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Properties

Label 10T6
10T6 1 6 1->6 2 4 2->4 7 2->7 3 8 3->8 4->6 9 4->9 5 10 5->10 6->8 8->10 10->2
Degree $10$
Order $50$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $D_5\times C_5$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(10, 6);
 
Copy content sage:G = TransitiveGroup(10, 6)
 
Copy content oscar:G = transitive_group(10, 6)
 
Copy content gap:G := TransitiveGroup(10, 6);
 

Group invariants

Abstract group:  $D_5\times C_5$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Copy content oscar:small_group_identification(G)
 
Copy content gap:IdGroup(G);
 
Order:  $50=2 \cdot 5^{2}$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Copy content gap: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)
 
Copy content gap:IsCyclic(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)
 
Copy content gap:IsAbelian(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)
 
Copy content gap:IsSolvable(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
 
Copy content gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;
 

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)
 
Copy content gap:NrMovedPoints(G);
 
Transitive number $t$:  $6$
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]
 
Copy content gap:TransitiveIdentification(G);
 
CHM label:   $[5^{2}]2$
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)
 
Copy content gap:ForAll(GeneratorsOfGroup(G), g -> SignPerm(g) = 1);
 
Transitivity:  1
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)
 
Copy content gap:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $5$
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])
 
Copy content gap:Order(Centralizer(SymmetricGroup(10), G));
 
Generators:  $(2,4,6,8,10)$, $(1,6)(2,7)(3,8)(4,9)(5,10)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 
Copy content gap:GeneratorsOfGroup(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$10$:  $D_{5}$, $C_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

10T6, 25T3

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 $2^{5}$ $5$ $2$ $5$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$
5A1 $5^{2}$ $1$ $5$ $8$ $( 1, 7, 3, 9, 5)( 2, 8, 4,10, 6)$
5A-1 $5^{2}$ $1$ $5$ $8$ $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)$
5A2 $5^{2}$ $1$ $5$ $8$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
5A-2 $5^{2}$ $1$ $5$ $8$ $( 1, 9, 7, 5, 3)( 2,10, 8, 6, 4)$
5B1 $5^{2}$ $2$ $5$ $8$ $( 1, 3, 5, 7, 9)( 2,10, 8, 6, 4)$
5B2 $5^{2}$ $2$ $5$ $8$ $( 1, 5, 9, 3, 7)( 2, 8, 4,10, 6)$
5C1 $5^{2}$ $2$ $5$ $8$ $( 1, 9, 7, 5, 3)( 2, 6,10, 4, 8)$
5C-1 $5^{2}$ $2$ $5$ $8$ $( 1, 7, 3, 9, 5)( 2, 4, 6, 8,10)$
5C2 $5^{2}$ $2$ $5$ $8$ $( 1, 7, 3, 9, 5)( 2,10, 8, 6, 4)$
5C-2 $5^{2}$ $2$ $5$ $8$ $( 1, 3, 5, 7, 9)( 2, 6,10, 4, 8)$
5D1 $5,1^{5}$ $2$ $5$ $4$ $( 2, 4, 6, 8,10)$
5D-1 $5,1^{5}$ $2$ $5$ $4$ $(1,9,7,5,3)$
5D2 $5,1^{5}$ $2$ $5$ $4$ $(1,5,9,3,7)$
5D-2 $5,1^{5}$ $2$ $5$ $4$ $( 2, 8, 4,10, 6)$
10A1 $10$ $5$ $10$ $9$ $( 1, 4, 7,10, 3, 6, 9, 2, 5, 8)$
10A-1 $10$ $5$ $10$ $9$ $( 1, 8, 5, 2, 9, 6, 3,10, 7, 4)$
10A3 $10$ $5$ $10$ $9$ $( 1,10, 9, 8, 7, 6, 5, 4, 3, 2)$
10A-3 $10$ $5$ $10$ $9$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$

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)
 
Copy content gap:ConjugacyClasses(G);
 

Character table

1A 2A 5A1 5A-1 5A2 5A-2 5B1 5B2 5C1 5C-1 5C2 5C-2 5D1 5D-1 5D2 5D-2 10A1 10A-1 10A3 10A-3
Size 1 5 1 1 1 1 2 2 2 2 2 2 2 2 2 2 5 5 5 5
2 P 1A 1A 5A2 5A-2 5A-1 5A1 5B2 5B1 5C2 5C-2 5C-1 5C1 5D2 5D-2 5D-1 5D1 5A1 5A-1 5A-2 5A2
5 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A
Type
50.3.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
50.3.1b R 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
50.3.1c1 C 1 1 ζ5−2 ζ52 ζ5 ζ5−1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2
50.3.1c2 C 1 1 ζ52 ζ5−2 ζ5−1 ζ5 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52
50.3.1c3 C 1 1 ζ5−1 ζ5 ζ5−2 ζ52 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1
50.3.1c4 C 1 1 ζ5 ζ5−1 ζ52 ζ5−2 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5
50.3.1d1 C 1 −1 ζ5−2 ζ52 ζ5 ζ5−1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2
50.3.1d2 C 1 −1 ζ52 ζ5−2 ζ5−1 ζ5 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52
50.3.1d3 C 1 −1 ζ5−1 ζ5 ζ5−2 ζ52 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1
50.3.1d4 C 1 −1 ζ5 ζ5−1 ζ52 ζ5−2 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5
50.3.2a1 R 2 0 2 2 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 0 0 0 0
50.3.2a2 R 2 0 2 2 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 0 0 0 0
50.3.2b1 C 2 0 2ζ5−2 2ζ52 2ζ5 2ζ5−1 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−1−ζ5 ζ5−2+ζ5 −1−ζ5−ζ52 ζ5+ζ52 1+ζ5 ζ5−1+1 1+ζ52 ζ5−2+1 0 0 0 0
50.3.2b2 C 2 0 2ζ52 2ζ5−2 2ζ5−1 2ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ5 −ζ5−2−1−ζ5 ζ5+ζ52 −1−ζ5−ζ52 ζ5−1+1 1+ζ5 ζ5−2+1 1+ζ52 0 0 0 0
50.3.2b3 C 2 0 2ζ5−1 2ζ5 2ζ5−2 2ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5+ζ52 −1−ζ5−ζ52 −ζ5−2−1−ζ5 ζ5−2+ζ5 ζ5−2+1 1+ζ52 1+ζ5 ζ5−1+1 0 0 0 0
50.3.2b4 C 2 0 2ζ5 2ζ5−1 2ζ52 2ζ5−2 ζ5−2+ζ52 ζ5−1+ζ5 −1−ζ5−ζ52 ζ5+ζ52 ζ5−2+ζ5 −ζ5−2−1−ζ5 1+ζ52 ζ5−2+1 ζ5−1+1 1+ζ5 0 0 0 0
50.3.2c1 C 2 0 2ζ5−2 2ζ52 2ζ5 2ζ5−1 ζ5−2+ζ52 ζ5−1+ζ5 1+ζ5 ζ5−1+1 1+ζ52 ζ5−2+1 −ζ5−2−1−ζ5 ζ5−2+ζ5 −1−ζ5−ζ52 ζ5+ζ52 0 0 0 0
50.3.2c2 C 2 0 2ζ52 2ζ5−2 2ζ5−1 2ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+1 1+ζ5 ζ5−2+1 1+ζ52 ζ5−2+ζ5 −ζ5−2−1−ζ5 ζ5+ζ52 −1−ζ5−ζ52 0 0 0 0
50.3.2c3 C 2 0 2ζ5−1 2ζ5 2ζ5−2 2ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+1 1+ζ52 1+ζ5 ζ5−1+1 ζ5+ζ52 −1−ζ5−ζ52 −ζ5−2−1−ζ5 ζ5−2+ζ5 0 0 0 0
50.3.2c4 C 2 0 2ζ5 2ζ5−1 2ζ52 2ζ5−2 ζ5−1+ζ5 ζ5−2+ζ52 1+ζ52 ζ5−2+1 ζ5−1+1 1+ζ5 −1−ζ5−ζ52 ζ5+ζ52 ζ5−2+ζ5 −ζ5−2−1−ζ5 0 0 0 0

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

Regular extensions

$f_{ 1 } =$ $x^{10} + \left(2 t + 2\right) x^{9} + \left(t^{2} + 6 t + 1\right) x^{8} + \left(6 t^{2} + 4 t - 2\right) x^{7} + \left(2 t^{3} + 6 t^{2} + 8\right) x^{6} + \left(4 t^{3} + 2 t^{2} + 4 t\right) x^{5} + \left(t^{4} + t^{3} + 14 t + 7\right) x^{4} + \left(8 t^{2} - 2 t - 10\right) x^{3} + \left(t^{3} - t^{2} - t + 13\right) x^{2} + \left(2 t - 2\right) x + 1$ Copy content Toggle raw display