# Properties

 Label 12T11 Degree $12$ Order $24$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $S_3 \times C_4$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(12, 11);

## Group action invariants

 Degree $n$: $12$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $11$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_3 \times C_4$ CHM label: $S(3)[x]C(4)$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $4$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,5)(2,10)(4,8)(7,11), (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
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$

Degree 6: $D_{6}$

## Low degree siblings

12T11, 24T12

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

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/4$

## 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: not nilpotent Label: 24.5 magma: IdentifyGroup(G); Character table:

 1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 6A 12A1 12A-1 Size 1 1 3 3 2 1 1 3 3 2 2 2 2 P 1A 1A 1A 1A 3A 2A 2A 2A 2A 3A 6A 6A 3 P 1A 2A 2B 2C 1A 4A-1 4A1 4B-1 4B1 2A 4A1 4A-1 Type 24.5.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 24.5.1b R $1$ $1$ $−1$ $−1$ $1$ $−1$ $−1$ $1$ $1$ $1$ $−1$ $−1$ 24.5.1c R $1$ $1$ $−1$ $−1$ $1$ $1$ $1$ $−1$ $−1$ $1$ $1$ $1$ 24.5.1d R $1$ $1$ $1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ $1$ $−1$ $−1$ 24.5.1e1 C $1$ $−1$ $−1$ $1$ $1$ $−i$ $i$ $i$ $−i$ $−1$ $i$ $−i$ 24.5.1e2 C $1$ $−1$ $−1$ $1$ $1$ $i$ $−i$ $−i$ $i$ $−1$ $−i$ $i$ 24.5.1f1 C $1$ $−1$ $1$ $−1$ $1$ $−i$ $i$ $−i$ $i$ $−1$ $i$ $−i$ 24.5.1f2 C $1$ $−1$ $1$ $−1$ $1$ $i$ $−i$ $i$ $−i$ $−1$ $−i$ $i$ 24.5.2a R $2$ $2$ $0$ $0$ $−1$ $2$ $2$ $0$ $0$ $−1$ $−1$ $−1$ 24.5.2b R $2$ $2$ $0$ $0$ $−1$ $−2$ $−2$ $0$ $0$ $−1$ $1$ $1$ 24.5.2c1 C $2$ $−2$ $0$ $0$ $−1$ $−2i$ $2i$ $0$ $0$ $1$ $−i$ $i$ 24.5.2c2 C $2$ $−2$ $0$ $0$ $−1$ $2i$ $−2i$ $0$ $0$ $1$ $i$ $−i$

magma: CharacterTable(G);