# Properties

 Label 9T30 Degree $9$ Order $648$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(9, 30);

## Group action invariants

 Degree $n$: $9$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $30$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ CHM label: $1/2[S(3)^{3}]S(3)$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,2,9), (4,5)(7,8), (1,4,7)(2,5,8)(3,6,9), (1,2)(3,6)(4,7)(5,8) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$24$:  $S_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $S_3$

## Low degree siblings

12T177 x 2, 12T178, 18T217, 18T218, 18T221, 18T222, 24T1529 x 2, 24T1530, 27T211, 27T216, 36T1121, 36T1122, 36T1123, 36T1124, 36T1125, 36T1130, 36T1140, 36T1239 x 2, 36T1240

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 $1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 1, 1, 1, 1, 1, 1$ $6$ $3$ $(1,2,9)$ $3, 3, 1, 1, 1$ $12$ $3$ $(1,2,9)(3,4,5)$ $3, 3, 3$ $8$ $3$ $(1,2,9)(3,4,5)(6,7,8)$ $2, 2, 1, 1, 1, 1, 1$ $27$ $2$ $(4,5)(7,8)$ $3, 2, 2, 1, 1$ $54$ $6$ $(1,2,9)(4,5)(7,8)$ $3, 3, 3$ $72$ $3$ $(1,4,7)(2,5,8)(3,6,9)$ $9$ $72$ $9$ $(1,4,7,2,5,8,9,3,6)$ $9$ $72$ $9$ $(1,4,7,9,3,6,2,5,8)$ $2, 2, 2, 2, 1$ $54$ $2$ $(2,9)(3,6)(4,7)(5,8)$ $6, 2, 1$ $108$ $6$ $(2,9)(3,6,4,7,5,8)$ $4, 2, 1, 1, 1$ $54$ $4$ $(3,6)(4,8,5,7)$ $4, 3, 2$ $54$ $12$ $(1,2,9)(3,6)(4,8,5,7)$ $4, 3, 2$ $54$ $12$ $(1,9,2)(3,6)(4,8,5,7)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $648=2^{3} \cdot 3^{4}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 648.703 magma: IdentifyGroup(G);
 Character table:  2 3 2 1 . 3 2 . . . 2 1 2 2 2 3 4 3 3 4 1 1 2 2 2 1 1 1 1 1 1a 3a 3b 3c 2a 6a 3d 9a 9b 2b 6b 4a 12a 12b 2P 1a 3a 3b 3c 1a 3a 3d 9a 9b 1a 3b 2a 6a 6a 3P 1a 1a 1a 1a 2a 2a 1a 3c 3c 2b 2b 4a 4a 4a 5P 1a 3a 3b 3c 2a 6a 3d 9a 9b 2b 6b 4a 12b 12a 7P 1a 3a 3b 3c 2a 6a 3d 9a 9b 2b 6b 4a 12b 12a 11P 1a 3a 3b 3c 2a 6a 3d 9a 9b 2b 6b 4a 12a 12b X.1 1 1 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 -1 -1 -1 X.3 2 2 2 2 2 2 -1 -1 -1 . . . . . X.4 3 3 3 3 -1 -1 . . . -1 -1 1 1 1 X.5 3 3 3 3 -1 -1 . . . 1 1 -1 -1 -1 X.6 6 3 . -3 2 -1 . . . . . -2 1 1 X.7 6 3 . -3 2 -1 . . . . . 2 -1 -1 X.8 6 3 . -3 -2 1 . . . . . . A -A X.9 6 3 . -3 -2 1 . . . . . . -A A X.10 8 -4 2 -1 . . 2 -1 -1 . . . . . X.11 8 -4 2 -1 . . -1 -1 2 . . . . . X.12 8 -4 2 -1 . . -1 2 -1 . . . . . X.13 12 . -3 3 . . . . . -2 1 . . . X.14 12 . -3 3 . . . . . 2 -1 . . . A = -E(12)^7+E(12)^11 = Sqrt(3) = r3 

magma: CharacterTable(G);