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Properties

Label 12T2
Degree $12$
Order $12$
Cyclic no
Abelian yes
Solvable yes
Primitive no
$p$-group no
Group: $C_6\times C_2$

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Show commands: Magma

magma: G := TransitiveGroup(12, 2);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $2$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_6\times C_2$
CHM label:   $E(4)[x]C(3)=6x2$
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:  (1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $C_3$

Degree 4: $C_2^2$

Degree 6: $C_6$ x 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 $2^{6}$ $1$ $2$ $6$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$
2B $2^{6}$ $1$ $2$ $6$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
2C $2^{6}$ $1$ $2$ $6$ $( 1,10)( 2, 5)( 3,12)( 4, 7)( 6, 9)( 8,11)$
3A1 $3^{4}$ $1$ $3$ $8$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
3A-1 $3^{4}$ $1$ $3$ $8$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
6A1 $6^{2}$ $1$ $6$ $10$ $( 1,12, 5, 4, 9, 8)( 2, 7, 6,11,10, 3)$
6A-1 $6^{2}$ $1$ $6$ $10$ $( 1, 6, 5,10, 9, 2)( 3, 8, 7,12,11, 4)$
6B1 $6^{2}$ $1$ $6$ $10$ $( 1, 2, 9,10, 5, 6)( 3, 4,11,12, 7, 8)$
6B-1 $6^{2}$ $1$ $6$ $10$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
6C1 $6^{2}$ $1$ $6$ $10$ $( 1, 8, 9, 4, 5,12)( 2, 3,10,11, 6, 7)$
6C-1 $6^{2}$ $1$ $6$ $10$ $( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $12=2^{2} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  12.5
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1
Size 1 1 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 1A 1A 3A-1 3A1 3A1 3A1 3A-1 3A1 3A-1 3A-1
3 P 1A 2B 2C 2A 1A 1A 2B 2A 2A 2C 2B 2C
Type
12.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1
12.5.1b R 1 −1 −1 1 1 1 −1 −1 −1 −1 1 1
12.5.1c R 1 −1 1 −1 1 1 −1 −1 1 1 −1 −1
12.5.1d R 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
12.5.1e1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
12.5.1e2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
12.5.1f1 C 1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3
12.5.1f2 C 1 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1
12.5.1g1 C 1 −1 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3
12.5.1g2 C 1 −1 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1
12.5.1h1 C 1 1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3
12.5.1h2 C 1 1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1

magma: CharacterTable(G);