# Properties

 Label 6T6 Degree $6$ Order $24$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $A_4\times C_2$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(6, 6);

## Group action invariants

 Degree $n$: $6$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $6$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $A_4\times C_2$ CHM label: $2A_{4}(6) = [2^{3}]3 = 2 wr 3$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $2$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,3,5)(2,4,6), (3,6) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 3: $C_3$

## Low degree siblings

8T13, 12T6, 12T7, 24T9

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 $1^{6}$ $1$ $1$ $()$ $2,1^{4}$ $3$ $2$ $(3,6)$ $2^{2},1^{2}$ $3$ $2$ $(2,5)(3,6)$ $3^{2}$ $4$ $3$ $(1,2,3)(4,5,6)$ $6$ $4$ $6$ $(1,2,3,4,5,6)$ $3^{2}$ $4$ $3$ $(1,3,2)(4,6,5)$ $6$ $4$ $6$ $(1,3,5,4,6,2)$ $2^{3}$ $1$ $2$ $(1,4)(2,5)(3,6)$

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: not nilpotent Label: 24.13 magma: IdentifyGroup(G); Character table:

 1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 Size 1 1 3 3 4 4 4 4 2 P 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 3 P 1A 2A 2B 2C 1A 1A 2A 2A Type 24.13.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 24.13.1b R $1$ $−1$ $1$ $−1$ $1$ $1$ $−1$ $−1$ 24.13.1c1 C $1$ $1$ $1$ $1$ $ζ3−1$ $ζ3$ $ζ3$ $ζ3−1$ 24.13.1c2 C $1$ $1$ $1$ $1$ $ζ3$ $ζ3−1$ $ζ3−1$ $ζ3$ 24.13.1d1 C $1$ $−1$ $1$ $−1$ $ζ3−1$ $ζ3$ $−ζ3$ $−ζ3−1$ 24.13.1d2 C $1$ $−1$ $1$ $−1$ $ζ3$ $ζ3−1$ $−ζ3−1$ $−ζ3$ 24.13.3a R $3$ $3$ $−1$ $−1$ $0$ $0$ $0$ $0$ 24.13.3b R $3$ $−3$ $−1$ $1$ $0$ $0$ $0$ $0$

magma: CharacterTable(G);