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Properties

Label 117600.d
Order \( 2^{5} \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Exponent \( 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Nilpotent no
Solvable no
$\card{G^{\mathrm{ab}}}$ \( 2 \)
$\card{Z(G)}$ 1
$\card{\Aut(G)}$ \( 2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
$\card{\mathrm{Out}(G)}$ \( 2 \)
Perm deg. $50$
Trans deg. not computed
Rank $2$

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

Group information

Description:$\POMinus(4,7)$
Order: \(117600\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
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: \(4200\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
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:$C_2$, $\PSL(2,49)$
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 12 14 24 25
Elements 1 1575 2450 9800 4704 22050 2400 19600 4900 16800 9800 23520 117600
Conjugacy classes   1 3 1 2 1 3 2 4 1 2 2 5 27
Divisions 1 3 1 2 1 3 2 2 1 2 1 1 20
Autjugacy classes 1 2 1 2 1 2 1 3 1 1 2 5 22

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 25 not computed not computed
Arbitrary not computed not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


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

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: $C_{2}$
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 $27 \times 27$ character table is not available for this group.

Rational character table

The $20 \times 20$ rational character table is not available for this group.