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Properties

Label 975.2
Order \( 3 \cdot 5^{2} \cdot 13 \)
Exponent \( 3 \cdot 5^{2} \cdot 13 \)
Abelian yes
$\card{\Aut(G)}$ \( 2^{5} \cdot 3 \cdot 5 \)
Perm deg. $41$
Trans deg. $975$
Rank $1$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := CyclicGroup(975);
 
Copy content gap:G := CyclicGroup(975);
 
Copy content sage:G = CyclicPermutationGroup(975)
 
Copy content sage_gap:G = libgap.SmallGroup(975, 2)
 
Copy content comment:Define the group as a permutation group
 

Group information

Description:$C_{975}$
Order: \(975\)\(\medspace = 3 \cdot 5^{2} \cdot 13 \)
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: \(975\)\(\medspace = 3 \cdot 5^{2} \cdot 13 \)
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:$C_2\times C_4\times C_{60}$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_3$, $C_5$ x 2, $C_{13}$
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 ($p = 3,5,13$), 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 3 5 13 15 25 39 65 75 195 325 975
Elements 1 2 4 12 8 20 24 48 40 96 240 480 975
Conjugacy classes   1 2 4 12 8 20 24 48 40 96 240 480 975
Divisions 1 1 1 1 1 1 1 1 1 1 1 1 12
Autjugacy classes 1 1 1 1 1 1 1 1 1 1 1 1 12

Minimal presentations

Permutation degree:$41$
Transitive degree:$975$
Rank: $1$
Inequivalent generators: $1$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 1 not computed not computed
Arbitrary 1 not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Presentation: $\langle a \mid a^{975}=1 \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([4, -3, -5, -5, -13, 12, 45, 70]); a := Explode([G.1]); AssignNames(~G, ["a", "a3", "a15", "a75"]);
 
Copy content gap:G := PcGroupCode(10227884507039,975); 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(10227884507039,975)'); 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(10227884507039,975)'); a = G.1;
 
Permutation group:Degree $41$ $\langle(1,3,2), (4,28,23,18,13,8,27,22,17,12,7,26,21,16,11,6,25,20,15,10,5,24,19,14,9) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 41 | (1,3,2), (4,28,23,18,13,8,27,22,17,12,7,26,21,16,11,6,25,20,15,10,5,24,19,14,9), (29,41,40,39,38,37,36,35,34,33,32,31,30), (4,8,7,6,5)(9,13,12,11,10)(14,18,17,16,15)(19,23,22,21,20)(24,28,27,26,25) >;
 
Copy content gap:G := Group( (1,3,2), (4,28,23,18,13,8,27,22,17,12,7,26,21,16,11,6,25,20,15,10,5,24,19,14,9), (29,41,40,39,38,37,36,35,34,33,32,31,30), (4,8,7,6,5)(9,13,12,11,10)(14,18,17,16,15)(19,23,22,21,20)(24,28,27,26,25) );
 
Copy content sage:G = PermutationGroup(['(1,3,2)', '(4,28,23,18,13,8,27,22,17,12,7,26,21,16,11,6,25,20,15,10,5,24,19,14,9)', '(29,41,40,39,38,37,36,35,34,33,32,31,30)', '(4,8,7,6,5)(9,13,12,11,10)(14,18,17,16,15)(19,23,22,21,20)(24,28,27,26,25)'])
 
Matrix group:$\left\langle \left(\begin{array}{rr} 85 & 469 \\ 67 & 85 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{599})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 2, GF(599) | [[85, 469, 67, 85]] >;
 
Copy content gap:G := Group([[[ Z(599)^224, Z(599)^319 ], [ Z(599)^318, Z(599)^224 ]]]);
 
Copy content sage:MS = MatrixSpace(GF(599), 2, 2) G = MatrixGroup([MS([[85, 469], [67, 85]])])
 
Direct product: $C_3$ $\, \times\, $ $C_{25}$ $\, \times\, $ $C_{13}$
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: $C_{195}$ . $C_5$ $C_{65}$ . $C_{15}$ $C_{15}$ . $C_{65}$ $C_5$ . $C_{195}$ more information

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

Homology

Primary decomposition: $C_{3} \times C_{25} \times C_{13}$
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 12 subgroups, all normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_{975}$ $G/Z \simeq$ $C_1$
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: $G' \simeq$ $C_1$ $G/G' \simeq$ $C_{975}$
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: $\Phi \simeq$ $C_5$ $G/\Phi \simeq$ $C_{195}$
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: $\operatorname{Fit} \simeq$ $C_{975}$ $G/\operatorname{Fit} \simeq$ $C_1$
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: $R \simeq$ $C_{975}$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_{195}$ $G/\operatorname{soc} \simeq$ $C_5$
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()
 
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup: $P_{ 5 } \simeq$ $C_{25}$
13-Sylow subgroup: $P_{ 13 } \simeq$ $C_{13}$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $C_{975}$ $\rhd$ $C_1$
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 $C_{975}$ $\rhd$ $C_{325}$ $\rhd$ $C_{65}$ $\rhd$ $C_{13}$ $\rhd$ $C_1$
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 $C_{975}$ $\rhd$ $C_1$
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 $C_1$ $\lhd$ $C_{975}$
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 8 larger groups in the database.

This group is a maximal quotient of 1 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 $975 \times 975$ character table is not available for this group.

Rational character table

The $12 \times 12$ rational character table is not available for this group.