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Properties

Label 10T1
10T1 1 2 1->2 3 2->3 4 3->4 5 4->5 6 5->6 7 6->7 8 7->8 9 8->9 10 9->10 10->1
Degree $10$
Order $10$
Cyclic yes
Abelian yes
Solvable yes
Primitive no
$p$-group no
Group: $C_{10}$

Related objects

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Show commands: Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(10, 1);
 
Copy content sage:G = TransitiveGroup(10, 1)
 
Copy content oscar:G = transitive_group(10, 1)
 

Group invariants

Abstract group:  $C_{10}$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Order:  $10=2 \cdot 5$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Cyclic:  yes
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:  yes
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:  yes
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:  $1$
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 $t$:  $1$
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:   $C(10)=5[x]2$
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:  no
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)
 
$\card{\Aut(F/K)}$:  $10$
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:  $(1,2,3,4,5,6,7,8,9,10)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: $C_5$

Low degree siblings

There are no siblings 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 $2^{5}$ $1$ $2$ $5$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$
5A1 $5^{2}$ $1$ $5$ $8$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
5A-1 $5^{2}$ $1$ $5$ $8$ $( 1, 9, 7, 5, 3)( 2,10, 8, 6, 4)$
5A2 $5^{2}$ $1$ $5$ $8$ $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)$
5A-2 $5^{2}$ $1$ $5$ $8$ $( 1, 7, 3, 9, 5)( 2, 8, 4,10, 6)$
10A1 $10$ $1$ $10$ $9$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$
10A-1 $10$ $1$ $10$ $9$ $( 1,10, 9, 8, 7, 6, 5, 4, 3, 2)$
10A3 $10$ $1$ $10$ $9$ $( 1, 4, 7,10, 3, 6, 9, 2, 5, 8)$
10A-3 $10$ $1$ $10$ $9$ $( 1, 8, 5, 2, 9, 6, 3,10, 7, 4)$

Malle's constant $a(G)$:     $1/5$

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 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3
Size 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 5A2 5A-2 5A-1 5A1 5A1 5A-1 5A-2 5A2
5 P 1A 2A 1A 1A 1A 1A 2A 2A 2A 2A
Type
10.2.1a R 1 1 1 1 1 1 1 1 1 1
10.2.1b R 1 −1 1 1 1 1 −1 −1 −1 −1
10.2.1c1 C 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ5−2 ζ52
10.2.1c2 C 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ52 ζ5−2
10.2.1c3 C 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5−1 ζ5
10.2.1c4 C 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5 ζ5−1
10.2.1d1 C 1 −1 ζ5−2 ζ52 ζ5 ζ5−1 −ζ5 −ζ5−1 −ζ5−2 −ζ52
10.2.1d2 C 1 −1 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5−1 −ζ5 −ζ52 −ζ5−2
10.2.1d3 C 1 −1 ζ5−1 ζ5 ζ5−2 ζ52 −ζ5−2 −ζ52 −ζ5−1 −ζ5
10.2.1d4 C 1 −1 ζ5 ζ5−1 ζ52 ζ5−2 −ζ52 −ζ5−2 −ζ5 −ζ5−1

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

Regular extensions

$f_{ 1 } =$ $x^{10} + \left(2 t^{2} - 4 t + 2\right) x^{9} + \left(t^{4} - 8 t^{3} + 6 t^{2} - 7 t - 7\right) x^{8} + \left(-4 t^{5} + 10 t^{4} - 2 t^{3} - 4 t^{2} + 24 t - 14\right) x^{7} + \left(6 t^{6} + 2 t^{5} - 6 t^{4} + 44 t^{3} - 22 t^{2} + 50 t + 16\right) x^{6} + \left(-4 t^{7} - 16 t^{6} + 8 t^{5} - 60 t^{4} - 8 t^{3} - 32 t^{2} - 28 t + 32\right) x^{5} + \left(t^{8} + 16 t^{7} + 5 t^{6} + 34 t^{5} + 54 t^{4} - 12 t^{3} + 30 t^{2} - 80 t - 13\right) x^{4} + \left(-8 t^{8} - 18 t^{7} - 18 t^{6} - 82 t^{5} - 24 t^{4} - 56 t^{3} + 40 t^{2} + 2 t - 26\right) x^{3} + \left(2 t^{9} + 14 t^{8} + 16 t^{7} + 63 t^{6} + 74 t^{5} + 94 t^{4} + 60 t^{3} + 35 t^{2} + 44 t + 3\right) x^{2} + \left(-4 t^{9} - 6 t^{8} - 16 t^{7} - 26 t^{6} - 16 t^{5} + 14 t^{4} + 56 t^{3} + 36 t^{2} + 16 t + 6\right) x + \left(t^{10} + 4 t^{9} + 12 t^{8} + 31 t^{7} + 58 t^{6} + 88 t^{5} + 99 t^{4} + 91 t^{3} + 57 t^{2} + 15 t + 1\right)$ Copy content Toggle raw display