# 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

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

 Label Cycle Type Size Order Representative 1A $1^{12}$ $1$ $1$ $()$ 2A $2^{6}$ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ 2B $2^{6}$ $2$ $2$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$ 2C $2^{3},1^{6}$ $2$ $2$ $( 1, 7)( 3, 9)( 5,11)$ 3A1 $3^{4}$ $1$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$ 3A-1 $3^{4}$ $1$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ 4A $4^{3}$ $2$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$ 6A1 $6^{2}$ $1$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ 6A-1 $6^{2}$ $1$ $6$ $( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$ 6B1 $6^{2}$ $2$ $6$ $( 1,12, 5, 4, 9, 8)( 2, 7, 6,11,10, 3)$ 6B-1 $6,3^{2}$ $2$ $6$ $( 1,11, 9, 7, 5, 3)( 2, 6,10)( 4, 8,12)$ 6C1 $6^{2}$ $2$ $6$ $( 1, 8, 9, 4, 5,12)( 2, 3,10,11, 6, 7)$ 6C-1 $6,3^{2}$ $2$ $6$ $( 1, 3, 5, 7, 9,11)( 2,10, 6)( 4,12, 8)$ 12A1 $12$ $2$ $12$ $( 1, 8, 3,10, 5,12, 7, 2, 9, 4,11, 6)$ 12A-1 $12$ $2$ $12$ $( 1,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$

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);