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Properties

Label 74412.a
Order \( 2^{2} \cdot 3^{3} \cdot 13 \cdot 53 \)
Exponent \( 2 \cdot 3^{3} \cdot 13 \cdot 53 \)
Simple yes
$\card{G^{\mathrm{ab}}}$ \( 1 \)
$\card{Z(G)}$ \( 1 \)
$\card{\Aut(G)}$ \( 2^{3} \cdot 3^{3} \cdot 13 \cdot 53 \)
$\card{\mathrm{Out}(G)}$ \( 2 \)
Perm deg. $54$
Trans deg. $54$
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := PSL(2, 53);
 
Copy content gap:G := PSL(2, 53);
 
Copy content sage:G = PSL(2, 53)
 
Copy content comment:Define the group as a permutation group
 

Group information

Description:$\PSL(2,53)$
Order: \(74412\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 13 \cdot 53 \)
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: \(37206\)\(\medspace = 2 \cdot 3^{3} \cdot 13 \cdot 53 \)
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:$\PGL(2,53)$, of order \(148824\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 13 \cdot 53 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$\PSL(2,53)$
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 9 13 26 27 53
Elements 1 1431 2756 8268 17172 17172 24804 2808 74412
Conjugacy classes   1 1 1 3 6 6 9 2 29
Divisions 1 1 1 1 1 1 1 1 8
Autjugacy classes 1 1 1 3 6 6 9 1 28

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 27 52 53 54 156 324 468
Irr. complex chars.   1 2 13 1 12 0 0 0 29
Irr. rational chars. 1 0 1 1 1 1 2 1 8

Minimal presentations

Permutation degree:$54$
Transitive degree:$54$
Rank: $2$
Inequivalent generating pairs: $36475$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 27 27 52
Arbitrary 27 27 52

Constructions

Show commands: Gap / Magma / SageMath


Groups of Lie type:$\PSL(2,53)$, $\PSU(2,53)$, $\Omega(3,53)$, $\POmega(3,53)$, $\PSigmaL(2,53)$
Permutation group:Degree $54$ $\langle(3,28,40,36,34,22,37,53,25,27,10,9,14,41,6,46,26,15,7,24,19,49,18,12,5,13) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 54 | (3,28,40,36,34,22,37,53,25,27,10,9,14,41,6,46,26,15,7,24,19,49,18,12,5,13)(4,20,35,23,21,17,29,54,44,52,45,39,8,38,33,50,42,31,11,51,16,43,48,47,30,32), (1,29,2)(3,51,28)(4,49,13)(5,34,11)(6,33,7)(8,40,37)(9,14,47)(10,19,42)(12,36,16)(15,48,53)(17,18,24)(20,43,52)(21,32,35)(22,54,25)(23,31,39)(26,50,46)(27,30,41)(38,44,45) >;
 
Copy content gap:G := Group( (3,28,40,36,34,22,37,53,25,27,10,9,14,41,6,46,26,15,7,24,19,49,18,12,5,13)(4,20,35,23,21,17,29,54,44,52,45,39,8,38,33,50,42,31,11,51,16,43,48,47,30,32), (1,29,2)(3,51,28)(4,49,13)(5,34,11)(6,33,7)(8,40,37)(9,14,47)(10,19,42)(12,36,16)(15,48,53)(17,18,24)(20,43,52)(21,32,35)(22,54,25)(23,31,39)(26,50,46)(27,30,41)(38,44,45) );
 
Copy content sage:G = PermutationGroup(['(3,28,40,36,34,22,37,53,25,27,10,9,14,41,6,46,26,15,7,24,19,49,18,12,5,13)(4,20,35,23,21,17,29,54,44,52,45,39,8,38,33,50,42,31,11,51,16,43,48,47,30,32)', '(1,29,2)(3,51,28)(4,49,13)(5,34,11)(6,33,7)(8,40,37)(9,14,47)(10,19,42)(12,36,16)(15,48,53)(17,18,24)(20,43,52)(21,32,35)(22,54,25)(23,31,39)(26,50,46)(27,30,41)(38,44,45)'])
 
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 $\SL(2,53)$.

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 43254 subgroups in 20 conjugacy classes, 2 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_1$ $G/Z \simeq$ $\PSL(2,53)$
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: $G' \simeq$ $\PSL(2,53)$ $G/G' \simeq$ $C_1$
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: $\Phi \simeq$ $C_1$ $G/\Phi \simeq$ $\PSL(2,53)$
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: $\operatorname{Fit} \simeq$ $C_1$ $G/\operatorname{Fit} \simeq$ $\PSL(2,53)$
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: $R \simeq$ $C_1$ $G/R \simeq$ $\PSL(2,53)$
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: $\operatorname{soc} \simeq$ $\PSL(2,53)$ $G/\operatorname{soc} \simeq$ $C_1$
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: $P_{ 3 } \simeq$ $C_{27}$
13-Sylow subgroup: $P_{ 13 } \simeq$ $C_{13}$
53-Sylow subgroup: $P_{ 53 } \simeq$ $C_{53}$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $\PSL(2,53)$
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 $\PSL(2,53)$ $\rhd$ $C_1$
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 $\PSL(2,53)$
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 $29 \times 29$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

1A 2A 3A 9A 13A 26A 27A 53A
Size 1 1431 2756 8268 17172 17172 24804 2808
2 P 1A 1A 3A 9A 13A 13A 27A 53A
3 P 1A 2A 1A 3A 13A 26A 9A 53A
13 P 1A 2A 3A 9A 1A 2A 27A 53A
53 P 1A 2A 3A 9A 13A 26A 27A 1A
74412.a.1a 1 1 1 1 1 1 1 1
74412.a.27a 54 −2 0 0 2 −2 0 1
74412.a.52a 52 0 −2 −2 0 0 1 −1
74412.a.52b 156 0 −6 3 0 0 0 −3
74412.a.52c 468 0 9 0 0 0 0 −9
74412.a.53a 53 1 −1 −1 1 1 −1 0
74412.a.54a 324 12 0 0 −1 −1 0 6
74412.a.54b 324 −12 0 0 −1 1 0 6