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Properties

Label 139968.df
Order \( 2^{6} \cdot 3^{7} \)
Exponent \( 2^{2} \cdot 3^{2} \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{3} \cdot 3 \)
$\card{Z(G)}$ \( 2 \cdot 3 \)
$\card{\Aut(G)}$ \( 2^{8} \cdot 3^{8} \)
$\card{\mathrm{Out}(G)}$ \( 2^{3} \cdot 3^{2} \)
Perm deg. $26$
Trans deg. $36$
Rank $3$

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Show commands: Gap / Magma / SageMath

Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 26 | (1,4,2)(3,7)(5,9)(6,8)(10,12,13,11)(14,15)(16,19,22,20,17,18)(23,25), (1,2,4)(3,6)(5,7)(8,9)(10,11)(12,13)(14,15)(16,18)(17,20)(19,22)(23,24,25,26), (1,3,4,8,2,5)(11,12)(14,16,15,17,21,22)(18,20,19)(24,25) >;
 
Copy content gap:G := Group( (1,4,2)(3,7)(5,9)(6,8)(10,12,13,11)(14,15)(16,19,22,20,17,18)(23,25), (1,2,4)(3,6)(5,7)(8,9)(10,11)(12,13)(14,15)(16,18)(17,20)(19,22)(23,24,25,26), (1,3,4,8,2,5)(11,12)(14,16,15,17,21,22)(18,20,19)(24,25) );
 
Copy content sage:G = PermutationGroup(['(1,4,2)(3,7)(5,9)(6,8)(10,12,13,11)(14,15)(16,19,22,20,17,18)(23,25)', '(1,2,4)(3,6)(5,7)(8,9)(10,11)(12,13)(14,15)(16,18)(17,20)(19,22)(23,24,25,26)', '(1,3,4,8,2,5)(11,12)(14,16,15,17,21,22)(18,20,19)(24,25)'])
 
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(3016211753084262315188979752036456207517039639577956802542183387963471800515793919942899158625969551065786871851446474832863326927768023926898012946873168303813609395834017985018613291378858975300895300432276119249090355703507627,139968)'); a = G.1; b = G.3; c = G.5; d = G.7; e = G.9; f = G.11; g = G.13;
 

Group information

Description:$C_6^4.(C_3\times S_3^2)$
Order: \(139968\)\(\medspace = 2^{6} \cdot 3^{7} \)
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: \(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
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()
 
Automorphism group:$(C_3\times C_6^2).C_3^5.C_2^6$, of order \(1679616\)\(\medspace = 2^{8} \cdot 3^{8} \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$ x 6, $C_3$ x 7
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:$3$
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 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 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 6 9 12 18 36
Elements 1 771 1376 1404 48072 5184 52056 23328 7776 139968
Conjugacy classes   1 11 92 8 1303 5 246 15 2 1683
Divisions 1 11 52 8 683 3 130 8 1 897
Autjugacy classes 1 10 20 7 187 3 40 7 1 276

Minimal presentations

Permutation degree:$26$
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 6 not computed not computed
Arbitrary not computed not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Presentation: ${\langle a, b, c, d, e, f, g \mid b^{6}=c^{6}=d^{6}=e^{6}=f^{6}=g^{3}=[a,g]= \!\cdots\! \rangle}$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([13, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 26, 303278, 350768, 2353236, 106, 952227, 1815844, 1307700, 186, 11237, 608431, 992634, 2312875, 765524, 494175, 187336, 118280, 266, 3055111, 3871316, 1453953, 49342, 338267, 68088, 15634952, 808738, 7106, 10980, 346, 10455129, 786275, 11007, 10240, 12231658, 834012, 7810, 11684, 426, 19408907]); a,b,c,d,e,f,g := Explode([G.1, G.3, G.5, G.7, G.9, G.11, G.13]); AssignNames(~G, ["a", "a2", "b", "b2", "c", "c2", "d", "d2", "e", "e2", "f", "f2", "g"]);
 
Copy content gap:G := PcGroupCode(3016211753084262315188979752036456207517039639577956802542183387963471800515793919942899158625969551065786871851446474832863326927768023926898012946873168303813609395834017985018613291378858975300895300432276119249090355703507627,139968); a := G.1; b := G.3; c := G.5; d := G.7; e := G.9; f := G.11; g := G.13;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(3016211753084262315188979752036456207517039639577956802542183387963471800515793919942899158625969551065786871851446474832863326927768023926898012946873168303813609395834017985018613291378858975300895300432276119249090355703507627,139968)'); a = G.1; b = G.3; c = G.5; d = G.7; e = G.9; f = G.11; g = G.13;
 
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(3016211753084262315188979752036456207517039639577956802542183387963471800515793919942899158625969551065786871851446474832863326927768023926898012946873168303813609395834017985018613291378858975300895300432276119249090355703507627,139968)'); a = G.1; b = G.3; c = G.5; d = G.7; e = G.9; f = G.11; g = G.13;
 
Permutation group:Degree $26$ $\langle(1,4,2)(3,7)(5,9)(6,8)(10,12,13,11)(14,15)(16,19,22,20,17,18)(23,25), (1,2,4) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 26 | (1,4,2)(3,7)(5,9)(6,8)(10,12,13,11)(14,15)(16,19,22,20,17,18)(23,25), (1,2,4)(3,6)(5,7)(8,9)(10,11)(12,13)(14,15)(16,18)(17,20)(19,22)(23,24,25,26), (1,3,4,8,2,5)(11,12)(14,16,15,17,21,22)(18,20,19)(24,25) >;
 
Copy content gap:G := Group( (1,4,2)(3,7)(5,9)(6,8)(10,12,13,11)(14,15)(16,19,22,20,17,18)(23,25), (1,2,4)(3,6)(5,7)(8,9)(10,11)(12,13)(14,15)(16,18)(17,20)(19,22)(23,24,25,26), (1,3,4,8,2,5)(11,12)(14,16,15,17,21,22)(18,20,19)(24,25) );
 
Copy content sage:G = PermutationGroup(['(1,4,2)(3,7)(5,9)(6,8)(10,12,13,11)(14,15)(16,19,22,20,17,18)(23,25)', '(1,2,4)(3,6)(5,7)(8,9)(10,11)(12,13)(14,15)(16,18)(17,20)(19,22)(23,24,25,26)', '(1,3,4,8,2,5)(11,12)(14,16,15,17,21,22)(18,20,19)(24,25)'])
 
Transitive group: 36T20914 more information
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: $C_3^6$ . $(D_4\times S_4)$ $(C_6^4.S_3^2)$ . $C_3$ $(C_3\times C_6^4)$ . $S_3^2$ $C_6^4$ . $(C_3\times S_3^2)$ all 146

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization: $C_{2}^{2} \times C_{6} \simeq C_{2}^{3} \times C_{3}$
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())
 
Schur multiplier: $C_{2}^{4}$
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()
 

There are 174 normal subgroups (166 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_6$ $G/Z \simeq$ $(C_3^2\times C_6^3):D_6$
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()
 
Commutator: $G' \simeq$ $C_2\times C_6^2.C_3^3.C_3$ $G/G' \simeq$ $C_2^2\times C_6$
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()
 
Frattini: $\Phi \simeq$ $C_3^3\times C_6$ $G/\Phi \simeq$ $C_6\times S_3\times S_4$
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()
 
Fitting: $\operatorname{Fit} \simeq$ $C_3^2\times C_6^4$ $G/\operatorname{Fit} \simeq$ $D_6$
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()
 
Radical: $R \simeq$ $C_6^4.(C_3\times S_3^2)$ $G/R \simeq$ $C_1$
Copy content comment:Radical of the group G
 
Copy content magma:Radical(G);
 
Copy content gap:SolvableRadical(G);
 
Copy content sage_gap:G.SolvableRadical()
 
Socle: $\operatorname{soc} \simeq$ $C_2\times C_6^2$ $G/\operatorname{soc} \simeq$ $C_2\times C_3^4:D_6$
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()
 
2-Sylow subgroup: $P_{ 2 } \simeq$ $D_4^2$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3^2\wr C_3$

Subgroup diagram and profile

Series

Derived series $C_6^4.(C_3\times S_3^2)$ $\rhd$ $C_2\times C_6^2.C_3^3.C_3$ $\rhd$ $C_3\times C_6^2$ $\rhd$ $C_1$
Copy content comment:Derived series of the group GF
 
Copy content magma:DerivedSeries(G);
 
Copy content gap:DerivedSeriesOfGroup(G);
 
Copy content sage:G.derived_series()
 
Copy content sage_gap:G.DerivedSeriesOfGroup()
 
Chief series $C_6^4.(C_3\times S_3^2)$ $\rhd$ $(C_3^2\times C_6^3).S_3^2$ $\rhd$ $C_3^5.(C_6\times S_4)$ $\rhd$ $C_2\times C_6^2.C_3^4:C_3$ $\rhd$ $C_2\times C_6^2.C_3^3.C_3$ $\rhd$ $C_6^2.C_3^3.C_3$ $\rhd$ $C_3^3\times C_6^2$ $\rhd$ $C_3^2\times C_6^2$ $\rhd$ $C_3\times C_6^2$ $\rhd$ $C_6^2$ $\rhd$ $C_2\times C_6$ $\rhd$ $C_2^2$ $\rhd$ $C_1$
Copy content comment:Chief series of the group G
 
Copy content magma:ChiefSeries(G);
 
Copy content gap:ChiefSeries(G);
 
Copy content sage_gap:G.ChiefSeries()
 
Lower central series $C_6^4.(C_3\times S_3^2)$ $\rhd$ $C_2\times C_6^2.C_3^3.C_3$ $\rhd$ $C_6^2.C_3^3.C_3$
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()
 
Upper central series $C_1$ $\lhd$ $C_6$ $\lhd$ $C_2\times C_6$
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()
 

Supergroups

This group is a maximal subgroup of 10 larger groups in the database.

This group is a maximal quotient of 4 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
 

Complex character table

The $1683 \times 1683$ character table is not available for this group.

Rational character table

The $897 \times 897$ rational character table is not available for this group.