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Properties

Label 15T47
15T47 1 2 1->2 9 1->9 3 2->3 15 2->15 3->1 14 3->14 4 5 4->5 6 5->6 11 5->11 6->2 7 6->7 7->5 12 7->12 8 8->4 10 8->10 9->8 9->10 10->3 10->9 13 11->13 12->6 12->14 13->8 13->12 14->1 14->13 15->7
Degree $15$
Order $2520$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $A_7$

Related objects

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Show commands: Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(15, 47);
 
Copy content sage:G = TransitiveGroup(15, 47)
 
Copy content oscar:G = transitive_group(15, 47)
 

Group invariants

Abstract group:  $A_7$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Order:  $2520=2^{3} \cdot 3^{2} \cdot 5 \cdot 7$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Cyclic:  no
Copy content comment:Determine if group is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content oscar:is_cyclic(G)
 
Abelian:  no
Copy content comment:Determine if group is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content oscar:is_abelian(G)
 
Solvable:  no
Copy content comment:Determine if group is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content sage:G.is_solvable()
 
Copy content oscar:is_solvable(G)
 
Nilpotency class:   not nilpotent
Copy content comment:Nilpotency class
 
Copy content magma:NilpotencyClass(G);
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 

Group action invariants

Degree $n$:  $15$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Transitive number $t$:  $47$
Copy content comment:Transitive number
 
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Copy content sage:G.transitive_number()
 
Copy content oscar:transitive_group_identification(G)[2]
 
CHM label:   $A_{7}(15)$
Parity:  $1$
Copy content comment:Parity
 
Copy content magma:IsEven(G);
 
Copy content sage:all(g.SignPerm() == 1 for g in libgap(G).GeneratorsOfGroup())
 
Copy content oscar:is_even(G)
 
Primitive:  yes
Copy content comment:Determine if group is primitive
 
Copy content magma:IsPrimitive(G);
 
Copy content sage:G.is_primitive()
 
Copy content oscar:is_primitive(G)
 
$\card{\Aut(F/K)}$:  $1$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(15).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(15), G)[1])
 
Generators:  $(1,9,10,3,14)(2,15,7,12,6)(4,5,11,13,8)$, $(1,2,3)(5,6,7)(8,10,9)(12,14,13)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: None

Low degree siblings

7T6, 15T47, 21T33, 35T28, 42T294, 42T299

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{15}$ $1$ $1$ $0$ $()$
2A $2^{6},1^{3}$ $105$ $2$ $6$ $( 1, 9)( 2,11)( 3,10)( 4, 5)( 8,15)(12,13)$
3A $3^{5}$ $70$ $3$ $10$ $( 1,11, 5)( 2,13,15)( 3, 4, 8)( 6, 7,14)( 9,10,12)$
3B $3^{4},1^{3}$ $280$ $3$ $8$ $( 1,12,13)( 2,14, 3)( 4, 9, 8)( 6, 7,11)$
4A $4^{3},2,1$ $630$ $4$ $10$ $( 1,12, 9,13)( 2,10,11, 3)( 4, 8, 5,15)( 7,14)$
5A $5^{3}$ $504$ $5$ $12$ $( 1, 6,13,11,14)( 2,12, 5, 7, 3)( 4, 9,10,15, 8)$
6A $6^{2},3$ $210$ $6$ $12$ $( 1, 7,11,14, 5, 6)( 2,15,13)( 3,10, 4,12, 8, 9)$
7A1 $7^{2},1$ $360$ $7$ $12$ $( 1,11,13,15, 7, 6,10)( 2, 8,14, 3, 4, 5, 9)$
7A-1 $7^{2},1$ $360$ $7$ $12$ $( 1,10, 6, 7,15,13,11)( 2, 9, 5, 4, 3,14, 8)$

Malle's constant $a(G)$:     $1/6$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 

Character table

1A 2A 3A 3B 4A 5A 6A 7A1 7A-1
Size 1 105 70 280 630 504 210 360 360
2 P 1A 1A 3A 3B 2A 5A 3A 7A1 7A-1
3 P 1A 2A 1A 1A 4A 5A 2A 7A-1 7A1
5 P 1A 2A 3A 3B 4A 1A 6A 7A-1 7A1
7 P 1A 2A 3A 3B 4A 5A 6A 1A 1A
Type
2520.a.1a R 1 1 1 1 1 1 1 1 1
2520.a.6a R 6 2 3 0 0 1 −1 −1 −1
2520.a.10a1 C 10 −2 1 1 0 0 1 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
2520.a.10a2 C 10 −2 1 1 0 0 1 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
2520.a.14a R 14 2 −1 2 0 −1 −1 0 0
2520.a.14b R 14 2 2 −1 0 −1 2 0 0
2520.a.15a R 15 −1 3 0 −1 0 −1 1 1
2520.a.21a R 21 1 −3 0 −1 1 1 0 0
2520.a.35a R 35 −1 −1 −1 1 0 −1 0 0

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 

Regular extensions

Data not computed