Properties

Label 12T57
12T57 1 2 1->2 5 1->5 12 1->12 3 2->3 6 2->6 2->12 3->1 7 3->7 4 4->7 8 4->8 5->6 9 5->9 10 6->10 11 7->11 8->9 8->12 9->1 10->2 11->3 12->3 12->4
Degree $12$
Order $96$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2^3.A_4$

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, 57);
 
Copy content sage:G = TransitiveGroup(12, 57)
 
Copy content oscar:G = transitive_group(12, 57)
 
Copy content gap:G := TransitiveGroup(12, 57);
 

Group invariants

Abstract group:  $C_2^3.A_4$
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:  $96=2^{5} \cdot 3$
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 $t$:  $57$
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:   $[(1/2.2^{2})^{3}]A_{4}(6)_{4}$
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);
 
$\card{\Aut(F/K)}$:  $2$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(12).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(12), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(12), G));
 
Generators:  $(1,12)(2,3)$, $(1,5,9)(2,6,10)(3,7,11)(4,8,12)$, $(1,2,12,3)(4,7)(5,6)(8,9)$
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)
$3$:  $C_3$
$12$:  $A_4$
$24$:  $\SL(2,3)$
$48$:  12T31

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: $A_4$

Low degree siblings

12T57, 24T179, 24T180 x 2, 32T417

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{12}$ $1$ $1$ $0$ $()$
2A $2^{6}$ $1$ $2$ $6$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$
2B $2^{4},1^{4}$ $3$ $2$ $4$ $( 4, 5)( 6, 7)( 8, 9)(10,11)$
2C $2^{2},1^{8}$ $3$ $2$ $2$ $( 1,12)( 2, 3)$
3A1 $3^{4}$ $16$ $3$ $8$ $( 1,11, 6)( 2, 9, 5)( 3, 8, 4)( 7,12,10)$
3A-1 $3^{4}$ $16$ $3$ $8$ $( 1, 6,11)( 2, 5, 9)( 3, 4, 8)( 7,10,12)$
4A1 $4,2^{3},1^{2}$ $6$ $4$ $6$ $( 1, 2,12, 3)( 4, 6)( 5, 7)( 8, 9)$
4A-1 $4,2^{3},1^{2}$ $6$ $4$ $6$ $( 1, 3,12, 2)( 4, 6)( 5, 7)( 8, 9)$
4B1 $4,2^{3},1^{2}$ $6$ $4$ $6$ $( 1, 2,12, 3)( 4, 7)( 5, 6)( 8, 9)$
4B-1 $4,2^{3},1^{2}$ $6$ $4$ $6$ $( 2, 3)( 4, 7, 5, 6)( 8,10)( 9,11)$
6A1 $6^{2}$ $16$ $6$ $10$ $( 1, 7,11,12, 6,10)( 2, 4, 9, 3, 5, 8)$
6A-1 $6^{2}$ $16$ $6$ $10$ $( 1,10, 6,12,11, 7)( 2, 8, 5, 3, 9, 4)$

Malle's constant $a(G)$:     $1/2$

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 3A1 3A-1 4A1 4A-1 4B1 4B-1 6A1 6A-1
Size 1 1 3 3 16 16 6 6 6 6 16 16
2 P 1A 1A 1A 1A 3A-1 3A1 2C 2C 2C 2C 3A1 3A-1
3 P 1A 2A 2B 2C 1A 1A 4A-1 4A1 4B-1 4B1 2A 2A
Type
96.3.1a R 1 1 1 1 1 1 1 1 1 1 1 1
96.3.1b1 C 1 1 1 1 ζ3−1 ζ3 1 1 1 1 ζ3 ζ3−1
96.3.1b2 C 1 1 1 1 ζ3 ζ3−1 1 1 1 1 ζ3−1 ζ3
96.3.2a S 2 −2 2 −2 −1 −1 0 0 0 0 1 1
96.3.2b1 C 2 −2 2 −2 −ζ3 −ζ3−1 0 0 0 0 ζ3−1 ζ3
96.3.2b2 C 2 −2 2 −2 −ζ3−1 −ζ3 0 0 0 0 ζ3 ζ3−1
96.3.3a R 3 3 3 3 0 0 −1 −1 −1 −1 0 0
96.3.3b1 C 3 3 −1 −1 0 0 −1−2i −1+2i 1 1 0 0
96.3.3b2 C 3 3 −1 −1 0 0 −1+2i −1−2i 1 1 0 0
96.3.3c1 C 3 3 −1 −1 0 0 1 1 −1−2i −1+2i 0 0
96.3.3c2 C 3 3 −1 −1 0 0 1 1 −1+2i −1−2i 0 0
96.3.6a R 6 −6 −2 2 0 0 0 0 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

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