↑

Properties

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

Related objects

Downloads

Learn more

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

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$8$:  $D_{4}$
$12$:  $C_6\times C_2$

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

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{12}$ $1$ $1$ $0$ $()$
2A $2^{6}$ $1$ $2$ $6$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
2B $2^{6}$ $2$ $2$ $6$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$
2C $2^{3},1^{6}$ $2$ $2$ $3$ $( 1, 7)( 3, 9)( 5,11)$
3A1 $3^{4}$ $1$ $3$ $8$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
3A-1 $3^{4}$ $1$ $3$ $8$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
4A $4^{3}$ $2$ $4$ $9$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$
6A1 $6^{2}$ $1$ $6$ $10$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
6A-1 $6^{2}$ $1$ $6$ $10$ $( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$
6B1 $6^{2}$ $2$ $6$ $10$ $( 1,12, 5, 4, 9, 8)( 2, 7, 6,11,10, 3)$
6B-1 $6,3^{2}$ $2$ $6$ $9$ $( 1,11, 9, 7, 5, 3)( 2, 6,10)( 4, 8,12)$
6C1 $6^{2}$ $2$ $6$ $10$ $( 1, 8, 9, 4, 5,12)( 2, 3,10,11, 6, 7)$
6C-1 $6,3^{2}$ $2$ $6$ $9$ $( 1, 3, 5, 7, 9,11)( 2,10, 6)( 4,12, 8)$
12A1 $12$ $2$ $12$ $11$ $( 1, 8, 3,10, 5,12, 7, 2, 9, 4,11, 6)$
12A-1 $12$ $2$ $12$ $11$ $( 1,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$

Malle's constant $a(G)$:     $1/3$

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:

1A 2A 2B 2C 3A1 3A-1 4A 6A1 6A-1 6B1 6B-1 6C1 6C-1 12A1 12A-1
Size 1 1 2 2 1 1 2 1 1 2 2 2 2 2 2
2 P 1A 1A 1A 1A 3A-1 3A1 2A 3A-1 3A1 3A-1 3A1 3A1 3A-1 6A1 6A-1
3 P 1A 2A 2B 2C 1A 1A 4A 2A 2A 2B 2C 2B 2C 4A 4A
Type
24.10.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
24.10.1b R 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 1 1
24.10.1c R 1 1 −1 1 1 1 −1 1 1 −1 −1 1 1 −1 −1
24.10.1d R 1 1 1 −1 1 1 −1 1 1 1 1 −1 −1 −1 −1
24.10.1e1 C 1 1 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3
24.10.1e2 C 1 1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1
24.10.1f1 C 1 1 −1 −1 ζ3−1 ζ3 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3−1 ζ3
24.10.1f2 C 1 1 −1 −1 ζ3 ζ3−1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3 ζ3−1
24.10.1g1 C 1 1 −1 1 ζ3−1 ζ3 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3
24.10.1g2 C 1 1 −1 1 ζ3 ζ3−1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1
24.10.1h1 C 1 1 1 −1 ζ3−1 ζ3 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3
24.10.1h2 C 1 1 1 −1 ζ3 ζ3−1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1
24.10.2a R 2 −2 0 0 2 2 0 −2 −2 0 0 0 0 0 0
24.10.2b1 C 2 −2 0 0 2ζ3−1 2ζ3 0 −2ζ3 −2ζ3−1 0 0 0 0 0 0
24.10.2b2 C 2 −2 0 0 2ζ3 2ζ3−1 0 −2ζ3−1 −2ζ3 0 0 0 0 0 0

magma: CharacterTable(G);