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Properties

Label 604...000.a
Order \( 2^{39} \cdot 3^{19} \cdot 5^{9} \cdot 7^{6} \cdot 11^{3} \cdot 13^{3} \cdot 17^{2} \cdot 19^{2} \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 41 \cdot 43 \)
Exponent \( 2^{5} \cdot 3^{3} \cdot 5^{2} \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 41 \cdot 43 \)
Nilpotent no
Solvable no
$\card{G^{\mathrm{ab}}}$ \( 2 \)
$\card{Z(G)}$ 1
$\card{\Aut(G)}$ \( 2^{39} \cdot 3^{19} \cdot 5^{9} \cdot 7^{6} \cdot 11^{3} \cdot 13^{3} \cdot 17^{2} \cdot 19^{2} \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 41 \cdot 43 \)
$\card{\mathrm{Out}(G)}$ \( 1 \)
Perm deg. $43$
Trans deg. not computed
Rank $2$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define group as a symmetric group
 
Copy content magma:G := SymmetricGroup(43);
 
Copy content gap:G := SymmetricGroup(43);
 
Copy content sage:G = SymmetricGroup(43)
 
Copy content sage_gap:G = libgap.eval('SymmetricGroup(43)')
 
Copy content oscar:G = symmetric_group(43)
 

Group information

Description:$S_{43}$
Order: \(604\!\cdots\!000\)\(\medspace = 2^{39} \cdot 3^{19} \cdot 5^{9} \cdot 7^{6} \cdot 11^{3} \cdot 13^{3} \cdot 17^{2} \cdot 19^{2} \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 41 \cdot 43 \)
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: \(9419588158802421600\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{2} \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 41 \cdot 43 \)
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:Group of order \(604\!\cdots\!000\)\(\medspace = 2^{39} \cdot 3^{19} \cdot 5^{9} \cdot 7^{6} \cdot 11^{3} \cdot 13^{3} \cdot 17^{2} \cdot 19^{2} \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 41 \cdot 43 \)
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:$C_2$, $A_{43}$
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:$1$
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, nonsolvable, and rational. Whether it is 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))
 

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Conjugacy classes   1 21 14 110 8 504 6 190 30 200 3 2084 3 110 97 114 2 657 2 648 54 48 1 1948 4 37 9 318 1 2317 1 14 24 22 23 1463 1 17 18 612 1 1099 1 126 94 10 616 27 11 90 48 12 312 9 7 5247 6 54 9 411 50 6 384 946 3 15 38 8 278 218 1 2369 6 4 128 1384 6 25 4 5 22 24 37 145 88 147 52 159 2 112 107 4 12 18 4 3039 9 670 111 822 3 10 46 63 703 3 216 2 2 81 35 10 86 2 543 8 2 61 1405 56 9 24 4 44 1869 60 20 1 16 2 6 42 44 237 20 2 256 5 28 2 2107 18 1 261 2 2 4 7 181 25 34 6 156 30 614 1 4 850 1 26 175 2 469 22 32 14 24 2 84 2 447 20 25 2 9 2 1 69 10 70 127 6 298 78 1 13 1 2 721 278 84 1 49 12 18 14 2 1 13 842 84 4 9 12 1 14 59 13 464 1 276 11 2 4 126 20 9 2748 1 10 1 2 4 1 160 7 14 10 81 27 337 2 172 40 4 6 5 33 388 4 206 103 3 6 84 28 16 217 4 30 1 96 6 138 32 6 24 3 6 3 18 2 67 79 2 50 4 683 1 13 3 2 1 866 4 5 1 44 40 56 18 4 10 1 96 5 113 52 3 1 14 6 3 28 12 122 527 1 120 63 3 2 16 13 2 12 9 1118 9 72 1 8 30 9 2 7 78 8 10 385 3 3 70 2 22 1 25 2 1 6 213 3 50 4 1 197 4 18 1 2 15 14 7 3 1 232 2 1 2 30 115 16 41 1 126 1 1 36 2 4 3 24 14 2 19 560 1 5 4 2 1 1 1 2 327 19 24 6 28 12 33 98 2 86 25 2 8 1 12 1 1 6 1 7 2 45 119 1 187 1 12 49 8 1 56 3 3 4 128 3 24 69 1 2 4 1 13 3 1 72 144 1 11 6 2 2 2 1 13 3 1 2 158 2 8 12 1 1 59 6 7 1 6 1 19 11 81 7 1 24 1 1 31 2 277 86 26 10 3 1 1 1 4 13 186 2 15 5 31 13 15 1 162 6 69 5 24 22 2 3 6 1 5 2 24 50 3 14 3 1 12 4 1 38 1 3 5 12 22 2 13 1 52 1 28 12 2 8 2 1 1 2 1 2 47 2 28 11 2 8 2 4 1 10 45 6 2 4 1 6 1 189 2 24 8 5 6 1 2 3 2 12 1 10 1 4 2 4 3 101 20 12 1 8 1 2 1 2 19 5 4 2 6 3 1 1 10 6 2 2 1 1 41 25 2 1 2 2 1 1 5 2 1 11 4 21 1 1 2 23 2 1 2 48 2 1 1 1 7 2 1 4 23 1 4 1 9 1 4 1 1 18 4 2 1 1 8 4 1 1 1 2 1 1 1 3 3 1 1 63261
Divisions 1 21 14 110 8 504 6 190 30 200 3 2084 3 110 97 114 2 657 2 648 54 48 1 1948 4 37 9 318 1 2317 1 14 24 22 23 1463 1 17 18 612 1 1099 1 126 94 10 616 27 11 90 48 12 312 9 7 5247 6 54 9 411 50 6 384 946 3 15 38 8 278 218 1 2369 6 4 128 1384 6 25 4 5 22 24 37 145 88 147 52 159 2 112 107 4 12 18 4 3039 9 670 111 822 3 10 46 63 703 3 216 2 2 81 35 10 86 2 543 8 2 61 1405 56 9 24 4 44 1869 60 20 1 16 2 6 42 44 237 20 2 256 5 28 2 2107 18 1 261 2 2 4 7 181 25 34 6 156 30 614 1 4 850 1 26 175 2 469 22 32 14 24 2 84 2 447 20 25 2 9 2 1 69 10 70 127 6 298 78 1 13 1 2 721 278 84 1 49 12 18 14 2 1 13 842 84 4 9 12 1 14 59 13 464 1 276 11 2 4 126 20 9 2748 1 10 1 2 4 1 160 7 14 10 81 27 337 2 172 40 4 6 5 33 388 4 206 103 3 6 84 28 16 217 4 30 1 96 6 138 32 6 24 3 6 3 18 2 67 79 2 50 4 683 1 13 3 2 1 866 4 5 1 44 40 56 18 4 10 1 96 5 113 52 3 1 14 6 3 28 12 122 527 1 120 63 3 2 16 13 2 12 9 1118 9 72 1 8 30 9 2 7 78 8 10 385 3 3 70 2 22 1 25 2 1 6 213 3 50 4 1 197 4 18 1 2 15 14 7 3 1 232 2 1 2 30 115 16 41 1 126 1 1 36 2 4 3 24 14 2 19 560 1 5 4 2 1 1 1 2 327 19 24 6 28 12 33 98 2 86 25 2 8 1 12 1 1 6 1 7 2 45 119 1 187 1 12 49 8 1 56 3 3 4 128 3 24 69 1 2 4 1 13 3 1 72 144 1 11 6 2 2 2 1 13 3 1 2 158 2 8 12 1 1 59 6 7 1 6 1 19 11 81 7 1 24 1 1 31 2 277 86 26 10 3 1 1 1 4 13 186 2 15 5 31 13 15 1 162 6 69 5 24 22 2 3 6 1 5 2 24 50 3 14 3 1 12 4 1 38 1 3 5 12 22 2 13 1 52 1 28 12 2 8 2 1 1 2 1 2 47 2 28 11 2 8 2 4 1 10 45 6 2 4 1 6 1 189 2 24 8 5 6 1 2 3 2 12 1 10 1 4 2 4 3 101 20 12 1 8 1 2 1 2 19 5 4 2 6 3 1 1 10 6 2 2 1 1 41 25 2 1 2 2 1 1 5 2 1 11 4 21 1 1 2 23 2 1 2 48 2 1 1 1 7 2 1 4 23 1 4 1 9 1 4 1 1 18 4 2 1 1 8 4 1 1 1 2 1 1 1 3 3 1 1 63261
Autjugacy classes 1 21 14 110 8 504 6 190 30 200 3 2084 3 110 97 114 2 657 2 648 54 48 1 1948 4 37 9 318 1 2317 1 14 24 22 23 1463 1 17 18 612 1 1099 1 126 94 10 616 27 11 90 48 12 312 9 7 5247 6 54 9 411 50 6 384 946 3 15 38 8 278 218 1 2369 6 4 128 1384 6 25 4 5 22 24 37 145 88 147 52 159 2 112 107 4 12 18 4 3039 9 670 111 822 3 10 46 63 703 3 216 2 2 81 35 10 86 2 543 8 2 61 1405 56 9 24 4 44 1869 60 20 1 16 2 6 42 44 237 20 2 256 5 28 2 2107 18 1 261 2 2 4 7 181 25 34 6 156 30 614 1 4 850 1 26 175 2 469 22 32 14 24 2 84 2 447 20 25 2 9 2 1 69 10 70 127 6 298 78 1 13 1 2 721 278 84 1 49 12 18 14 2 1 13 842 84 4 9 12 1 14 59 13 464 1 276 11 2 4 126 20 9 2748 1 10 1 2 4 1 160 7 14 10 81 27 337 2 172 40 4 6 5 33 388 4 206 103 3 6 84 28 16 217 4 30 1 96 6 138 32 6 24 3 6 3 18 2 67 79 2 50 4 683 1 13 3 2 1 866 4 5 1 44 40 56 18 4 10 1 96 5 113 52 3 1 14 6 3 28 12 122 527 1 120 63 3 2 16 13 2 12 9 1118 9 72 1 8 30 9 2 7 78 8 10 385 3 3 70 2 22 1 25 2 1 6 213 3 50 4 1 197 4 18 1 2 15 14 7 3 1 232 2 1 2 30 115 16 41 1 126 1 1 36 2 4 3 24 14 2 19 560 1 5 4 2 1 1 1 2 327 19 24 6 28 12 33 98 2 86 25 2 8 1 12 1 1 6 1 7 2 45 119 1 187 1 12 49 8 1 56 3 3 4 128 3 24 69 1 2 4 1 13 3 1 72 144 1 11 6 2 2 2 1 13 3 1 2 158 2 8 12 1 1 59 6 7 1 6 1 19 11 81 7 1 24 1 1 31 2 277 86 26 10 3 1 1 1 4 13 186 2 15 5 31 13 15 1 162 6 69 5 24 22 2 3 6 1 5 2 24 50 3 14 3 1 12 4 1 38 1 3 5 12 22 2 13 1 52 1 28 12 2 8 2 1 1 2 1 2 47 2 28 11 2 8 2 4 1 10 45 6 2 4 1 6 1 189 2 24 8 5 6 1 2 3 2 12 1 10 1 4 2 4 3 101 20 12 1 8 1 2 1 2 19 5 4 2 6 3 1 1 10 6 2 2 1 1 41 25 2 1 2 2 1 1 5 2 1 11 4 21 1 1 2 23 2 1 2 48 2 1 1 1 7 2 1 4 23 1 4 1 9 1 4 1 1 18 4 2 1 1 8 4 1 1 1 2 1 1 1 3 3 1 1 63261

Minimal presentations

Permutation degree:$43$
Transitive degree:not computed
Rank: $2$
Inequivalent generating pairs: not computed

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

Constructions

Show commands: Gap / Magma / Oscar / SageMath


Permutation group:Degree $43$ $\langle(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43), (1,2)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 43 | (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43), (1,2) >;
 
Copy content gap:G := Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43), (1,2) );
 
Copy content sage:G = PermutationGroup(['(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43)', '(1,2)'])
 
Copy content sage_gap:G = gap.new('Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43), (1,2) )')
 
Copy content oscar:G = @permutation_group(43, (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43), (1,2))
 
Transitive group: 43T10 more information
Copy content magma:G := TransitiveGroup(43, 10);
 
Copy content gap:G := TransitiveGroup(43, 10);
 
Copy content sage:G = TransitiveGroup(43, 10)
 
Copy content sage_gap:G = libgap.TransitiveGroup(43, 10)
 
Copy content oscar:G = transitive_group(43, 10)
 
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 permutations of degree 43.

Homology

Abelianization: $C_{2} $
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: not computed
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: not computed
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

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

The $63261 \times 63261$ rational character table is not available for this group.