# Properties

 Label 9T6 Degree $9$ Order $27$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_9:C_3$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(9, 6);

## Group action invariants

 Degree $n$: $9$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $6$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_9:C_3$ CHM label: $1/3[3^{3}]3$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $3$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,4,7)(2,8,5), (1,2,3,4,5,6,7,8,9) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$ x 4
$9$:  $C_3^2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$

## Low degree siblings

27T5

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 1, 1, 1$ $3$ $3$ $(2,5,8)(3,9,6)$ $3, 3, 1, 1, 1$ $3$ $3$ $(2,8,5)(3,6,9)$ $9$ $3$ $9$ $(1,2,3,4,5,6,7,8,9)$ $9$ $3$ $9$ $(1,2,6,4,5,9,7,8,3)$ $9$ $3$ $9$ $(1,2,9,4,5,3,7,8,6)$ $9$ $3$ $9$ $(1,3,8,7,9,5,4,6,2)$ $9$ $3$ $9$ $(1,3,5,7,9,2,4,6,8)$ $9$ $3$ $9$ $(1,3,2,7,9,8,4,6,5)$ $3, 3, 3$ $1$ $3$ $(1,4,7)(2,5,8)(3,6,9)$ $3, 3, 3$ $1$ $3$ $(1,7,4)(2,8,5)(3,9,6)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $27=3^{3}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: $2$ Label: 27.4 magma: IdentifyGroup(G);
 Character table:  3 3 2 2 2 2 2 2 2 2 3 3 1a 3a 3b 9a 9b 9c 9d 9e 9f 3c 3d 2P 1a 3b 3a 9e 9d 9f 9b 9a 9c 3d 3c 3P 1a 1a 1a 3c 3c 3c 3d 3d 3d 1a 1a 5P 1a 3b 3a 9e 9d 9f 9b 9a 9c 3d 3c 7P 1a 3a 3b 9a 9b 9c 9d 9e 9f 3c 3d X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 A A A /A /A /A 1 1 X.3 1 1 1 /A /A /A A A A 1 1 X.4 1 A /A 1 A /A /A 1 A 1 1 X.5 1 /A A 1 /A A A 1 /A 1 1 X.6 1 A /A A /A 1 A /A 1 1 1 X.7 1 /A A /A A 1 /A A 1 1 1 X.8 1 A /A /A 1 A 1 A /A 1 1 X.9 1 /A A A 1 /A 1 /A A 1 1 X.10 3 . . . . . . . . B /B X.11 3 . . . . . . . . /B B A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 3*E(3)^2 = (-3-3*Sqrt(-3))/2 = -3-3b3 

magma: CharacterTable(G);