Properties

Label 12T1
Degree $12$
Order $12$
Cyclic yes
Abelian yes
Solvable yes
Primitive no
$p$-group no
Group: $C_{12}$

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Show commands: Magma

magma: G := TransitiveGroup(12, 1);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{12}$
CHM label:   $C(4)[x]C(3)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $12$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: $C_4$

Degree 6: $C_6$

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 TypeSizeOrderRepresentative
1A $1^{12}$ $1$ $1$ $()$
2A $2^{6}$ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
3A1 $3^{4}$ $1$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
3A-1 $3^{4}$ $1$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
4A1 $4^{3}$ $1$ $4$ $( 1,10, 7, 4)( 2,11, 8, 5)( 3,12, 9, 6)$
4A-1 $4^{3}$ $1$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$
6A1 $6^{2}$ $1$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
6A-1 $6^{2}$ $1$ $6$ $( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$
12A1 $12$ $1$ $12$ $( 1,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
12A-1 $12$ $1$ $12$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$
12A5 $12$ $1$ $12$ $( 1, 8, 3,10, 5,12, 7, 2, 9, 4,11, 6)$
12A-5 $12$ $1$ $12$ $( 1, 6,11, 4, 9, 2, 7,12, 5,10, 3, 8)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $12=2^{2} \cdot 3$
magma: Order(G);
 
Cyclic:  yes
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  12.2
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 4A1 4A-1 6A1 6A-1 12A1 12A-1 12A5 12A-5
Size 1 1 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 3A-1 3A1 2A 2A 3A1 3A-1 6A-1 6A1 6A1 6A-1
3 P 1A 2A 1A 1A 4A-1 4A1 2A 2A 4A1 4A-1 4A1 4A-1
Type
12.2.1a R 1 1 1 1 1 1 1 1 1 1 1 1
12.2.1b R 1 1 1 1 1 1 1 1 1 1 1 1
12.2.1c1 C 1 1 ζ31 ζ3 1 1 ζ3 ζ31 ζ31 ζ3 ζ3 ζ31
12.2.1c2 C 1 1 ζ3 ζ31 1 1 ζ31 ζ3 ζ3 ζ31 ζ31 ζ3
12.2.1d1 C 1 1 1 1 i i 1 1 i i i i
12.2.1d2 C 1 1 1 1 i i 1 1 i i i i
12.2.1e1 C 1 1 ζ31 ζ3 1 1 ζ3 ζ31 ζ31 ζ3 ζ3 ζ31
12.2.1e2 C 1 1 ζ3 ζ31 1 1 ζ31 ζ3 ζ3 ζ31 ζ31 ζ3
12.2.1f1 C 1 1 ζ122 ζ124 ζ123 ζ123 ζ124 ζ122 ζ125 ζ12 ζ12 ζ125
12.2.1f2 C 1 1 ζ124 ζ122 ζ123 ζ123 ζ122 ζ124 ζ12 ζ125 ζ125 ζ12
12.2.1f3 C 1 1 ζ122 ζ124 ζ123 ζ123 ζ124 ζ122 ζ125 ζ12 ζ12 ζ125
12.2.1f4 C 1 1 ζ124 ζ122 ζ123 ζ123 ζ122 ζ124 ζ12 ζ125 ζ125 ζ12

magma: CharacterTable(G);