$
Copy content comment:The lower central series of the group G
 
Copy content magma:LowerCentralSeries(G);
 
Copy content gap:LowerCentralSeriesOfGroup(G);
 
Copy content sage:G.lower_central_series()
 
Copy content sage_gap:G.LowerCentralSeriesOfGroup()
 
Copy content oscar:lower_central_series(G)
 
Upper central series $C_1$
Copy content comment:The upper central series of the group G
 
Copy content magma:UpperCentralSeries(G);
 
Copy content gap:UpperCentralSeriesOfGroup(G);
 
Copy content sage:G.upper_central_series()
 
Copy content sage_gap:G.UpperCentralSeriesOfGroup()
 
Copy content oscar:upper_central_series(G)
 

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

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
 
Copy content oscar:character_table(G) # Output not guaranteed to exactly match the LMFDB table
 

Complex character table

The $378 \times 378$ character table is not available for this group.

Rational character table

The $306 \times 306$ rational character table is not available for this group.