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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

|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

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 6, 6 $ $1$ $6$ $( 1, 2, 9,10, 5, 6)( 3, 4,11,12, 7, 8)$
$ 6, 6 $ $1$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$
$ 3, 3, 3, 3 $ $1$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 6, 6 $ $1$ $6$ $( 1, 6, 5,10, 9, 2)( 3, 8, 7,12,11, 4)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
$ 6, 6 $ $1$ $6$ $( 1, 8, 9, 4, 5,12)( 2, 3,10,11, 6, 7)$
$ 3, 3, 3, 3 $ $1$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,10)( 2, 5)( 3,12)( 4, 7)( 6, 9)( 8,11)$
$ 6, 6 $ $1$ $6$ $( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$
$ 6, 6 $ $1$ $6$ $( 1,12, 5, 4, 9, 8)( 2, 7, 6,11,10, 3)$

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:    
      2  2   2   2  2   2   2  2   2   2  2   2   2
      3  1   1   1  1   1   1  1   1   1  1   1   1

        1a  6a  6b 2a  3a  6c 2b  6d  3b 2c  6e  6f
     2P 1a  3b  3a 1a  3b  3a 1a  3b  3a 1a  3b  3a
     3P 1a  2c  2b 2a  1a  2c 2b  2a  1a 2c  2b  2a
     5P 1a  6c  6e 2a  3b  6a 2b  6f  3a 2c  6b  6d

X.1      1   1   1  1   1   1  1   1   1  1   1   1
X.2      1  -1  -1  1   1  -1 -1   1   1 -1  -1   1
X.3      1  -1   1 -1   1  -1  1  -1   1 -1   1  -1
X.4      1   1  -1 -1   1   1 -1  -1   1  1  -1  -1
X.5      1   A  /A  1  -A  /A -1  -A -/A -1   A -/A
X.6      1  /A   A  1 -/A   A -1 -/A  -A -1  /A  -A
X.7      1   A -/A -1  -A  /A  1   A -/A -1  -A  /A
X.8      1  /A  -A -1 -/A   A  1  /A  -A -1 -/A   A
X.9      1 -/A   A -1 -/A  -A -1  /A  -A  1  /A   A
X.10     1  -A  /A -1  -A -/A -1   A -/A  1   A  /A
X.11     1 -/A  -A  1 -/A  -A  1 -/A  -A  1 -/A  -A
X.12     1  -A -/A  1  -A -/A  1  -A -/A  1  -A -/A

A = -E(3)
  = (1-Sqrt(-3))/2 = -b3

magma: CharacterTable(G);