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Properties

Label 12T14
Degree $12$
Order $24$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_4 \times C_3$

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Show commands: Magma

magma: G := TransitiveGroup(12, 14);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $14$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_4 \times C_3$
CHM label:  $D(4)[x]C(3)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,7)(3,9)(5,11), (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)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$8$:  $D_{4}$
$12$:  $C_6\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: $D_{4}$

Degree 6: $C_6$

Low degree siblings

12T14, 24T15

Siblings are shown 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 $ $1$ $1$ $()$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 2, 8)( 4,10)( 6,12)$
$ 12 $ $2$ $12$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$
$ 6, 6 $ $2$ $6$ $( 1, 2, 9,10, 5, 6)( 3, 4,11,12, 7, 8)$
$ 6, 6 $ $1$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
$ 6, 3, 3 $ $2$ $6$ $( 1, 3, 5, 7, 9,11)( 2,10, 6)( 4,12, 8)$
$ 4, 4, 4 $ $2$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$
$ 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$
$ 3, 3, 3, 3 $ $1$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 6, 3, 3 $ $2$ $6$ $( 1, 5, 9)( 2,12,10, 8, 6, 4)( 3, 7,11)$
$ 6, 6 $ $2$ $6$ $( 1, 6, 5,10, 9, 2)( 3, 8, 7,12,11, 4)$
$ 12 $ $2$ $12$ $( 1, 6,11, 4, 9, 2, 7,12, 5,10, 3, 8)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
$ 3, 3, 3, 3 $ $1$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
$ 6, 6 $ $1$ $6$ $( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $24=2^{3} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $2$
Label:  24.10
magma: IdentifyGroup(G);
 
Character table:    
      2  3  2   2   2   3   2  2  2   3   2   2   2  3   3   3
      3  1  1   1   1   1   1  1  1   1   1   1   1  1   1   1

        1a 2a 12a  6a  6b  6c 4a 2b  3a  6d  6e 12b 2c  3b  6f
     2P 1a 1a  6b  3b  3a  3a 2c 1a  3b  3b  3a  6f 1a  3a  3b
     3P 1a 2a  4a  2b  2c  2a 4a 2b  1a  2a  2b  4a 2c  1a  2c
     5P 1a 2a 12b  6e  6f  6d 4a 2b  3b  6c  6a 12a 2c  3a  6b
     7P 1a 2a 12a  6a  6b  6c 4a 2b  3a  6d  6e 12b 2c  3b  6f
    11P 1a 2a 12b  6e  6f  6d 4a 2b  3b  6c  6a 12a 2c  3a  6b

X.1      1  1   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  1   1   1
X.3      1 -1   1  -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  1   1   1
X.5      1 -1   A  -A -/A  /A -1  1  -A   A -/A  /A  1 -/A  -A
X.6      1 -1  /A -/A  -A   A -1  1 -/A  /A  -A   A  1  -A -/A
X.7      1 -1 -/A  /A  -A   A  1 -1 -/A  /A   A  -A  1  -A -/A
X.8      1 -1  -A   A -/A  /A  1 -1  -A   A  /A -/A  1 -/A  -A
X.9      1  1   A   A -/A -/A -1 -1  -A  -A  /A  /A  1 -/A  -A
X.10     1  1  /A  /A  -A  -A -1 -1 -/A -/A   A   A  1  -A -/A
X.11     1  1 -/A -/A  -A  -A  1  1 -/A -/A  -A  -A  1  -A -/A
X.12     1  1  -A  -A -/A -/A  1  1  -A  -A -/A -/A  1 -/A  -A
X.13     2  .   .   .  -2   .  .  .   2   .   .   . -2   2  -2
X.14     2  .   .   .   B   .  .  . -/B   .   .   . -2  -B  /B
X.15     2  .   .   .  /B   .  .  .  -B   .   .   . -2 -/B   B

A = -E(3)
  = (1-Sqrt(-3))/2 = -b3
B = -2*E(3)
  = 1-Sqrt(-3) = 1-i3

magma: CharacterTable(G);