Properties

Label 12T4
12T4 1 9 1->9 11 1->11 2 4 2->4 2->9 3 3->2 10 3->10 4->3 8 4->8 5 5->1 5->3 6 6->1 6->8 7 7->2 7->6 8->7 12 8->12 9->5 9->7 10->5 10->12 11->6 11->10 12->4 12->11
Degree 1212
Order 1212
Cyclic no
Abelian no
Solvable yes
Primitive no
pp-group no
Group: A4A_4

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Show commands: Magma

Copy content magma:G := TransitiveGroup(12, 4);
 

Group invariants

Abstract group:  A4A_4
Copy content magma:IdentifyGroup(G);
 
Order:  12=22312=2^{2} \cdot 3
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree nn:  1212
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number tt:  44
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   A4(12)A_{4}(12)
Parity:  11
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
#Aut(F/K)\card{\Aut(F/K)}:  1212
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,9,5)(2,4,3)(6,8,7)(10,12,11)(1,9,5)(2,4,3)(6,8,7)(10,12,11), (1,11,6)(2,9,7)(3,10,5)(4,8,12)(1,11,6)(2,9,7)(3,10,5)(4,8,12)
Copy content magma:Generators(G);
 

Low degree resolvents

#(G/N)\card{(G/N)}Galois groups for stem field(s)
33C3C_3

Resolvents shown for degrees 47\leq 47

Subfields

Degree 2: None

Degree 3: C3C_3

Degree 4: A4A_4

Degree 6: A4A_4

Low degree siblings

4T4, 6T4

Siblings are shown with degree 47\leq 47

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A 1121^{12} 11 11 00 ()()
2A 262^{6} 33 22 66 (1,10)(2,5)(3,12)(4,7)(6,9)(8,11)( 1,10)( 2, 5)( 3,12)( 4, 7)( 6, 9)( 8,11)
3A1 343^{4} 44 33 88 (1,9,5)(2,4,3)(6,8,7)(10,12,11)( 1, 9, 5)( 2, 4, 3)( 6, 8, 7)(10,12,11)
3A-1 343^{4} 44 33 88 (1,5,9)(2,3,4)(6,7,8)(10,11,12)( 1, 5, 9)( 2, 3, 4)( 6, 7, 8)(10,11,12)

Malle's constant a(G)a(G):     1/61/6

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1
Size 1 3 4 4
2 P 1A 1A 3A-1 3A1
3 P 1A 2A 1A 1A
Type
12.3.1a R 1 1 1 1
12.3.1b1 C 1 1 ζ31 ζ3
12.3.1b2 C 1 1 ζ3 ζ31
12.3.3a R 3 1 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed