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Properties

Label 107...000.cz
Order \( 2^{22} \cdot 3^{8} \cdot 5^{8} \)
Exponent \( 2^{3} \cdot 3 \cdot 5 \)
Nilpotent no
Solvable no
$\card{G^{\mathrm{ab}}}$ \( 2^{3} \)
$\card{Z(G)}$ 1
$\card{\Aut(G)}$ \( 2^{26} \cdot 3^{8} \cdot 5^{8} \)
$\card{\mathrm{Out}(G)}$ \( 2^{4} \)
Perm deg. $40$
Trans deg. $40$
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 40 | (1,8,14,38,4,6,13,37,5,10,11,39,3,7,12,40,2,9,15,36)(16,23,29,33,20,25,28,35)(17,22,30,31,18,21,26,32)(19,24,27,34), (1,36,32,30)(2,40,34,28,4,38,35,26)(3,37,33,27,5,39,31,29)(6,25,18,12,8,23,20,13)(7,22,17,15,10,24,16,14)(9,21,19,11) >;
 
Copy content gap:G := Group( (1,8,14,38,4,6,13,37,5,10,11,39,3,7,12,40,2,9,15,36)(16,23,29,33,20,25,28,35)(17,22,30,31,18,21,26,32)(19,24,27,34), (1,36,32,30)(2,40,34,28,4,38,35,26)(3,37,33,27,5,39,31,29)(6,25,18,12,8,23,20,13)(7,22,17,15,10,24,16,14)(9,21,19,11) );
 
Copy content sage:G = PermutationGroup(['(1,8,14,38,4,6,13,37,5,10,11,39,3,7,12,40,2,9,15,36)(16,23,29,33,20,25,28,35)(17,22,30,31,18,21,26,32)(19,24,27,34)', '(1,36,32,30)(2,40,34,28,4,38,35,26)(3,37,33,27,5,39,31,29)(6,25,18,12,8,23,20,13)(7,22,17,15,10,24,16,14)(9,21,19,11)'])
 

Group information

Description:$A_5^8.(C_4.D_4).C_2$
Order: \(10749542400000000\)\(\medspace = 2^{22} \cdot 3^{8} \cdot 5^{8} \)
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: \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
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:Group of order \(171992678400000000\)\(\medspace = 2^{26} \cdot 3^{8} \cdot 5^{8} \)
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, $A_5$ x 8
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:$2$
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 nonsolvable. Whether it is rational 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))
 

Statistics about orders of elements in this group have not been computed.

Minimal presentations

Permutation degree:$40$
Transitive degree:$40$
Rank: $2$
Inequivalent generating pairs: not computed

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

Constructions

Show commands: Gap / Magma / SageMath


Permutation group:Degree $40$ $\langle(1,8,14,38,4,6,13,37,5,10,11,39,3,7,12,40,2,9,15,36)(16,23,29,33,20,25,28,35) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 40 | (1,8,14,38,4,6,13,37,5,10,11,39,3,7,12,40,2,9,15,36)(16,23,29,33,20,25,28,35)(17,22,30,31,18,21,26,32)(19,24,27,34), (1,36,32,30)(2,40,34,28,4,38,35,26)(3,37,33,27,5,39,31,29)(6,25,18,12,8,23,20,13)(7,22,17,15,10,24,16,14)(9,21,19,11) >;
 
Copy content gap:G := Group( (1,8,14,38,4,6,13,37,5,10,11,39,3,7,12,40,2,9,15,36)(16,23,29,33,20,25,28,35)(17,22,30,31,18,21,26,32)(19,24,27,34), (1,36,32,30)(2,40,34,28,4,38,35,26)(3,37,33,27,5,39,31,29)(6,25,18,12,8,23,20,13)(7,22,17,15,10,24,16,14)(9,21,19,11) );
 
Copy content sage:G = PermutationGroup(['(1,8,14,38,4,6,13,37,5,10,11,39,3,7,12,40,2,9,15,36)(16,23,29,33,20,25,28,35)(17,22,30,31,18,21,26,32)(19,24,27,34)', '(1,36,32,30)(2,40,34,28,4,38,35,26)(3,37,33,27,5,39,31,29)(6,25,18,12,8,23,20,13)(7,22,17,15,10,24,16,14)(9,21,19,11)'])
 
Transitive group: 40T313786 more information
Direct product: not computed
Semidirect product: not computed
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Possibly split product: $(A_5^8.C_2^3.C_2^2)$ . $C_2$ (2) $(A_5^4.A_5^4.C_4^2.C_2)$ . $C_2$ $(A_5^4.A_5^4.C_2.C_2^3)$ . $C_2^2$ $(A_5^2.A_5^2.A_5^2.A_5^2.C_2^3)$ . $D_4$ (2) all 9

Elements of the group are displayed as permutations of degree 40.

Homology

Abelianization: $C_{2} \times C_{4} $
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: 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()
 

There are 14 normal subgroups (10 characteristic).

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

Special subgroups

Center: a subgroup isomorphic to $C_1$
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: not computed
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: a subgroup isomorphic to $C_1$
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: not computed
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: not computed
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: not computed
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: $P_{ 2 } \simeq$ $C_2^{11}.C_2^6.C_2^4.C_2$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3^8$
5-Sylow subgroup: $P_{ 5 } \simeq$ $C_5^8$

Subgroup diagram and profile

Series

Derived series not computed
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 not computed
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 not computed
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 not computed
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 7 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
 

The character tables for this group have not been computed.