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Properties

Label 120960.a
Order \( 2^{7} \cdot 3^{3} \cdot 5 \cdot 7 \)
Exponent \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Nilpotent no
Solvable no
$\card{G^{\mathrm{ab}}}$ \( 2 \)
$\card{Z(G)}$ 1
$\card{\Aut(G)}$ \( 2^{8} \cdot 3^{3} \cdot 5 \cdot 7 \)
$\card{\mathrm{Out}(G)}$ \( 2 \)
Perm deg. $21$
Trans deg. not computed
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := PGammaL(3,4);
 
Copy content gap:G := PGammaL(3,4);
 
Copy content comment:Define the group as a permutation group
 
Copy content sage:G = PermutationGroup(['(1,2,12)(3,5,17,7,21,8)(4,13,18)(6,10)(9,16,15,14,11,20)', '(1,6,8,5,19,20,10,3,12,18,14,21,16,13,4,7,15,17,2,9,11)'])
 

Group information

Description:$\PSL(3,4).S_3$
Order: \(120960\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \cdot 7 \)
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: \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
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:$\PSL(3,4):D_6$, of order \(241920\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5 \cdot 7 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$, $C_3$, $\PSL(3,4)$
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:$2$
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 nonsolvable. 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 7 8 14 15 21
Elements 1 675 4832 11340 8064 30240 5760 15120 17280 16128 11520 120960
Conjugacy classes   1 2 3 2 1 2 2 1 2 2 2 20
Divisions 1 2 3 2 1 2 1 1 1 1 1 16
Autjugacy classes 1 2 3 2 1 2 1 1 1 1 1 16

Minimal presentations

Permutation degree:$21$
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 20 not computed not computed
Arbitrary not computed not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Groups of Lie type:$\PGammaL(3,4)$
Copy content magma:G := PGammaL(3,4);
 
Copy content gap:G := PGammaL(3,4);
 
Copy content sage:F = GF(4); al = F.0; MS = MatrixSpace(F, 3, 3) G = MatrixGroup([MS([[1, 0, 1], [1, 0, 0], [0, 1, 0]]), MS([[al^1, 0, 0], [0, 1, 0], [0, 0, 1]])])
 
Permutation group:Degree $21$ $\langle(1,2,12)(3,5,17,7,21,8)(4,13,18)(6,10)(9,16,15,14,11,20), (1,6,8,5,19,20,10,3,12,18,14,21,16,13,4,7,15,17,2,9,11)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 21 | (1,2,12)(3,5,17,7,21,8)(4,13,18)(6,10)(9,16,15,14,11,20), (1,6,8,5,19,20,10,3,12,18,14,21,16,13,4,7,15,17,2,9,11) >;
 
Copy content gap:G := Group( (1,2,12)(3,5,17,7,21,8)(4,13,18)(6,10)(9,16,15,14,11,20), (1,6,8,5,19,20,10,3,12,18,14,21,16,13,4,7,15,17,2,9,11) );
 
Copy content sage:G = PermutationGroup(['(1,2,12)(3,5,17,7,21,8)(4,13,18)(6,10)(9,16,15,14,11,20)', '(1,6,8,5,19,20,10,3,12,18,14,21,16,13,4,7,15,17,2,9,11)'])
 
Transitive group: 21T103 42T1277 more information
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 equivalence classes (represented by square brackets) of matrices in $\GammaL(3,4)$.

Homology

Abelianization: $C_{2} $
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: $C_1$
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 $20 \times 20$ character table is not available for this group.

Rational character table

The $16 \times 16$ rational character table is not available for this group.