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Properties

Label 456288.a
Order \( 2^{5} \cdot 3 \cdot 7^{2} \cdot 97 \)
Exponent \( 2^{4} \cdot 3 \cdot 7^{2} \cdot 97 \)
Simple yes
$\card{G^{\mathrm{ab}}}$ \( 1 \)
$\card{Z(G)}$ \( 1 \)
$\card{\Aut(G)}$ \( 2^{6} \cdot 3 \cdot 7^{2} \cdot 97 \)
$\card{\mathrm{Out}(G)}$ \( 2 \)
Perm deg. $98$
Trans deg. $98$
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := PSL(2, 97);
 
Copy content gap:G := PSL(2, 97);
 
Copy content sage:G = PSL(2, 97)
 
Copy content comment:Define the group as a permutation group
 

Group information

Description:$\PSL(2,97)$
Order: \(456288\)\(\medspace = 2^{5} \cdot 3 \cdot 7^{2} \cdot 97 \)
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: \(228144\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{2} \cdot 97 \)
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:$\PGL(2,97)$, of order \(912576\)\(\medspace = 2^{6} \cdot 3 \cdot 7^{2} \cdot 97 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$\PSL(2,97)$
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 and simple (hence nonsolvable, perfect, quasisimple, and almost simple).

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 6 7 8 12 16 24 48 49 97
Elements 1 4753 9506 9506 9506 27936 19012 19012 38024 38024 76048 195552 9408 456288
Conjugacy classes   1 1 1 1 1 3 2 2 4 4 8 21 2 51
Divisions 1 1 1 1 1 1 1 1 1 1 1 1 1 13
Autjugacy classes 1 1 1 1 1 3 2 2 4 4 8 21 1 50

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 49 96 97 98 196 288 392 784 2016
Irr. complex chars.   1 2 24 1 23 0 0 0 0 0 51
Irr. rational chars. 1 0 0 1 4 2 1 2 1 1 13

Minimal presentations

Permutation degree:$98$
Transitive degree:$98$
Rank: $2$
Inequivalent generating pairs: $225732$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 49 49 97
Arbitrary 49 49 97

Constructions

Show commands: Gap / Magma / SageMath


Groups of Lie type:$\PSL(2,97)$, $\PSU(2,97)$, $\Omega(3,97)$, $\POmega(3,97)$, $\PSigmaL(2,97)$
Permutation group:Degree $98$ $\langle(3,38,52,6,71,74,34,12,82,41,48,64,5,90,47,39,77,51,84,16,40,59,86,94,69,66,43,9,65,93,29,33,56,21,23,18,44,55,4,75,20,73,91,87,70,76,88,79) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 98 | (3,38,52,6,71,74,34,12,82,41,48,64,5,90,47,39,77,51,84,16,40,59,86,94,69,66,43,9,65,93,29,33,56,21,23,18,44,55,4,75,20,73,91,87,70,76,88,79)(7,15,42,61,85,17,50,24,62,54,11,96,37,53,60,19,89,67,27,30,95,49,63,98,22,13,25,31,14,10,28,81,26,97,46,57,83,78,80,45,68,72,8,36,92,58,35,32), (1,98,2)(3,21,53)(4,59,57)(5,82,25)(6,30,69)(7,65,84)(8,15,42)(9,60,74)(10,33,23)(11,17,28)(12,80,38)(13,51,71)(14,73,66)(16,35,93)(18,95,75)(20,88,62)(22,37,54)(24,55,52)(26,40,91)(27,86,34)(29,49,87)(31,70,94)(32,68,50)(36,39,44)(41,96,43)(45,76,48)(46,63,78)(47,79,97)(56,61,64)(58,85,92)(67,90,77)(72,83,89) >;
 
Copy content gap:G := Group( (3,38,52,6,71,74,34,12,82,41,48,64,5,90,47,39,77,51,84,16,40,59,86,94,69,66,43,9,65,93,29,33,56,21,23,18,44,55,4,75,20,73,91,87,70,76,88,79)(7,15,42,61,85,17,50,24,62,54,11,96,37,53,60,19,89,67,27,30,95,49,63,98,22,13,25,31,14,10,28,81,26,97,46,57,83,78,80,45,68,72,8,36,92,58,35,32), (1,98,2)(3,21,53)(4,59,57)(5,82,25)(6,30,69)(7,65,84)(8,15,42)(9,60,74)(10,33,23)(11,17,28)(12,80,38)(13,51,71)(14,73,66)(16,35,93)(18,95,75)(20,88,62)(22,37,54)(24,55,52)(26,40,91)(27,86,34)(29,49,87)(31,70,94)(32,68,50)(36,39,44)(41,96,43)(45,76,48)(46,63,78)(47,79,97)(56,61,64)(58,85,92)(67,90,77)(72,83,89) );
 
Copy content sage:G = PermutationGroup(['(3,38,52,6,71,74,34,12,82,41,48,64,5,90,47,39,77,51,84,16,40,59,86,94,69,66,43,9,65,93,29,33,56,21,23,18,44,55,4,75,20,73,91,87,70,76,88,79)(7,15,42,61,85,17,50,24,62,54,11,96,37,53,60,19,89,67,27,30,95,49,63,98,22,13,25,31,14,10,28,81,26,97,46,57,83,78,80,45,68,72,8,36,92,58,35,32)', '(1,98,2)(3,21,53)(4,59,57)(5,82,25)(6,30,69)(7,65,84)(8,15,42)(9,60,74)(10,33,23)(11,17,28)(12,80,38)(13,51,71)(14,73,66)(16,35,93)(18,95,75)(20,88,62)(22,37,54)(24,55,52)(26,40,91)(27,86,34)(29,49,87)(31,70,94)(32,68,50)(36,39,44)(41,96,43)(45,76,48)(46,63,78)(47,79,97)(56,61,64)(58,85,92)(67,90,77)(72,83,89)'])
 
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 $\SL(2,97)$.

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 451547 subgroups in 45 conjugacy classes, 2 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_1$ $G/Z \simeq$ $\PSL(2,97)$
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: $G' \simeq$ $\PSL(2,97)$ $G/G' \simeq$ $C_1$
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: $\Phi \simeq$ $C_1$ $G/\Phi \simeq$ $\PSL(2,97)$
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: $\operatorname{Fit} \simeq$ $C_1$ $G/\operatorname{Fit} \simeq$ $\PSL(2,97)$
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: $R \simeq$ $C_1$ $G/R \simeq$ $\PSL(2,97)$
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: $\operatorname{soc} \simeq$ $\PSL(2,97)$ $G/\operatorname{soc} \simeq$ $C_1$
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: $P_{ 2 } \simeq$ $D_{16}$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup: $P_{ 7 } \simeq$ $C_{49}$
97-Sylow subgroup: $P_{ 97 } \simeq$ $C_{97}$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $\PSL(2,97)$
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 $\PSL(2,97)$ $\rhd$ $C_1$
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 $\PSL(2,97)$
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 $51 \times 51$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

1A 2A 3A 4A 6A 7A 8A 12A 16A 24A 48A 49A 97A
Size 1 4753 9506 9506 9506 27936 19012 19012 38024 38024 76048 195552 9408
2 P 1A 1A 3A 2A 3A 7A 4A 6A 8A 12A 24A 49A 97A
3 P 1A 2A 1A 4A 2A 7A 8A 4A 16A 8A 16A 49A 97A
7 P 1A 2A 3A 4A 6A 1A 8A 12A 16A 24A 48A 7A 97A
97 P 1A 2A 3A 4A 6A 7A 8A 12A 16A 24A 48A 49A 1A
456288.a.1a 1 1 1 1 1 1 1 1 1 1 1 1 1
456288.a.49a 98 2 2 2 2 0 2 2 −2 2 −2 0 1
456288.a.96a 288 0 0 0 0 −6 0 0 0 0 0 1 −3
456288.a.96b 2016 0 0 0 0 7 0 0 0 0 0 0 −21
456288.a.97a 97 1 1 1 1 −1 1 1 1 1 1 −1 0
456288.a.98a 98 2 −1 2 −1 0 2 −1 2 −1 −1 0 1
456288.a.98b 98 2 2 2 2 0 −2 2 0 −2 0 0 1
456288.a.98c 98 2 −1 2 −1 0 2 −1 −2 −1 1 0 1
456288.a.98d 196 4 4 −4 4 0 0 −4 0 0 0 0 2
456288.a.98e 196 4 −2 4 −2 0 −4 −2 0 2 0 0 2
456288.a.98f 392 −8 8 0 −8 0 0 0 0 0 0 0 4
456288.a.98g 392 8 −4 −8 −4 0 0 4 0 0 0 0 4
456288.a.98h 784 −16 −8 0 8 0 0 0 0 0 0 0 8