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Properties

Label 235200.c
Order \( 2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Exponent \( 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Nilpotent no
Solvable no
$\card{G^{\mathrm{ab}}}$ \( 2^{2} \)
$\card{Z(G)}$ 1
$\card{\Aut(G)}$ \( 2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
$\card{\mathrm{Out}(G)}$ \( 1 \)
Perm deg. $50$
Trans deg. not computed
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := PGammaL(2,49);
 
Copy content gap:G := PGammaL(2,49);
 
Copy content comment:Define the group as a permutation group
 
Copy content sage:G = PermutationGroup(['(3,13,32,20,33,21,40,50,30,6,8,48,35,44,22,29,27,10,41,46,38,34,42,36,37,49,28,14,39,25,4,16,17,11,19,15,7,12,43,26,24,31,9,18,5,45,47,23)', '(3,4)(5,32)(6,45)(7,9)(8,33)(10,21)(11,34)(12,13)(16,50)(17,22)(19,39)(23,36)(24,42)(25,31)(26,29)(27,47)(28,38)(30,43)(35,41)(37,40)(44,49)', '(1,14,2)(3,37,11)(4,40,34)(5,22,23)(6,38,39)(7,27,21)(8,50,24)(9,47,10)(12,30,35)(13,43,41)(16,42,33)(17,36,32)(18,48,20)(19,45,28)(25,44,29)(26,31,49)'])
 

Group information

Description:$\PSL(2,49).C_2^2$
Order: \(235200\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
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: \(8400\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
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:Group of order \(235200\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$ x 2, $\PSL(2,49)$
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:$1$
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 10 12 14 16 24 25 48 50
Elements 1 2751 2450 19600 4704 22050 2400 19600 4704 24500 16800 39200 9800 23520 19600 23520 235200
Conjugacy classes   1 3 1 3 1 2 1 3 1 2 1 4 2 5 4 5 39
Divisions 1 3 1 3 1 2 1 2 1 2 1 2 1 1 1 1 24
Autjugacy classes 1 3 1 3 1 2 1 3 1 2 1 4 2 5 4 5 39

Minimal presentations

Permutation degree:$50$
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 49 not computed not computed
Arbitrary not computed not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Groups of Lie type:$\PGammaL(2,49)$
Copy content magma:G := PGammaL(2,49);
 
Copy content gap:G := PGammaL(2,49);
 
Copy content sage:F = GF(49); al = F.0; MS = MatrixSpace(F, 2, 2) G = MatrixGroup([MS([[al^1, 0], [0, 1]]), MS([[al^24, 1], [al^24, 0]])])
 
Permutation group:Degree $50$ $\langle(3,13,32,20,33,21,40,50,30,6,8,48,35,44,22,29,27,10,41,46,38,34,42,36,37,49,28,14,39,25,4,16,17,11,19,15,7,12,43,26,24,31,9,18,5,45,47,23) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 50 | (3,13,32,20,33,21,40,50,30,6,8,48,35,44,22,29,27,10,41,46,38,34,42,36,37,49,28,14,39,25,4,16,17,11,19,15,7,12,43,26,24,31,9,18,5,45,47,23), (3,4)(5,32)(6,45)(7,9)(8,33)(10,21)(11,34)(12,13)(16,50)(17,22)(19,39)(23,36)(24,42)(25,31)(26,29)(27,47)(28,38)(30,43)(35,41)(37,40)(44,49), (1,14,2)(3,37,11)(4,40,34)(5,22,23)(6,38,39)(7,27,21)(8,50,24)(9,47,10)(12,30,35)(13,43,41)(16,42,33)(17,36,32)(18,48,20)(19,45,28)(25,44,29)(26,31,49) >;
 
Copy content gap:G := Group( (3,13,32,20,33,21,40,50,30,6,8,48,35,44,22,29,27,10,41,46,38,34,42,36,37,49,28,14,39,25,4,16,17,11,19,15,7,12,43,26,24,31,9,18,5,45,47,23), (3,4)(5,32)(6,45)(7,9)(8,33)(10,21)(11,34)(12,13)(16,50)(17,22)(19,39)(23,36)(24,42)(25,31)(26,29)(27,47)(28,38)(30,43)(35,41)(37,40)(44,49), (1,14,2)(3,37,11)(4,40,34)(5,22,23)(6,38,39)(7,27,21)(8,50,24)(9,47,10)(12,30,35)(13,43,41)(16,42,33)(17,36,32)(18,48,20)(19,45,28)(25,44,29)(26,31,49) );
 
Copy content sage:G = PermutationGroup(['(3,13,32,20,33,21,40,50,30,6,8,48,35,44,22,29,27,10,41,46,38,34,42,36,37,49,28,14,39,25,4,16,17,11,19,15,7,12,43,26,24,31,9,18,5,45,47,23)', '(3,4)(5,32)(6,45)(7,9)(8,33)(10,21)(11,34)(12,13)(16,50)(17,22)(19,39)(23,36)(24,42)(25,31)(26,29)(27,47)(28,38)(30,43)(35,41)(37,40)(44,49)', '(1,14,2)(3,37,11)(4,40,34)(5,22,23)(6,38,39)(7,27,21)(8,50,24)(9,47,10)(12,30,35)(13,43,41)(16,42,33)(17,36,32)(18,48,20)(19,45,28)(25,44,29)(26,31,49)'])
 
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 $\GammaL(2,49)$.

Homology

Abelianization: $C_{2}^{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: $C_{2}$
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()
 

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 $39 \times 39$ character table is not available for this group.

Rational character table

The $24 \times 24$ rational character table is not available for this group.