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Properties

Label 559872.bi
Order \( 2^{8} \cdot 3^{7} \)
Exponent \( 2^{2} \cdot 3^{2} \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{3} \cdot 3 \)
$\card{Z(G)}$ \( 2 \cdot 3 \)
$\card{\Aut(G)}$ \( 2^{11} \cdot 3^{8} \)
$\card{\mathrm{Out}(G)}$ \( 2^{4} \cdot 3^{2} \)
Perm deg. $30$
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< 30 | (1,2,4)(3,7,9,6,5,8)(10,12)(11,13)(14,17)(15,16)(20,24,25,26,22,23)(27,28)(29,30), (1,3,8)(2,5,7,4,9,6)(10,12,15,11,14,17)(13,16)(18,20,24,19,22,23,21,25,26)(27,29,28,30), (3,6)(5,7)(8,9)(10,11)(12,13)(14,16)(15,17)(18,19,21)(20,23,25,24,22,26)(27,28) >;
 
Copy content gap:G := Group( (1,2,4)(3,7,9,6,5,8)(10,12)(11,13)(14,17)(15,16)(20,24,25,26,22,23)(27,28)(29,30), (1,3,8)(2,5,7,4,9,6)(10,12,15,11,14,17)(13,16)(18,20,24,19,22,23,21,25,26)(27,29,28,30), (3,6)(5,7)(8,9)(10,11)(12,13)(14,16)(15,17)(18,19,21)(20,23,25,24,22,26)(27,28) );
 
Copy content sage:G = PermutationGroup(['(1,2,4)(3,7,9,6,5,8)(10,12)(11,13)(14,17)(15,16)(20,24,25,26,22,23)(27,28)(29,30)', '(1,3,8)(2,5,7,4,9,6)(10,12,15,11,14,17)(13,16)(18,20,24,19,22,23,21,25,26)(27,29,28,30)', '(3,6)(5,7)(8,9)(10,11)(12,13)(14,16)(15,17)(18,19,21)(20,23,25,24,22,26)(27,28)'])
 
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(488825928176712750775774816461662600258426314310097112530527890561048203287574733555487603005105645965254913493597383812830699803134292081077724750079915743934387486630312093356326804046502026232759639245877246954451513335038213935899507396117625132220556886563697004802262847791921116607983268700094844132903624912789362433311623632683477902207317,559872)'); a = G.1; b = G.3; c = G.5; d = G.8; e = G.10; f = G.12; g = G.14;
 

Group information

Description:$C_6\wr D_6$
Order: \(559872\)\(\medspace = 2^{8} \cdot 3^{7} \)
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:$(C_2\times C_6^3).C_3^5.C_2^6.C_2$, of order \(13436928\)\(\medspace = 2^{11} \cdot 3^{8} \)
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 8, $C_3$ 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 1359 3320 7344 126216 20736 276480 93312 31104 559872
Conjugacy classes   1 17 92 15 4384 5 635 15 2 5166
Divisions 1 17 52 15 2244 3 325 8 1 2666
Autjugacy classes 1 15 20 12 416 3 71 7 1 546

Minimal presentations

Permutation degree:$30$
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: ${\langle a, b, c, d, e, f, g \mid a^{6}=b^{6}=c^{12}=d^{6}=e^{6}=f^{6}=g^{6}= \!\cdots\! \rangle}$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([15, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 30, 19596872, 122, 13065123, 29822404, 9333019, 6823384, 2721874, 214, 48245765, 548660, 6669035, 1633550, 260, 17812626, 3863811, 4629276, 3811551, 51321607, 1585507, 258562, 123217, 352, 25194248, 1399748, 372683, 175058, 56505609, 19869, 217884, 306999, 444, 31933450, 31750, 443605, 12970, 80170571, 1313351, 1113566, 23321, 536, 42793932, 2255832, 594447, 50412, 60782413, 1378513, 844288, 564583, 628, 194414, 2370674, 2789, 310604]); a,b,c,d,e,f,g := Explode([G.1, G.3, G.5, G.8, G.10, G.12, G.14]); AssignNames(~G, ["a", "a2", "b", "b2", "c", "c2", "c4", "d", "d2", "e", "e2", "f", "f2", "g", "g2"]);
 
Copy content gap:G := PcGroupCode(488825928176712750775774816461662600258426314310097112530527890561048203287574733555487603005105645965254913493597383812830699803134292081077724750079915743934387486630312093356326804046502026232759639245877246954451513335038213935899507396117625132220556886563697004802262847791921116607983268700094844132903624912789362433311623632683477902207317,559872); a := G.1; b := G.3; c := G.5; d := G.8; e := G.10; f := G.12; g := G.14;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(488825928176712750775774816461662600258426314310097112530527890561048203287574733555487603005105645965254913493597383812830699803134292081077724750079915743934387486630312093356326804046502026232759639245877246954451513335038213935899507396117625132220556886563697004802262847791921116607983268700094844132903624912789362433311623632683477902207317,559872)'); a = G.1; b = G.3; c = G.5; d = G.8; e = G.10; f = G.12; g = G.14;
 
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(488825928176712750775774816461662600258426314310097112530527890561048203287574733555487603005105645965254913493597383812830699803134292081077724750079915743934387486630312093356326804046502026232759639245877246954451513335038213935899507396117625132220556886563697004802262847791921116607983268700094844132903624912789362433311623632683477902207317,559872)'); a = G.1; b = G.3; c = G.5; d = G.8; e = G.10; f = G.12; g = G.14;
 
Permutation group:Degree $30$ $\langle(1,2,4)(3,7,9,6,5,8)(10,12)(11,13)(14,17)(15,16)(20,24,25,26,22,23)(27,28) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 30 | (1,2,4)(3,7,9,6,5,8)(10,12)(11,13)(14,17)(15,16)(20,24,25,26,22,23)(27,28)(29,30), (1,3,8)(2,5,7,4,9,6)(10,12,15,11,14,17)(13,16)(18,20,24,19,22,23,21,25,26)(27,29,28,30), (3,6)(5,7)(8,9)(10,11)(12,13)(14,16)(15,17)(18,19,21)(20,23,25,24,22,26)(27,28) >;
 
Copy content gap:G := Group( (1,2,4)(3,7,9,6,5,8)(10,12)(11,13)(14,17)(15,16)(20,24,25,26,22,23)(27,28)(29,30), (1,3,8)(2,5,7,4,9,6)(10,12,15,11,14,17)(13,16)(18,20,24,19,22,23,21,25,26)(27,29,28,30), (3,6)(5,7)(8,9)(10,11)(12,13)(14,16)(15,17)(18,19,21)(20,23,25,24,22,26)(27,28) );
 
Copy content sage:G = PermutationGroup(['(1,2,4)(3,7,9,6,5,8)(10,12)(11,13)(14,17)(15,16)(20,24,25,26,22,23)(27,28)(29,30)', '(1,3,8)(2,5,7,4,9,6)(10,12,15,11,14,17)(13,16)(18,20,24,19,22,23,21,25,26)(27,29,28,30)', '(3,6)(5,7)(8,9)(10,11)(12,13)(14,16)(15,17)(18,19,21)(20,23,25,24,22,26)(27,28)'])
 
Transitive group: 36T30406 more information
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: not computed
Trans. wreath product: $C_6$$\ \wr\ $$D_6$
Possibly split product: $C_6^6$ . $D_6$ (6) $C_6^5$ . $(S_3\times D_6)$ $C_6^5$ . $(C_6\times D_6)$ (2) $(C_2\times C_6^5)$ . $S_3^2$ all 187

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: $C_{2}^{5}$
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 222 normal subgroups (214 characteristic).

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

Special subgroups

Center: $Z \simeq$ $C_6$ $G/Z \simeq$ $C_2\times C_6^4.S_3^2$
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: $G' \simeq$ $C_6^4.C_3.C_6$ $G/G' \simeq$ $C_2^2\times C_6$
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: $\Phi \simeq$ $C_3\times C_6^3$ $G/\Phi \simeq$ $C_6\times S_3\times S_4$
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: $\operatorname{Fit} \simeq$ $C_6^6$ $G/\operatorname{Fit} \simeq$ $D_6$
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: $R \simeq$ $C_6\wr D_6$ $G/R \simeq$ $C_1$
Copy content comment:Radical of the group G
 
Copy content magma:Radical(G);
 
Copy content gap:SolvableRadical(G);
 
Copy content sage_gap:G.SolvableRadical()
 
Socle: $\operatorname{soc} \simeq$ $C_2\times C_6^2$ $G/\operatorname{soc} \simeq$ $C_6^3.S_3^2$
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()
 
2-Sylow subgroup: $P_{ 2 } \simeq$ $C_2^5:D_4$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3^2\wr C_3$

Subgroup diagram and profile

Series

Derived series $C_6\wr D_6$ $\rhd$ $C_6^4.C_3.C_6$ $\rhd$ $C_2\times C_6^3$ $\rhd$ $C_1$
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 $C_6\wr D_6$ $\rhd$ $C_6^4.C_3^3.C_2^3$ $\rhd$ $C_6^4.C_3.C_6^2$ $\rhd$ $C_6^4.C_3^3.C_2$ $\rhd$ $C_6^4.C_3.C_6$ $\rhd$ $C_6^4.C_3^2$ $\rhd$ $C_2^4.C_3^4:C_3$ $\rhd$ $C_6^4$ $\rhd$ $C_2\times C_6^3$ $\rhd$ $C_2^2\times C_6^2$ $\rhd$ $C_2^3\times C_6$ $\rhd$ $C_2^4$ $\rhd$ $C_2^2$ $\rhd$ $C_1$
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 $C_6\wr D_6$ $\rhd$ $C_6^4.C_3.C_6$ $\rhd$ $C_6^4.C_3^2$
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()
 
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 9 larger groups in the database.

This group is a maximal quotient of 3 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 $5166 \times 5166$ character table is not available for this group.

Rational character table

The $2666 \times 2666$ rational character table is not available for this group.