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Properties

Label 976500.a
Order \( 2^{2} \cdot 3^{2} \cdot 5^{3} \cdot 7 \cdot 31 \)
Exponent \( 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \)
Simple yes
$\card{G^{\mathrm{ab}}}$ \( 1 \)
$\card{Z(G)}$ \( 1 \)
$\card{\Aut(G)}$ \( 2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7 \cdot 31 \)
$\card{\mathrm{Out}(G)}$ \( 2 \cdot 3 \)
Perm deg. $126$
Trans deg. $126$
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := PSL(2, 125);
 
Copy content gap:G := PSL(2, 125);
 
Copy content sage:G = PSL(2, 125)
 
Copy content comment:Define the group as a permutation group
 

Group information

Description:$\PSL(2,125)$
Order: \(976500\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5^{3} \cdot 7 \cdot 31 \)
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: \(19530\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \)
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:$\PSL(2,125).C_6$, of order \(5859000\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7 \cdot 31 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$\PSL(2,125)$
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:$0$
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, simple (hence nonsolvable, perfect, quasisimple, and almost simple), 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 5 7 9 21 31 62 63
Elements 1 7875 15500 15624 46500 46500 93000 236250 236250 279000 976500
Conjugacy classes   1 1 1 2 3 3 6 15 15 18 65
Divisions 1 1 1 1 1 1 1 1 1 1 10
Autjugacy classes 1 1 1 1 1 1 2 5 5 6 24

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 63 124 125 126 372 744 1890 2232
Irr. complex chars.   1 2 31 1 30 0 0 0 0 65
Irr. rational chars. 1 0 1 1 1 2 1 2 1 10

Minimal presentations

Permutation degree:$126$
Transitive degree:$126$
Rank: $2$
Inequivalent generating pairs: $161420$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 63 63 124
Arbitrary 63 63 124

Constructions

Show commands: Gap / Magma / SageMath


Groups of Lie type:$\PSL(2,125)$, $\PSU(2,125)$, $\Omega(3,125)$, $\POmega(3,125)$
Permutation group:Degree $126$ $\langle(3,58,40,62,43,125,92,20,85,118,70,91,87,19,7,10,107,60,104,24,75,29,14,6,98,84,33,78,57,106,79,21,34,101,17,81,83,80,64,114,82,77,12,53,13,32,8,68,93,120,124,102,66,90,41,103,94,99,16,111,55,56) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 126 | (3,58,40,62,43,125,92,20,85,118,70,91,87,19,7,10,107,60,104,24,75,29,14,6,98,84,33,78,57,106,79,21,34,101,17,81,83,80,64,114,82,77,12,53,13,32,8,68,93,120,124,102,66,90,41,103,94,99,16,111,55,56)(4,86,67,89,71,126,73,74,18,113,30,35,26,88,39,63,27,5,9,36,61,121,97,116,76,117,52,47,15,65,49,46,48,112,28,95,108,50,23,72,51,96,45,31,123,115,100,54,105,25,69,22,119,122,110,42,38,59,11,44,109,37), (1,51,2)(3,41,18)(4,60,32)(5,123,15)(6,94,115)(7,53,126)(8,34,100)(9,85,83)(10,84,97)(11,57,64)(12,91,113)(13,63,62)(14,73,17)(16,87,110)(19,81,39)(20,28,29)(21,24,26)(22,33,68)(23,61,46)(25,109,45)(27,69,116)(30,56,90)(31,42,122)(35,36,44)(37,74,121)(38,40,43)(47,117,50)(48,80,103)(49,67,59)(52,77,99)(54,93,106)(55,107,105)(58,75,96)(65,92,78)(66,72,101)(70,114,88)(71,104,82)(76,120,102)(79,95,111)(86,119,108)(89,118,124)(98,125,112) >;
 
Copy content gap:G := Group( (3,58,40,62,43,125,92,20,85,118,70,91,87,19,7,10,107,60,104,24,75,29,14,6,98,84,33,78,57,106,79,21,34,101,17,81,83,80,64,114,82,77,12,53,13,32,8,68,93,120,124,102,66,90,41,103,94,99,16,111,55,56)(4,86,67,89,71,126,73,74,18,113,30,35,26,88,39,63,27,5,9,36,61,121,97,116,76,117,52,47,15,65,49,46,48,112,28,95,108,50,23,72,51,96,45,31,123,115,100,54,105,25,69,22,119,122,110,42,38,59,11,44,109,37), (1,51,2)(3,41,18)(4,60,32)(5,123,15)(6,94,115)(7,53,126)(8,34,100)(9,85,83)(10,84,97)(11,57,64)(12,91,113)(13,63,62)(14,73,17)(16,87,110)(19,81,39)(20,28,29)(21,24,26)(22,33,68)(23,61,46)(25,109,45)(27,69,116)(30,56,90)(31,42,122)(35,36,44)(37,74,121)(38,40,43)(47,117,50)(48,80,103)(49,67,59)(52,77,99)(54,93,106)(55,107,105)(58,75,96)(65,92,78)(66,72,101)(70,114,88)(71,104,82)(76,120,102)(79,95,111)(86,119,108)(89,118,124)(98,125,112) );
 
Copy content sage:G = PermutationGroup(['(3,58,40,62,43,125,92,20,85,118,70,91,87,19,7,10,107,60,104,24,75,29,14,6,98,84,33,78,57,106,79,21,34,101,17,81,83,80,64,114,82,77,12,53,13,32,8,68,93,120,124,102,66,90,41,103,94,99,16,111,55,56)(4,86,67,89,71,126,73,74,18,113,30,35,26,88,39,63,27,5,9,36,61,121,97,116,76,117,52,47,15,65,49,46,48,112,28,95,108,50,23,72,51,96,45,31,123,115,100,54,105,25,69,22,119,122,110,42,38,59,11,44,109,37)', '(1,51,2)(3,41,18)(4,60,32)(5,123,15)(6,94,115)(7,53,126)(8,34,100)(9,85,83)(10,84,97)(11,57,64)(12,91,113)(13,63,62)(14,73,17)(16,87,110)(19,81,39)(20,28,29)(21,24,26)(22,33,68)(23,61,46)(25,109,45)(27,69,116)(30,56,90)(31,42,122)(35,36,44)(37,74,121)(38,40,43)(47,117,50)(48,80,103)(49,67,59)(52,77,99)(54,93,106)(55,107,105)(58,75,96)(65,92,78)(66,72,101)(70,114,88)(71,104,82)(76,120,102)(79,95,111)(86,119,108)(89,118,124)(98,125,112)'])
 
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: not isomorphic to a non-trivial semidirect product
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\SL(2,125)$.

Homology

Abelianization: $C_1 $
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: $C_{2}$
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()
 

There are 708273 subgroups in 29 conjugacy classes, 2 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_1$ $G/Z \simeq$ $\PSL(2,125)$
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: $G' \simeq$ $\PSL(2,125)$ $G/G' \simeq$ $C_1$
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: $\Phi \simeq$ $C_1$ $G/\Phi \simeq$ $\PSL(2,125)$
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: $\operatorname{Fit} \simeq$ $C_1$ $G/\operatorname{Fit} \simeq$ $\PSL(2,125)$
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: $R \simeq$ $C_1$ $G/R \simeq$ $\PSL(2,125)$
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: $\operatorname{soc} \simeq$ $\PSL(2,125)$ $G/\operatorname{soc} \simeq$ $C_1$
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: $P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_9$
5-Sylow subgroup: $P_{ 5 } \simeq$ $C_5^3$
7-Sylow subgroup: $P_{ 7 } \simeq$ $C_7$
31-Sylow subgroup: $P_{ 31 } \simeq$ $C_{31}$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $\PSL(2,125)$
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 $\PSL(2,125)$ $\rhd$ $C_1$
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 $\PSL(2,125)$
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 $C_1$
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 1 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
 

Complex character table

See the $65 \times 65$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

1A 2A 3A 5A 7A 9A 21A 31A 62A 63A
Size 1 7875 15500 15624 46500 46500 93000 236250 236250 279000
2 P 1A 1A 3A 5A 7A 9A 21A 31A 31A 63A
3 P 1A 2A 1A 5A 7A 3A 7A 31A 62A 21A
5 P 1A 2A 3A 1A 7A 9A 21A 31A 62A 63A
7 P 1A 2A 3A 5A 1A 9A 3A 31A 62A 9A
31 P 1A 2A 3A 5A 7A 9A 21A 1A 2A 63A
976500.a.1a 1 1 1 1 1 1 1 1 1 1
976500.a.63a 126 −2 0 1 0 0 0 2 −2 0
976500.a.124a 124 0 −2 −1 −2 1 −2 0 0 1
976500.a.124b 372 0 −6 −3 1 −6 1 0 0 1
976500.a.124c 372 0 3 −3 −6 0 3 0 0 0
976500.a.124d 744 0 −12 −6 2 6 2 0 0 −1
976500.a.124e 2232 0 18 −18 6 0 −3 0 0 0
976500.a.125a 125 1 −1 0 −1 −1 −1 1 1 −1
976500.a.126a 1890 30 0 15 0 0 0 −1 −1 0
976500.a.126b 1890 −30 0 15 0 0 0 −1 1 0