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Properties

Label 300696.a
Order \( 2^{3} \cdot 3 \cdot 11 \cdot 17 \cdot 67 \)
Exponent \( 2^{2} \cdot 3 \cdot 11 \cdot 17 \cdot 67 \)
Nilpotent no
Solvable no
$\card{G^{\mathrm{ab}}}$ \( 1 \)
$\card{Z(G)}$ 2
$\card{\Aut(G)}$ \( 2^{3} \cdot 3 \cdot 11 \cdot 17 \cdot 67 \)
$\card{\mathrm{Out}(G)}$ \( 2 \)
Perm deg. $136$
Trans deg. not computed
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SL(2, 67);
 
Copy content gap:G := SL(2, 67);
 
Copy content sage:G = SL(2, 67)
 
Copy content comment:Define the group as a permutation group
 

Group information

Description:$\SL(2,67)$
Order: \(300696\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \cdot 17 \cdot 67 \)
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: \(150348\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \cdot 17 \cdot 67 \)
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:$\PGL(2,67)$, of order \(300696\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \cdot 17 \cdot 67 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$, $\PSL(2,67)$
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 6 11 17 22 33 34 66 67 68 134
Elements 1 1 4556 4422 4556 22780 35376 22780 45560 35376 45560 4488 70752 4488 300696
Conjugacy classes   1 1 1 1 1 5 8 5 10 8 10 2 16 2 71
Divisions 1 1 1 1 1 1 1 1 1 1 1 1 1 1 14
Autjugacy classes 1 1 1 1 1 5 8 5 10 8 10 1 16 1 69

Minimal presentations

Permutation degree:$136$
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 34 not computed not computed
Arbitrary not computed not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Groups of Lie type:$\SL(2,67)$, $\SU(2,67)$
Permutation group:Degree $136$ $\langle(1,3,4)(2,5,6)(7,12,13)(8,14,16)(9,18,19)(10,20,22)(11,23,24)(15,29,30) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 136 | (1,3,4)(2,5,6)(7,12,13)(8,14,16)(9,18,19)(10,20,22)(11,23,24)(15,29,30)(17,33,34)(21,39,40)(25,48,49)(26,50,52)(27,53,55)(28,57,45)(31,62,63)(32,64,66)(35,72,73)(36,74,75)(37,76,78)(38,80,69)(41,84,85)(42,86,88)(43,90,91)(44,65,93)(46,95,96)(47,59,97)(51,102,103)(54,92,105)(56,109,107)(58,110,111)(60,106,112)(61,113,114)(67,120,121)(68,87,99)(70,123,94)(71,82,117)(77,122,125)(79,129,127)(81,104,130)(83,126,131)(89,118,133)(98,108,135)(100,128,136)(116,134,119), (1,2)(3,7,11,20,38,79,128,83,40,82,127,78,126,93,72,100,49,86,118,66,117,91,57,92,44,23,43,89,54,27,13,21,10,5,9,17,14,28,56,108,60,30,59,107,55,106,99,48,98,73,64,116,88,97,121,80,122,68,33,67,119,77,37,19,15,8)(4,6)(12,25,47,46,24,45,94,52,63,115,135,130,112,125,103,113,129,90,123,133,101,50,62,104,53,29,58,65,32,16,31,61,36,18,35,71,70,34,69,96,75,85,132,136,111,131,105,114,102,109,120,95,134,124,74,84,110,76,39,81,87,42,22,41,51,26) >;
 
Copy content gap:G := Group( (1,3,4)(2,5,6)(7,12,13)(8,14,16)(9,18,19)(10,20,22)(11,23,24)(15,29,30)(17,33,34)(21,39,40)(25,48,49)(26,50,52)(27,53,55)(28,57,45)(31,62,63)(32,64,66)(35,72,73)(36,74,75)(37,76,78)(38,80,69)(41,84,85)(42,86,88)(43,90,91)(44,65,93)(46,95,96)(47,59,97)(51,102,103)(54,92,105)(56,109,107)(58,110,111)(60,106,112)(61,113,114)(67,120,121)(68,87,99)(70,123,94)(71,82,117)(77,122,125)(79,129,127)(81,104,130)(83,126,131)(89,118,133)(98,108,135)(100,128,136)(116,134,119), (1,2)(3,7,11,20,38,79,128,83,40,82,127,78,126,93,72,100,49,86,118,66,117,91,57,92,44,23,43,89,54,27,13,21,10,5,9,17,14,28,56,108,60,30,59,107,55,106,99,48,98,73,64,116,88,97,121,80,122,68,33,67,119,77,37,19,15,8)(4,6)(12,25,47,46,24,45,94,52,63,115,135,130,112,125,103,113,129,90,123,133,101,50,62,104,53,29,58,65,32,16,31,61,36,18,35,71,70,34,69,96,75,85,132,136,111,131,105,114,102,109,120,95,134,124,74,84,110,76,39,81,87,42,22,41,51,26) );
 
Copy content sage:G = PermutationGroup(['(1,3,4)(2,5,6)(7,12,13)(8,14,16)(9,18,19)(10,20,22)(11,23,24)(15,29,30)(17,33,34)(21,39,40)(25,48,49)(26,50,52)(27,53,55)(28,57,45)(31,62,63)(32,64,66)(35,72,73)(36,74,75)(37,76,78)(38,80,69)(41,84,85)(42,86,88)(43,90,91)(44,65,93)(46,95,96)(47,59,97)(51,102,103)(54,92,105)(56,109,107)(58,110,111)(60,106,112)(61,113,114)(67,120,121)(68,87,99)(70,123,94)(71,82,117)(77,122,125)(79,129,127)(81,104,130)(83,126,131)(89,118,133)(98,108,135)(100,128,136)(116,134,119)', '(1,2)(3,7,11,20,38,79,128,83,40,82,127,78,126,93,72,100,49,86,118,66,117,91,57,92,44,23,43,89,54,27,13,21,10,5,9,17,14,28,56,108,60,30,59,107,55,106,99,48,98,73,64,116,88,97,121,80,122,68,33,67,119,77,37,19,15,8)(4,6)(12,25,47,46,24,45,94,52,63,115,135,130,112,125,103,113,129,90,123,133,101,50,62,104,53,29,58,65,32,16,31,61,36,18,35,71,70,34,69,96,75,85,132,136,111,131,105,114,102,109,120,95,134,124,74,84,110,76,39,81,87,42,22,41,51,26)'])
 
Matrix group:$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{67})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 2, GF(67) | [[1, 1, 0, 1], [1, 0, 1, 1]] >;
 
Copy content gap:G := Group([[[ Z(67)^0, Z(67)^0 ], [ 0*Z(67), Z(67)^0 ]], [[ Z(67)^0, 0*Z(67) ], [ Z(67)^0, Z(67)^0 ]]]);
 
Copy content sage:MS = MatrixSpace(GF(67), 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 $\SL(2,67)$.

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 $71 \times 71$ character table is not available for this group.

Rational character table

The $14 \times 14$ rational character table is not available for this group.