↑

Properties

Label 492960.a
Order \( 2^{5} \cdot 3 \cdot 5 \cdot 13 \cdot 79 \)
Exponent \( 2^{4} \cdot 3 \cdot 5 \cdot 13 \cdot 79 \)
Nilpotent no
Solvable no
$\card{G^{\mathrm{ab}}}$ \( 1 \)
$\card{Z(G)}$ 2
$\card{\Aut(G)}$ \( 2^{5} \cdot 3 \cdot 5 \cdot 13 \cdot 79 \)
$\card{\mathrm{Out}(G)}$ \( 2 \)
Perm deg. $160$
Trans deg. not computed
Rank $2$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / SageMath

Copy content magma:G := SL(2, 79);
 
Copy content gap:G := SL(2, 79);
 
Copy content sage:G = SL(2, 79)
 
Copy content comment:Define the group as a permutation group
 

Group information

Description:$\SL(2,79)$
Order: \(492960\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 13 \cdot 79 \)
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: \(246480\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 13 \cdot 79 \)
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:$\PGL(2,79)$, of order \(492960\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 13 \cdot 79 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$, $\PSL(2,79)$
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 quasisimple (hence nonsolvable and perfect). Whether it is almost simple has not been computed.

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 13 16 20 26 39 40 78 79 80 158
Elements 1 1 6320 6162 12324 6320 12324 12324 37920 24648 24648 37920 75840 49296 75840 6240 98592 6240 492960
Conjugacy classes   1 1 1 1 2 1 2 2 6 4 4 6 12 8 12 2 16 2 83
Divisions 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 18
Autjugacy classes 1 1 1 1 2 1 2 2 6 4 4 6 12 8 12 1 16 1 81

Minimal presentations

Permutation degree:$160$
Transitive degree:not computed
Rank: $2$
Inequivalent generating pairs: not computed

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 40 not computed not computed
Arbitrary not computed not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Groups of Lie type:$\SL(2,79)$, $\SU(2,79)$
Permutation group:Degree $160$ $\langle(1,2)(3,7,11,23,37,18,36,73,128,72,127,154,104,153,116,61,115,85,131,76,69,108,57,107,82,136,158,140,160,123,67,100,51,34,16,33,42,21,10,5,9,17,35,26,12,25,49,96,46,95,145,133,155,138,84,117,62,102,53,91,135,80,111,59,110,152,119,156,143,89,129,74,44,22,43,32,15,8) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 160 | (1,2)(3,7,11,23,37,18,36,73,128,72,127,154,104,153,116,61,115,85,131,76,69,108,57,107,82,136,158,140,160,123,67,100,51,34,16,33,42,21,10,5,9,17,35,26,12,25,49,96,46,95,145,133,155,138,84,117,62,102,53,91,135,80,111,59,110,152,119,156,143,89,129,74,44,22,43,32,15,8)(4,6)(13,27,54,103,122,66,31,50,99,141,87,41,20,40,83,114,65,121,68,124,106,56,75,130,150,147,159,149,137,92,63,118,94,45,93,55,105,81,39,19,38,77,132,98,48,24,47,97,120,64,30,14,29,60,112,88,142,90,144,134,79,52,101,146,151,148,157,113,70,86,139,126,71,125,78,109,58,28), (1,3,4)(2,5,6)(7,12,13)(8,14,16)(9,18,19)(10,20,22)(11,21,24)(15,31,17)(23,45,46)(25,50,51)(26,52,53)(27,55,56)(28,57,59)(29,61,62)(30,63,65)(32,67,68)(33,69,70)(34,49,48)(35,71,72)(36,47,74)(37,75,76)(38,78,79)(39,80,82)(40,84,85)(41,86,88)(42,89,90)(43,91,92)(44,73,66)(54,100,104)(58,105,101)(60,113,114)(64,119,96)(77,129,133)(81,109,130)(83,137,112)(87,140,128)(93,108,143)(95,131,142)(97,146,147)(98,148,149)(99,150,151)(102,121,127)(103,115,152)(106,154,116)(107,141,155)(110,156,118)(111,120,153)(117,158,132)(122,159,157)(123,125,135)(134,145,138)(136,160,139) >;
 
Copy content gap:G := Group( (1,2)(3,7,11,23,37,18,36,73,128,72,127,154,104,153,116,61,115,85,131,76,69,108,57,107,82,136,158,140,160,123,67,100,51,34,16,33,42,21,10,5,9,17,35,26,12,25,49,96,46,95,145,133,155,138,84,117,62,102,53,91,135,80,111,59,110,152,119,156,143,89,129,74,44,22,43,32,15,8)(4,6)(13,27,54,103,122,66,31,50,99,141,87,41,20,40,83,114,65,121,68,124,106,56,75,130,150,147,159,149,137,92,63,118,94,45,93,55,105,81,39,19,38,77,132,98,48,24,47,97,120,64,30,14,29,60,112,88,142,90,144,134,79,52,101,146,151,148,157,113,70,86,139,126,71,125,78,109,58,28), (1,3,4)(2,5,6)(7,12,13)(8,14,16)(9,18,19)(10,20,22)(11,21,24)(15,31,17)(23,45,46)(25,50,51)(26,52,53)(27,55,56)(28,57,59)(29,61,62)(30,63,65)(32,67,68)(33,69,70)(34,49,48)(35,71,72)(36,47,74)(37,75,76)(38,78,79)(39,80,82)(40,84,85)(41,86,88)(42,89,90)(43,91,92)(44,73,66)(54,100,104)(58,105,101)(60,113,114)(64,119,96)(77,129,133)(81,109,130)(83,137,112)(87,140,128)(93,108,143)(95,131,142)(97,146,147)(98,148,149)(99,150,151)(102,121,127)(103,115,152)(106,154,116)(107,141,155)(110,156,118)(111,120,153)(117,158,132)(122,159,157)(123,125,135)(134,145,138)(136,160,139) );
 
Copy content sage:G = PermutationGroup(['(1,2)(3,7,11,23,37,18,36,73,128,72,127,154,104,153,116,61,115,85,131,76,69,108,57,107,82,136,158,140,160,123,67,100,51,34,16,33,42,21,10,5,9,17,35,26,12,25,49,96,46,95,145,133,155,138,84,117,62,102,53,91,135,80,111,59,110,152,119,156,143,89,129,74,44,22,43,32,15,8)(4,6)(13,27,54,103,122,66,31,50,99,141,87,41,20,40,83,114,65,121,68,124,106,56,75,130,150,147,159,149,137,92,63,118,94,45,93,55,105,81,39,19,38,77,132,98,48,24,47,97,120,64,30,14,29,60,112,88,142,90,144,134,79,52,101,146,151,148,157,113,70,86,139,126,71,125,78,109,58,28)', '(1,3,4)(2,5,6)(7,12,13)(8,14,16)(9,18,19)(10,20,22)(11,21,24)(15,31,17)(23,45,46)(25,50,51)(26,52,53)(27,55,56)(28,57,59)(29,61,62)(30,63,65)(32,67,68)(33,69,70)(34,49,48)(35,71,72)(36,47,74)(37,75,76)(38,78,79)(39,80,82)(40,84,85)(41,86,88)(42,89,90)(43,91,92)(44,73,66)(54,100,104)(58,105,101)(60,113,114)(64,119,96)(77,129,133)(81,109,130)(83,137,112)(87,140,128)(93,108,143)(95,131,142)(97,146,147)(98,148,149)(99,150,151)(102,121,127)(103,115,152)(106,154,116)(107,141,155)(110,156,118)(111,120,153)(117,158,132)(122,159,157)(123,125,135)(134,145,138)(136,160,139)'])
 
Matrix group:$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{79})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 2, GF(79) | [[1, 1, 0, 1], [1, 0, 1, 1]] >;
 
Copy content gap:G := Group([[[ Z(79)^0, Z(79)^0 ], [ 0*Z(79), Z(79)^0 ]], [[ Z(79)^0, 0*Z(79) ], [ Z(79)^0, Z(79)^0 ]]]);
 
Copy content sage:MS = MatrixSpace(GF(79), 2, 2) G = MatrixGroup([MS([[1, 1], [0, 1]]), MS([[1, 0], [1, 1]])])
 
Direct product: not computed
Semidirect product: not computed
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\SL(2,79)$.

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: not computed
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()
 

Subgroup data has not been computed.

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 $83 \times 83$ character table is not available for this group.

Rational character table

The $18 \times 18$ rational character table is not available for this group.