# Properties

 Label 9T1 Degree $9$ Order $9$ Cyclic yes Abelian yes Solvable yes Primitive no $p$-group yes Group: $C_9$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(9, 1);

## Group action invariants

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

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

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

 Label Cycle Type Size Order Index Representative 1A $1^{9}$ $1$ $1$ $0$ $()$ 3A1 $3^{3}$ $1$ $3$ $6$ $(1,4,7)(2,5,8)(3,6,9)$ 3A-1 $3^{3}$ $1$ $3$ $6$ $(1,7,4)(2,8,5)(3,9,6)$ 9A1 $9$ $1$ $9$ $8$ $(1,2,3,4,5,6,7,8,9)$ 9A-1 $9$ $1$ $9$ $8$ $(1,5,9,4,8,3,7,2,6)$ 9A2 $9$ $1$ $9$ $8$ $(1,6,2,7,3,8,4,9,5)$ 9A-2 $9$ $1$ $9$ $8$ $(1,8,6,4,2,9,7,5,3)$ 9A4 $9$ $1$ $9$ $8$ $(1,9,8,7,6,5,4,3,2)$ 9A-4 $9$ $1$ $9$ $8$ $(1,3,5,7,9,2,4,6,8)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/6$

## Group invariants

 Order: $9=3^{2}$ magma: Order(G); Cyclic: yes magma: IsCyclic(G); Abelian: yes magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: $1$ Label: 9.1 magma: IdentifyGroup(G); Character table:

 1A 3A1 3A-1 9A1 9A-1 9A2 9A-2 9A4 9A-4 Size 1 1 1 1 1 1 1 1 1 3 P 1A 3A-1 3A1 9A2 9A-1 9A1 9A-4 9A-2 9A4 Type 9.1.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 9.1.1b1 C $1$ $1$ $1$ $ζ3−1$ $ζ3$ $ζ3$ $ζ3−1$ $ζ3−1$ $ζ3$ 9.1.1b2 C $1$ $1$ $1$ $ζ3$ $ζ3−1$ $ζ3−1$ $ζ3$ $ζ3$ $ζ3−1$ 9.1.1c1 C $1$ $ζ9−3$ $ζ93$ $ζ9−4$ $ζ94$ $ζ9$ $ζ9−1$ $ζ92$ $ζ9−2$ 9.1.1c2 C $1$ $ζ93$ $ζ9−3$ $ζ94$ $ζ9−4$ $ζ9−1$ $ζ9$ $ζ9−2$ $ζ92$ 9.1.1c3 C $1$ $ζ9−3$ $ζ93$ $ζ92$ $ζ9−2$ $ζ94$ $ζ9−4$ $ζ9−1$ $ζ9$ 9.1.1c4 C $1$ $ζ93$ $ζ9−3$ $ζ9−2$ $ζ92$ $ζ9−4$ $ζ94$ $ζ9$ $ζ9−1$ 9.1.1c5 C $1$ $ζ9−3$ $ζ93$ $ζ9−1$ $ζ9$ $ζ9−2$ $ζ92$ $ζ9−4$ $ζ94$ 9.1.1c6 C $1$ $ζ93$ $ζ9−3$ $ζ9$ $ζ9−1$ $ζ92$ $ζ9−2$ $ζ94$ $ζ9−4$

magma: CharacterTable(G);