# Properties

 Label 12T1 Degree $12$ Order $12$ Cyclic yes Abelian yes Solvable yes Primitive no $p$-group no Group: $C_{12}$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(12, 1);

## Group action invariants

 Degree $n$: $12$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $1$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_{12}$ CHM label: $C(4)[x]C(3)$ 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: (1,5,9)(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)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: $C_4$

Degree 6: $C_6$

## 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

 Label Cycle Type Size Order Representative 1A $1^{12}$ $1$ $1$ $()$ 2A $2^{6}$ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ 3A1 $3^{4}$ $1$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ 3A-1 $3^{4}$ $1$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$ 4A1 $4^{3}$ $1$ $4$ $( 1,10, 7, 4)( 2,11, 8, 5)( 3,12, 9, 6)$ 4A-1 $4^{3}$ $1$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$ 6A1 $6^{2}$ $1$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ 6A-1 $6^{2}$ $1$ $6$ $( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$ 12A1 $12$ $1$ $12$ $( 1,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$ 12A-1 $12$ $1$ $12$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$ 12A5 $12$ $1$ $12$ $( 1, 8, 3,10, 5,12, 7, 2, 9, 4,11, 6)$ 12A-5 $12$ $1$ $12$ $( 1, 6,11, 4, 9, 2, 7,12, 5,10, 3, 8)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $12=2^{2} \cdot 3$ magma: Order(G); Cyclic: yes magma: IsCyclic(G); Abelian: yes magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: $1$ Label: 12.2 magma: IdentifyGroup(G); Character table:

 1A 2A 3A1 3A-1 4A1 4A-1 6A1 6A-1 12A1 12A-1 12A5 12A-5 Size 1 1 1 1 1 1 1 1 1 1 1 1 2 P 1A 1A 3A-1 3A1 2A 2A 3A1 3A-1 6A-1 6A1 6A1 6A-1 3 P 1A 2A 1A 1A 4A-1 4A1 2A 2A 4A1 4A-1 4A1 4A-1 Type 12.2.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 12.2.1b R $1$ $1$ $1$ $1$ $−1$ $−1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ 12.2.1c1 C $1$ $1$ $ζ3−1$ $ζ3$ $1$ $1$ $ζ3$ $ζ3−1$ $ζ3−1$ $ζ3$ $ζ3$ $ζ3−1$ 12.2.1c2 C $1$ $1$ $ζ3$ $ζ3−1$ $1$ $1$ $ζ3−1$ $ζ3$ $ζ3$ $ζ3−1$ $ζ3−1$ $ζ3$ 12.2.1d1 C $1$ $−1$ $1$ $1$ $−i$ $i$ $−1$ $−1$ $i$ $−i$ $i$ $−i$ 12.2.1d2 C $1$ $−1$ $1$ $1$ $i$ $−i$ $−1$ $−1$ $−i$ $i$ $−i$ $i$ 12.2.1e1 C $1$ $1$ $ζ3−1$ $ζ3$ $−1$ $−1$ $ζ3$ $ζ3−1$ $−ζ3−1$ $−ζ3$ $−ζ3$ $−ζ3−1$ 12.2.1e2 C $1$ $1$ $ζ3$ $ζ3−1$ $−1$ $−1$ $ζ3−1$ $ζ3$ $−ζ3$ $−ζ3−1$ $−ζ3−1$ $−ζ3$ 12.2.1f1 C $1$ $−1$ $−ζ122$ $ζ124$ $−ζ123$ $ζ123$ $−ζ124$ $ζ122$ $−ζ125$ $ζ12$ $−ζ12$ $ζ125$ 12.2.1f2 C $1$ $−1$ $ζ124$ $−ζ122$ $ζ123$ $−ζ123$ $ζ122$ $−ζ124$ $ζ12$ $−ζ125$ $ζ125$ $−ζ12$ 12.2.1f3 C $1$ $−1$ $−ζ122$ $ζ124$ $ζ123$ $−ζ123$ $−ζ124$ $ζ122$ $ζ125$ $−ζ12$ $ζ12$ $−ζ125$ 12.2.1f4 C $1$ $−1$ $ζ124$ $−ζ122$ $−ζ123$ $ζ123$ $ζ122$ $−ζ124$ $−ζ12$ $ζ125$ $−ζ125$ $ζ12$

magma: CharacterTable(G);