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Properties

Label 10T7
10T7 1 3 1->3 9 1->9 2 4 2->4 3->4 5 3->5 6 4->6 7 5->7 10 5->10 6->7 8 6->8 7->9 8->10 9->1 10->2
Degree $10$
Order $60$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $A_{5}$

Related objects

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Show commands: Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(10, 7);
 
Copy content sage:G = TransitiveGroup(10, 7)
 
Copy content oscar:G = transitive_group(10, 7)
 

Group invariants

Abstract group:  $A_{5}$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Order:  $60=2^{2} \cdot 3 \cdot 5$
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 $n$:  $10$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Transitive number $t$:  $7$
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:   $A_{5}(10)$
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(10).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(10), G)[1])
 
Generators:  $(1,9)(3,4)(5,10)(6,7)$, $(1,3,5,7,9)(2,4,6,8,10)$
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 2: None

Degree 5: None

Low degree siblings

5T4, 6T12, 12T33, 15T5, 20T15, 30T9

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{10}$ $1$ $1$ $0$ $()$
2A $2^{4},1^{2}$ $15$ $2$ $4$ $( 2, 8)( 3, 6)( 4, 7)( 5,10)$
3A $3^{3},1$ $20$ $3$ $6$ $( 2, 4, 5)( 3, 6, 9)( 7, 8,10)$
5A1 $5^{2}$ $12$ $5$ $8$ $( 1,10, 9, 6, 4)( 2, 8, 3, 5, 7)$
5A2 $5^{2}$ $12$ $5$ $8$ $( 1, 9, 4,10, 6)( 2, 3, 7, 8, 5)$

Malle's constant $a(G)$:     $1/4$

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 5A1 5A2
Size 1 15 20 12 12
2 P 1A 1A 3A 5A2 5A1
3 P 1A 2A 1A 5A2 5A1
5 P 1A 2A 3A 1A 1A
Type
60.5.1a R 1 1 1 1 1
60.5.3a1 R 3 −1 0 −ζ5−1−ζ5 −ζ5−2−ζ52
60.5.3a2 R 3 −1 0 −ζ5−2−ζ52 −ζ5−1−ζ5
60.5.4a R 4 0 1 −1 −1
60.5.5a R 5 1 −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

$f_{ 1 } =$ $x^{10} + 4 x^{9} + 12 x^{8} + \left(t + 20\right) x^{7} + \left(3 t + 24\right) x^{6} + \left(6 t + 14\right) x^{5} + \left(7 t - 1\right) x^{4} + \left(6 t - 10\right) x^{3} + \left(3 t - 7\right) x^{2} + t x + 1$ Copy content Toggle raw display