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Properties

Label 12T11
12T11 1 4 1->4 5 1->5 1->5 2 2->5 6 2->6 10 2->10 3 3->6 7 3->7 4->7 8 4->8 4->8 5->8 9 5->9 6->9 6->10 7->10 11 7->11 7->11 8->11 12 8->12 9->1 9->12 10->1 10->2 11->2 11->3 12->3 12->4
Degree $12$
Order $24$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $S_3 \times C_4$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(12, 11);
 
Copy content sage:G = TransitiveGroup(12, 11)
 
Copy content oscar:G = transitive_group(12, 11)
 
Copy content gap:G := TransitiveGroup(12, 11);
 

Group invariants

Abstract group:  $S_3 \times C_4$
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:  $24=2^{3} \cdot 3$
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 $n$:  $12$
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 $t$:  $11$
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:   $S(3)[x]C(4)$
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)
 
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);
 
$\card{\Aut(F/K)}$:  $4$
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])
 
Copy content gap:Order(Centralizer(SymmetricGroup(12), G));
 
Generators:  $(1,5)(2,10)(4,8)(7,11)$, $(1,5,9)(2,6,10)(3,7,11)(4,8,12)$, $(1,4,7,10)(2,5,8,11)(3,6,9,12)$
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)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$

Degree 6: $D_{6}$

Low degree siblings

12T11, 24T12

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{12}$ $1$ $1$ $0$ $()$
2A $2^{6}$ $1$ $2$ $6$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
2B $2^{4},1^{4}$ $3$ $2$ $4$ $( 2, 6)( 3,11)( 5, 9)( 8,12)$
2C $2^{6}$ $3$ $2$ $6$ $( 1, 3)( 2, 8)( 4, 6)( 5,11)( 7, 9)(10,12)$
3A $3^{4}$ $2$ $3$ $8$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
4A1 $4^{3}$ $1$ $4$ $9$ $( 1,10, 7, 4)( 2,11, 8, 5)( 3,12, 9, 6)$
4A-1 $4^{3}$ $1$ $4$ $9$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$
4B1 $4^{3}$ $3$ $4$ $9$ $( 1, 8, 7, 2)( 3, 6, 9,12)( 4,11,10, 5)$
4B-1 $4^{3}$ $3$ $4$ $9$ $( 1,10, 7, 4)( 2, 3, 8, 9)( 5, 6,11,12)$
6A $6^{2}$ $2$ $6$ $10$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
12A1 $12$ $2$ $12$ $11$ $( 1, 8, 3,10, 5,12, 7, 2, 9, 4,11, 6)$
12A-1 $12$ $2$ $12$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$

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 2B 2C 3A 4A1 4A-1 4B1 4B-1 6A 12A1 12A-1
Size 1 1 3 3 2 1 1 3 3 2 2 2
2 P 1A 1A 1A 1A 3A 2A 2A 2A 2A 3A 6A 6A
3 P 1A 2A 2B 2C 1A 4A-1 4A1 4B-1 4B1 2A 4A1 4A-1
Type
24.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1
24.5.1b R 1 1 −1 −1 1 −1 −1 1 1 1 −1 −1
24.5.1c R 1 1 −1 −1 1 1 1 −1 −1 1 1 1
24.5.1d R 1 1 1 1 1 −1 −1 −1 −1 1 −1 −1
24.5.1e1 C 1 −1 −1 1 1 −i i −i i −1 i −i
24.5.1e2 C 1 −1 −1 1 1 i −i i −i −1 −i i
24.5.1f1 C 1 −1 1 −1 1 −i i i −i −1 i −i
24.5.1f2 C 1 −1 1 −1 1 i −i −i i −1 −i i
24.5.2a R 2 2 0 0 −1 2 2 0 0 −1 −1 −1
24.5.2b R 2 2 0 0 −1 −2 −2 0 0 −1 1 1
24.5.2c1 C 2 −2 0 0 −1 −2i 2i 0 0 1 −i i
24.5.2c2 C 2 −2 0 0 −1 2i −2i 0 0 1 i −i

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

Data not computed