# Properties

 Label 9T14 Degree $9$ Order $72$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_3^2:Q_8$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(9, 14);

## Group action invariants

 Degree $n$: $9$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $14$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_3^2:Q_8$ CHM label: $M(9)=E(9):Q_{8}$ Parity: $1$ magma: IsEven(G); Primitive: yes magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,2,9)(3,4,5)(6,7,8), (1,8,2,4)(3,5,6,7), (1,6,2,3)(4,7,8,5), (1,4,7)(2,5,8)(3,6,9) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $Q_8$

Resolvents shown for degrees $\leq 47$

Degree 3: None

## Low degree siblings

12T47, 18T35 x 3, 24T82, 36T55

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 $1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 4, 1$ $18$ $4$ $(2,3,9,8)(4,5,7,6)$ $4, 4, 1$ $18$ $4$ $(2,4,9,7)(3,6,8,5)$ $4, 4, 1$ $18$ $4$ $(2,5,9,6)(3,4,8,7)$ $2, 2, 2, 2, 1$ $9$ $2$ $(2,9)(3,8)(4,7)(5,6)$ $3, 3, 3$ $8$ $3$ $(1,2,9)(3,4,5)(6,7,8)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $72=2^{3} \cdot 3^{2}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 72.41 magma: IdentifyGroup(G);
 Character table:  2 3 2 2 2 3 . 3 2 . . . . 2 1a 4a 4b 4c 2a 3a 2P 1a 2a 2a 2a 1a 3a 3P 1a 4a 4b 4c 2a 1a X.1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 1 X.3 1 -1 1 -1 1 1 X.4 1 1 -1 -1 1 1 X.5 2 . . . -2 2 X.6 8 . . . . -1 

magma: CharacterTable(G);