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Properties

Label 314928.py
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
$\card{\Aut(G)}$ \( 2^{6} \cdot 3^{11} \)
$\card{\mathrm{Out}(G)}$ \( 2^{2} \cdot 3^{2} \)
Perm deg. $36$
Trans deg. not computed
Rank $3$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Construction of abstract group
 
Copy content magma:G := PermutationGroup< 36 | (1,12,14,34,27,24,3,10,13,35,26,22,2,11,15,36,25,23)(4,20,30,31,16,8,6,21,29,32,18,9,5,19,28,33,17,7), (1,31,2,33,3,32)(4,35)(5,36)(6,34)(7,15,8,14,9,13)(10,28,12,30,11,29)(16,23,17,24,18,22)(19,25)(20,27)(21,26), (1,13,25,3,15,27,2,14,26)(4,17)(5,16)(6,18)(7,33,19,9,32,21,8,31,20)(10,24)(11,23)(12,22)(28,30)(34,35) >;
 
Copy content gap:G := Group( (1,12,14,34,27,24,3,10,13,35,26,22,2,11,15,36,25,23)(4,20,30,31,16,8,6,21,29,32,18,9,5,19,28,33,17,7), (1,31,2,33,3,32)(4,35)(5,36)(6,34)(7,15,8,14,9,13)(10,28,12,30,11,29)(16,23,17,24,18,22)(19,25)(20,27)(21,26), (1,13,25,3,15,27,2,14,26)(4,17)(5,16)(6,18)(7,33,19,9,32,21,8,31,20)(10,24)(11,23)(12,22)(28,30)(34,35) );
 
Copy content sage:G = PermutationGroup(['(1,12,14,34,27,24,3,10,13,35,26,22,2,11,15,36,25,23)(4,20,30,31,16,8,6,21,29,32,18,9,5,19,28,33,17,7)', '(1,31,2,33,3,32)(4,35)(5,36)(6,34)(7,15,8,14,9,13)(10,28,12,30,11,29)(16,23,17,24,18,22)(19,25)(20,27)(21,26)', '(1,13,25,3,15,27,2,14,26)(4,17)(5,16)(6,18)(7,33,19,9,32,21,8,31,20)(10,24)(11,23)(12,22)(28,30)(34,35)'])
 
Copy content sage_gap:G = gap.new('Group( (1,12,14,34,27,24,3,10,13,35,26,22,2,11,15,36,25,23)(4,20,30,31,16,8,6,21,29,32,18,9,5,19,28,33,17,7), (1,31,2,33,3,32)(4,35)(5,36)(6,34)(7,15,8,14,9,13)(10,28,12,30,11,29)(16,23,17,24,18,22)(19,25)(20,27)(21,26), (1,13,25,3,15,27,2,14,26)(4,17)(5,16)(6,18)(7,33,19,9,32,21,8,31,20)(10,24)(11,23)(12,22)(28,30)(34,35) )')
 
Copy content oscar:G = @permutation_group(36, (1,12,14,34,27,24,3,10,13,35,26,22,2,11,15,36,25,23)(4,20,30,31,16,8,6,21,29,32,18,9,5,19,28,33,17,7), (1,31,2,33,3,32)(4,35)(5,36)(6,34)(7,15,8,14,9,13)(10,28,12,30,11,29)(16,23,17,24,18,22)(19,25)(20,27)(21,26), (1,13,25,3,15,27,2,14,26)(4,17)(5,16)(6,18)(7,33,19,9,32,21,8,31,20)(10,24)(11,23)(12,22)(28,30)(34,35))
 

Group information

Description:$C_9^4.C_6.C_2^3$
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:$C_3^7.S_3\wr C_2^2$, of order \(11337408\)\(\medspace = 2^{6} \cdot 3^{11} \)
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 or almost simple 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 18
Elements 1 8019 242 78732 33534 19440 174960 314928
Conjugacy classes   1 7 11 2 23 178 84 306
Divisions 1 7 11 2 21 86 42 170
Autjugacy classes 1 5 7 1 13 30 22 79

Copy content comment:Compute statistics about the characters of G
 
Copy content 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);
 
Copy content 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);
 
Copy content 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)]
 
Copy content sage_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 G.CharacterDegrees()
 
Copy content oscar:# Outputs an MSet containing the absolutely irreducible degrees of G and their multiplicities. character_degrees(G)
 

Dimension 1 2 4 8 12 16 24 32 36 48 72 144
Irr. complex chars.   8 10 32 20 32 8 84 0 0 112 0 0 306
Irr. rational chars. 8 6 18 20 8 8 18 2 8 22 22 30 170

Minimal presentations

Permutation degree:$36$
Transitive degree:not computed
Rank: $3$
Inequivalent generating triples: $783820800$

Minimal degrees of faithful linear representations

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

Constructions

Show commands: Gap / Magma / Oscar / SageMath


Presentation: ${\langle a, b, c, d, e, f \mid b^{6}=c^{18}=d^{18}=e^{9}=f^{9}=[c,e]=[e,f]= \!\cdots\! \rangle}$ 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, 3, 2, 3, 3, 3, 3, 3, 3, 2678832, 4928145, 66, 8188442, 12668763, 7953832, 2747345, 146, 13703044, 382997, 2760, 251, 202181, 5634, 5499318, 9621631, 63914, 1266219, 45103, 266, 2438599, 157268, 44961, 29998, 410, 16856, 25319, 18195849, 1010902, 884555, 18807, 516, 15011578, 15532, 4245707, 5458776, 5458789, 3639218, 202265, 622, 1971228, 17740969, 2956875, 164358]); a,b,c,d,e,f := Explode([G.1, G.2, G.4, G.7, G.10, G.12]); AssignNames(~G, ["a", "b", "b2", "c", "c2", "c6", "d", "d2", "d6", "e", "e3", "f", "f3"]);
 
Copy content gap:G := PcGroupCode(2686465571530319074572734014821484939986554241202366020200089459041204682010326746823284002636178478550073661901690872675239573332383272302989894983141697413714529682025340084887907605566660684024347820084260039274618911929362932410560147938980110744448214689788945855,314928); a := G.1; b := G.2; c := G.4; d := G.7; e := G.10; f := 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(2686465571530319074572734014821484939986554241202366020200089459041204682010326746823284002636178478550073661901690872675239573332383272302989894983141697413714529682025340084887907605566660684024347820084260039274618911929362932410560147938980110744448214689788945855,314928)'); a = G.1; b = G.2; c = G.4; d = G.7; e = G.10; f = 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(2686465571530319074572734014821484939986554241202366020200089459041204682010326746823284002636178478550073661901690872675239573332383272302989894983141697413714529682025340084887907605566660684024347820084260039274618911929362932410560147938980110744448214689788945855,314928)'); a = G.1; b = G.2; c = G.4; d = G.7; e = G.10; f = G.12;
 
Permutation group:Degree $36$ $\langle(1,12,14,34,27,24,3,10,13,35,26,22,2,11,15,36,25,23)(4,20,30,31,16,8,6,21,29,32,18,9,5,19,28,33,17,7) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 36 | (1,12,14,34,27,24,3,10,13,35,26,22,2,11,15,36,25,23)(4,20,30,31,16,8,6,21,29,32,18,9,5,19,28,33,17,7), (1,31,2,33,3,32)(4,35)(5,36)(6,34)(7,15,8,14,9,13)(10,28,12,30,11,29)(16,23,17,24,18,22)(19,25)(20,27)(21,26), (1,13,25,3,15,27,2,14,26)(4,17)(5,16)(6,18)(7,33,19,9,32,21,8,31,20)(10,24)(11,23)(12,22)(28,30)(34,35) >;
 
Copy content gap:G := Group( (1,12,14,34,27,24,3,10,13,35,26,22,2,11,15,36,25,23)(4,20,30,31,16,8,6,21,29,32,18,9,5,19,28,33,17,7), (1,31,2,33,3,32)(4,35)(5,36)(6,34)(7,15,8,14,9,13)(10,28,12,30,11,29)(16,23,17,24,18,22)(19,25)(20,27)(21,26), (1,13,25,3,15,27,2,14,26)(4,17)(5,16)(6,18)(7,33,19,9,32,21,8,31,20)(10,24)(11,23)(12,22)(28,30)(34,35) );
 
Copy content sage:G = PermutationGroup(['(1,12,14,34,27,24,3,10,13,35,26,22,2,11,15,36,25,23)(4,20,30,31,16,8,6,21,29,32,18,9,5,19,28,33,17,7)', '(1,31,2,33,3,32)(4,35)(5,36)(6,34)(7,15,8,14,9,13)(10,28,12,30,11,29)(16,23,17,24,18,22)(19,25)(20,27)(21,26)', '(1,13,25,3,15,27,2,14,26)(4,17)(5,16)(6,18)(7,33,19,9,32,21,8,31,20)(10,24)(11,23)(12,22)(28,30)(34,35)'])
 
Copy content sage_gap:G = gap.new('Group( (1,12,14,34,27,24,3,10,13,35,26,22,2,11,15,36,25,23)(4,20,30,31,16,8,6,21,29,32,18,9,5,19,28,33,17,7), (1,31,2,33,3,32)(4,35)(5,36)(6,34)(7,15,8,14,9,13)(10,28,12,30,11,29)(16,23,17,24,18,22)(19,25)(20,27)(21,26), (1,13,25,3,15,27,2,14,26)(4,17)(5,16)(6,18)(7,33,19,9,32,21,8,31,20)(10,24)(11,23)(12,22)(28,30)(34,35) )')
 
Copy content oscar:G = @permutation_group(36, (1,12,14,34,27,24,3,10,13,35,26,22,2,11,15,36,25,23)(4,20,30,31,16,8,6,21,29,32,18,9,5,19,28,33,17,7), (1,31,2,33,3,32)(4,35)(5,36)(6,34)(7,15,8,14,9,13)(10,28,12,30,11,29)(16,23,17,24,18,22)(19,25)(20,27)(21,26), (1,13,25,3,15,27,2,14,26)(4,17)(5,16)(6,18)(7,33,19,9,32,21,8,31,20)(10,24)(11,23)(12,22)(28,30)(34,35))
 
Transitive group: 36T27822 more information
Copy content magma:G := TransitiveGroup(36, 27822);
 
Copy content gap:G := TransitiveGroup(36, 27822);
 
Copy content sage:G = TransitiveGroup(36, 27822)
 
Copy content sage_gap:G = libgap.TransitiveGroup(36, 27822)
 
Copy content oscar:G = transitive_group(36, 27822)
 
Direct product: not computed
Semidirect product: not computed
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product

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

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)
 

Subgroup data has not been computed.

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

See the $306 \times 306$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $170 \times 170$ rational character table (warning: may be slow to load).