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Properties

Label 240.4
Order \( 2^{4} \cdot 3 \cdot 5 \)
Exponent \( 2^{4} \cdot 3 \cdot 5 \)
Abelian yes
$\card{\Aut(G)}$ \( 2^{6} \)
Perm deg. $24$
Trans deg. $240$
Rank $1$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := CyclicGroup(240);
 
Copy content gap:G := CyclicGroup(240);
 
Copy content sage:G = CyclicPermutationGroup(240)
 
Copy content sage_gap:G = libgap.SmallGroup(240, 4)
 
Copy content comment:Define the group as a permutation group
 

Group information

Description:$C_{240}$
Order: \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \)
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: \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \)
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:$C_2^2\times C_4^2$, of order \(64\)\(\medspace = 2^{6} \)
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 4, $C_3$, $C_5$
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()
 
Nilpotency class:$1$
Copy content comment:Nilpotency class of the group
 
Copy content magma:NilpotencyClass(G);
 
Copy content gap:NilpotencyClassOfGroup(G);
 
Copy content sage_gap:G.NilpotencyClassOfGroup()
 
Derived length:$1$
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 cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

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 5 6 8 10 12 15 16 20 24 30 40 48 60 80 120 240
Elements 1 1 2 2 4 2 4 4 4 8 8 8 8 8 16 16 16 32 32 64 240
Conjugacy classes   1 1 2 2 4 2 4 4 4 8 8 8 8 8 16 16 16 32 32 64 240
Divisions 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 20
Autjugacy classes 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 20

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:G.CharacterDegrees()
 

Dimension 1 2 4 8 16 32 64
Irr. complex chars.   240 0 0 0 0 0 0 240
Irr. rational chars. 2 3 4 5 3 2 1 20

Minimal presentations

Permutation degree:$24$
Transitive degree:$240$
Rank: $1$
Inequivalent generators: $1$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 1 2 64
Arbitrary 1 2 14

Constructions

Show commands: Gap / Magma / SageMath


Presentation: $\langle a \mid a^{240}=1 \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([6, -2, -2, -2, -2, -3, -5, 12, 31, 50, 69, 118]); a := Explode([G.1]); AssignNames(~G, ["a", "a2", "a4", "a8", "a16", "a48"]);
 
Copy content gap:G := PcGroupCode(14550280868682919943,240); a := G.1;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(14550280868682919943,240)'); a = G.1;
 
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(14550280868682919943,240)'); a = G.1;
 
Permutation group:Degree $24$ $\langle(1,16,8,12,4,14,6,10,2,15,7,11,3,13,5,9), (17,19,18), (20,24,23,22,21), (1,8,4,6,2,7,3,5) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 24 | (1,16,8,12,4,14,6,10,2,15,7,11,3,13,5,9), (17,19,18), (20,24,23,22,21), (1,8,4,6,2,7,3,5)(9,16,12,14,10,15,11,13), (1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16) >;
 
Copy content gap:G := Group( (1,16,8,12,4,14,6,10,2,15,7,11,3,13,5,9), (17,19,18), (20,24,23,22,21), (1,8,4,6,2,7,3,5)(9,16,12,14,10,15,11,13), (1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16) );
 
Copy content sage:G = PermutationGroup(['(1,16,8,12,4,14,6,10,2,15,7,11,3,13,5,9)', '(17,19,18)', '(20,24,23,22,21)', '(1,8,4,6,2,7,3,5)(9,16,12,14,10,15,11,13)', '(1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15)', '(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)'])
 
Matrix group:$\left\langle \left(\begin{array}{rr} 2 & 12 \\ 4 & 2 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{31})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 2, GF(31) | [[2, 12, 4, 2]] >;
 
Copy content gap:G := Group([[[ Z(31)^24, Z(31)^19 ], [ Z(31)^18, Z(31)^24 ]]]);
 
Copy content sage:MS = MatrixSpace(GF(31), 2, 2) G = MatrixGroup([MS([[2, 12], [4, 2]])])
 
Direct product: $C_{16}$ $\, \times\, $ $C_3$ $\, \times\, $ $C_5$
Semidirect product: not isomorphic to a non-trivial semidirect product
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $C_{60}$ . $C_4$ $C_{40}$ . $C_6$ $C_{30}$ . $C_8$ $C_8$ . $C_{30}$ all 12
Aut. group: $\Aut(C_{241})$ $\Aut(C_{482})$

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

Homology

Primary decomposition: $C_{16} \times C_{3} \times C_{5}$
Copy content comment:The primary decomposition of the group
 
Copy content magma:PrimaryInvariants(G);
 
Copy content gap:AbelianInvariants(G);
 
Copy content sage_gap:G.AbelianInvariants()
 
Schur multiplier: $C_1$
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: $0$
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 20 subgroups, all normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_{240}$ $G/Z \simeq$ $C_1$
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: $G' \simeq$ $C_1$ $G/G' \simeq$ $C_{240}$
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: $\Phi \simeq$ $C_8$ $G/\Phi \simeq$ $C_{30}$
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: $\operatorname{Fit} \simeq$ $C_{240}$ $G/\operatorname{Fit} \simeq$ $C_1$
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: $R \simeq$ $C_{240}$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_{30}$ $G/\operatorname{soc} \simeq$ $C_8$
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: $P_{ 2 } \simeq$ $C_{16}$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup: $P_{ 5 } \simeq$ $C_5$

Subgroup diagram and profile

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
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Subgroup information

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Series

Derived series $C_{240}$ $\rhd$ $C_1$
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 $C_{240}$ $\rhd$ $C_{120}$ $\rhd$ $C_{60}$ $\rhd$ $C_{30}$ $\rhd$ $C_{15}$ $\rhd$ $C_5$ $\rhd$ $C_1$
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 $C_{240}$ $\rhd$ $C_1$
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 $C_1$ $\lhd$ $C_{240}$
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 75 larger groups in the database.

This group is a maximal quotient of 52 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

See the $240 \times 240$ 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 $20 \times 20$ rational character table.