# Properties

 Label 9T26 Degree $9$ Order $432$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $((C_3^2:Q_8):C_3):C_2$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(9, 26);

## Group action invariants

 Degree $n$: $9$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $26$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $((C_3^2:Q_8):C_3):C_2$ CHM label: $E(9):2S_{4}$ 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: (1,6,4,5,2,3,8,7), (1,2,9)(3,4,5)(6,7,8), (3,4,5)(6,8,7), (1,4,7)(2,5,8)(3,6,9) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$24$:  $S_4$
$48$:  $\textrm{GL(2,3)}$

Resolvents shown for degrees $\leq 47$

Degree 3: None

## Low degree siblings

12T157, 18T157, 24T1325, 24T1326, 24T1327, 24T1334, 27T139, 36T689, 36T709

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 $1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 1, 1, 1$ $24$ $3$ $(3,4,5)(6,8,7)$ $2, 2, 2, 1, 1, 1$ $36$ $2$ $(3,6)(4,7)(5,8)$ $8, 1$ $54$ $8$ $(2,3,4,6,9,8,7,5)$ $6, 2, 1$ $72$ $6$ $(2,3,5,9,8,6)(4,7)$ $8, 1$ $54$ $8$ $(2,3,6,7,9,8,5,4)$ $4, 4, 1$ $54$ $4$ $(2,3,9,8)(4,5,7,6)$ $2, 2, 2, 2, 1$ $9$ $2$ $(2,9)(3,8)(4,7)(5,6)$ $3, 3, 3$ $48$ $3$ $(1,2,3)(4,5,6)(7,8,9)$ $6, 3$ $72$ $6$ $(1,2,3,4,8,6)(5,9,7)$ $3, 3, 3$ $8$ $3$ $(1,2,9)(3,4,5)(6,7,8)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $432=2^{4} \cdot 3^{3}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 432.734 magma: IdentifyGroup(G);
 Character table:  2 4 1 2 3 1 3 3 4 . 1 1 3 3 2 1 . 1 . . 1 2 1 3 1a 3a 2a 8a 6a 8b 4a 2b 3b 6b 3c 2P 1a 3a 1a 4a 3a 4a 2b 1a 3b 3c 3c 3P 1a 1a 2a 8a 2b 8b 4a 2b 1a 2a 1a 5P 1a 3a 2a 8b 6a 8a 4a 2b 3b 6b 3c 7P 1a 3a 2a 8b 6a 8a 4a 2b 3b 6b 3c X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 -1 1 1 1 -1 1 X.3 2 -1 . . -1 . 2 2 -1 . 2 X.4 2 -1 . A 1 -A . -2 -1 . 2 X.5 2 -1 . -A 1 A . -2 -1 . 2 X.6 3 . -1 1 . 1 -1 3 . -1 3 X.7 3 . 1 -1 . -1 -1 3 . 1 3 X.8 4 1 . . -1 . . -4 1 . 4 X.9 8 2 -2 . . . . . -1 1 -1 X.10 8 2 2 . . . . . -1 -1 -1 X.11 16 -2 . . . . . . 1 . -2 A = -E(8)-E(8)^3 = -Sqrt(-2) = -i2 

magma: CharacterTable(G);