# Properties

 Label 6T1 Degree $6$ Order $6$ Cyclic yes Abelian yes Solvable yes Primitive no $p$-group no Group: $C_6$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(6, 1);

## Group action invariants

 Degree $n$: $6$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $1$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_6$ CHM label: $C(6) = 6 = 3[x]2$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $1$ magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $6$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,2,3,4,5,6) magma: Generators(G);

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $C_3$

## 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

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $6$ $1$ $6$ $(1,2,3,4,5,6)$ $3, 3$ $1$ $3$ $(1,3,5)(2,4,6)$ $2, 2, 2$ $1$ $2$ $(1,4)(2,5)(3,6)$ $3, 3$ $1$ $3$ $(1,5,3)(2,6,4)$ $6$ $1$ $6$ $(1,6,5,4,3,2)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $6=2 \cdot 3$ magma: Order(G); Cyclic: yes magma: IsCyclic(G); Abelian: yes magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 6.2 magma: IdentifyGroup(G);
 Character table:  2 1 1 1 1 1 1 3 1 1 1 1 1 1 1a 6a 3a 2a 3b 6b X.1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 -1 X.3 1 A /A 1 A /A X.4 1 -A /A -1 A -/A X.5 1 /A A 1 /A A X.6 1 -/A A -1 /A -A A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 

magma: CharacterTable(G);