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Properties

Label 12T5
12T5 1 5 1->5 7 1->7 8 1->8 2 2->1 6 2->6 2->8 3 3->6 3->7 9 3->9 4 4->8 10 4->10 11 4->11 5->4 5->9 5->11 6->9 6->10 12 6->12 7->2 7->11 8->7 8->12 9->1 9->12 10->2 10->5 11->3 11->10 12->3 12->4
Degree $12$
Order $12$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3 : C_4$

Related objects

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Show commands: Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(12, 5);
 
Copy content sage:G = TransitiveGroup(12, 5)
 
Copy content oscar:G = transitive_group(12, 5)
 

Group invariants

Abstract group:  $C_3 : C_4$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Order:  $12=2^{2} \cdot 3$
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:  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:   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 $n$:  $12$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Transitive number $t$:  $5$
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:   $1/2[3:2]4$
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)
 
$\card{\Aut(F/K)}$:  $12$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(12).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(12), G)[1])
 
Generators:  $(1,8,7,2)(3,6,9,12)(4,11,10,5)$, $(1,5,9)(2,6,10)(3,7,11)(4,8,12)$, $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$
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)
$2$:  $C_2$
$4$:  $C_4$
$6$:  $S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$

Degree 6: $S_3$

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 $1^{12}$ $1$ $1$ $0$ $()$
2A $2^{6}$ $1$ $2$ $6$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
3A $3^{4}$ $2$ $3$ $8$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
4A1 $4^{3}$ $3$ $4$ $9$ $( 1, 8, 7, 2)( 3, 6, 9,12)( 4,11,10, 5)$
4A-1 $4^{3}$ $3$ $4$ $9$ $( 1, 2, 7, 8)( 3,12, 9, 6)( 4, 5,10,11)$
6A $6^{2}$ $2$ $6$ $10$ $( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$

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 4A1 4A-1 6A
Size 1 1 2 3 3 2
2 P 1A 1A 3A 2A 2A 3A
3 P 1A 2A 1A 4A-1 4A1 2A
Type
12.1.1a R 1 1 1 1 1 1
12.1.1b R 1 1 1 −1 −1 1
12.1.1c1 C 1 −1 1 −i i −1
12.1.1c2 C 1 −1 1 i −i −1
12.1.2a R 2 2 −1 0 0 −1
12.1.2b S 2 −2 −1 0 0 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

Data not computed