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Properties

Label 10T36
10T36 1 3 1->3 2 4 2->4 2->4 5 3->5 6 4->6 10 4->10 7 5->7 5->7 5->10 8 6->8 9 7->9 7->9 8->10 9->1 9->5 10->2 10->2
Degree $10$
Order $1920$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2 \wr A_5$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(10, 36);
 
Copy content sage:G = TransitiveGroup(10, 36)
 
Copy content oscar:G = transitive_group(10, 36)
 
Copy content gap:G := TransitiveGroup(10, 36);
 

Group invariants

Abstract group:  $C_2 \wr A_5$
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:  $1920=2^{7} \cdot 3 \cdot 5$
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:  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)
 
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 $t$:  $36$
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:   $[2^{5}]A(5)$
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);
 
$\card{\Aut(F/K)}$:  $2$
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:  $(5,10)$, $(2,4,10)(5,7,9)$, $(1,3,5,7,9)(2,4,6,8,10)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 
Copy content gap:GeneratorsOfGroup(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$60$:  $A_5$
$120$:  $A_5\times C_2$
$960$:  $C_2^4 : A_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: $A_5$

Low degree siblings

20T224, 20T225, 20T230, 30T344, 30T354, 32T97741, 40T1576, 40T1578, 40T1585, 40T1586, 40T1597, 40T1598, 40T1644

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 $2^{5}$ $1$ $2$ $5$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$
2B $2^{4},1^{2}$ $5$ $2$ $4$ $( 1, 6)( 2, 7)( 3, 8)( 5,10)$
2C $2,1^{8}$ $5$ $2$ $1$ $(3,8)$
2D $2^{3},1^{4}$ $10$ $2$ $3$ $( 1, 6)( 4, 9)( 5,10)$
2E $2^{2},1^{6}$ $10$ $2$ $2$ $( 3, 8)( 5,10)$
2F $2^{5}$ $60$ $2$ $5$ $( 1, 3)( 2, 7)( 4,10)( 5, 9)( 6, 8)$
2G $2^{4},1^{2}$ $60$ $2$ $4$ $( 1, 8)( 3, 6)( 4,10)( 5, 9)$
3A $3^{2},1^{4}$ $80$ $3$ $4$ $( 1,10, 2)( 5, 7, 6)$
4A $4^{2},2$ $60$ $4$ $7$ $( 1, 7, 6, 2)( 3, 5, 8,10)( 4, 9)$
4B $4^{2},1^{2}$ $60$ $4$ $6$ $( 1,10, 6, 5)( 3, 4, 8, 9)$
4C $4,2^{2},1^{2}$ $120$ $4$ $5$ $( 1, 7)( 2, 6)( 3, 5, 8,10)$
4D $4,2^{3}$ $120$ $4$ $6$ $( 1, 3, 6, 8)( 2, 5)( 4, 9)( 7,10)$
5A1 $5^{2}$ $192$ $5$ $8$ $( 1, 2, 4,10, 8)( 3, 6, 7, 9, 5)$
5A2 $5^{2}$ $192$ $5$ $8$ $( 1, 4, 8, 2,10)( 3, 7, 5, 6, 9)$
6A $6,1^{4}$ $80$ $6$ $5$ $( 1,10, 4, 6, 5, 9)$
6B $3^{2},2^{2}$ $80$ $6$ $6$ $( 1, 6)( 2, 7)( 3, 4, 5)( 8, 9,10)$
6C $6,2^{2}$ $80$ $6$ $7$ $( 1,10, 3, 6, 5, 8)( 2, 7)( 4, 9)$
6D1 $3^{2},2,1^{2}$ $80$ $6$ $5$ $( 1, 2,10)( 3, 8)( 5, 6, 7)$
6D-1 $3^{2},2,1^{2}$ $80$ $6$ $5$ $( 1,10, 2)( 3, 8)( 5, 7, 6)$
6E1 $6,2,1^{2}$ $80$ $6$ $6$ $( 1, 5, 8, 6,10, 3)( 4, 9)$
6E-1 $6,2,1^{2}$ $80$ $6$ $6$ $( 1, 3,10, 6, 8, 5)( 4, 9)$
10A1 $10$ $192$ $10$ $9$ $( 1, 5, 2, 3, 4, 6,10, 7, 8, 9)$
10A3 $10$ $192$ $10$ $9$ $( 1, 3,10, 9, 2, 6, 8, 5, 4, 7)$

Malle's constant $a(G)$:     $1$

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 2B 2C 2D 2E 2F 2G 3A 4A 4B 4C 4D 5A1 5A2 6A 6B 6C 6D1 6D-1 6E1 6E-1 10A1 10A3
Size 1 1 5 5 10 10 60 60 80 60 60 120 120 192 192 80 80 80 80 80 80 80 192 192
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A 2B 2B 2E 2E 5A2 5A1 3A 3A 3A 3A 3A 3A 3A 5A1 5A2
3 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 4A 4B 4C 4D 5A2 5A1 2D 2E 2A 2C 2C 2B 2B 10A3 10A1
5 P 1A 2A 2B 2C 2D 2E 2F 2G 3A 4A 4B 4C 4D 1A 1A 6A 6B 6C 6D-1 6D1 6E-1 6E1 2A 2A
Type
1920.240997.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1920.240997.1b R 1 −1 1 −1 −1 1 −1 1 1 −1 1 −1 1 1 1 −1 1 −1 −1 −1 1 1 −1 −1
1920.240997.3a1 R 3 3 3 3 3 3 −1 −1 0 −1 −1 −1 −1 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 0 0 0 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5
1920.240997.3a2 R 3 3 3 3 3 3 −1 −1 0 −1 −1 −1 −1 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 0 0 0 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52
1920.240997.3b1 R 3 −3 3 −3 −3 3 1 −1 0 1 −1 1 −1 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 0 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5
1920.240997.3b2 R 3 −3 3 −3 −3 3 1 −1 0 1 −1 1 −1 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 0 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52
1920.240997.4a R 4 4 4 4 4 4 0 0 1 0 0 0 0 −1 −1 1 1 1 1 1 1 1 −1 −1
1920.240997.4b R 4 −4 4 −4 −4 4 0 0 1 0 0 0 0 −1 −1 −1 1 −1 −1 −1 1 1 1 1
1920.240997.5a R 5 5 5 5 5 5 1 1 −1 1 1 1 1 0 0 −1 −1 −1 −1 −1 −1 −1 0 0
1920.240997.5b R 5 5 −3 −3 1 1 1 1 2 1 1 −1 −1 0 0 −2 −2 2 0 0 0 0 0 0
1920.240997.5c R 5 −5 −3 3 −1 1 −1 1 2 −1 1 1 −1 0 0 2 −2 −2 0 0 0 0 0 0
1920.240997.5d R 5 −5 5 −5 −5 5 −1 1 −1 −1 1 −1 1 0 0 1 −1 1 1 1 −1 −1 0 0
1920.240997.5e1 C 5 5 −3 −3 1 1 1 1 −1 1 1 −1 −1 0 0 1 1 −1 −1−2ζ3 1+2ζ3 −1−2ζ3 1+2ζ3 0 0
1920.240997.5e2 C 5 5 −3 −3 1 1 1 1 −1 1 1 −1 −1 0 0 1 1 −1 1+2ζ3 −1−2ζ3 1+2ζ3 −1−2ζ3 0 0
1920.240997.5f1 C 5 −5 −3 3 −1 1 −1 1 −1 −1 1 1 −1 0 0 −1 1 1 −1−2ζ3 1+2ζ3 1+2ζ3 −1−2ζ3 0 0
1920.240997.5f2 C 5 −5 −3 3 −1 1 −1 1 −1 −1 1 1 −1 0 0 −1 1 1 1+2ζ3 −1−2ζ3 −1−2ζ3 1+2ζ3 0 0
1920.240997.10a R 10 10 2 2 −2 −2 2 2 1 −2 −2 0 0 0 0 1 1 1 −1 −1 −1 −1 0 0
1920.240997.10b R 10 10 2 2 −2 −2 −2 −2 1 2 2 0 0 0 0 1 1 1 −1 −1 −1 −1 0 0
1920.240997.10c R 10 −10 2 −2 2 −2 2 −2 1 −2 2 0 0 0 0 −1 1 −1 1 1 −1 −1 0 0
1920.240997.10d R 10 −10 2 −2 2 −2 −2 2 1 2 −2 0 0 0 0 −1 1 −1 1 1 −1 −1 0 0
1920.240997.15a R 15 15 −9 −9 3 3 −1 −1 0 −1 −1 1 1 0 0 0 0 0 0 0 0 0 0 0
1920.240997.15b R 15 −15 −9 9 −3 3 1 −1 0 1 −1 −1 1 0 0 0 0 0 0 0 0 0 0 0
1920.240997.20a R 20 20 4 4 −4 −4 0 0 −1 0 0 0 0 0 0 −1 −1 −1 1 1 1 1 0 0
1920.240997.20b R 20 −20 4 −4 4 −4 0 0 −1 0 0 0 0 0 0 1 −1 1 −1 −1 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)
 
Copy content gap:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{10} + 75 x^{6} + t x^{4} + 3 t$ Copy content Toggle raw display