↑

Properties

Label 9T24
Degree $9$
Order $324$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $((C_3^3:C_3):C_2):C_2$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(9, 24);
 

Group action invariants

Degree $n$:  $9$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $24$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $((C_3^3:C_3):C_2):C_2$
CHM label:   $[3^{3}:2]S(3)$
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), (3,6)(4,7)(5,8), (1,2)(4,5)(7,8), (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
$4$:  $C_2^2$
$6$:  $S_3$ x 2
$12$:  $D_{6}$ x 2
$36$:  $S_3^2$
$108$:  $C_3^2 : D_{6} $

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Low degree siblings

9T24 x 2, 18T129 x 3, 18T136 x 3, 18T137 x 3, 27T121, 27T128 x 3, 27T129, 36T502 x 3

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 1, 1, 1, 1, 1, 1 $ $6$ $3$ $(6,7,8)$
$ 3, 3, 1, 1, 1 $ $6$ $3$ $(3,4,5)(6,7,8)$
$ 3, 3, 1, 1, 1 $ $6$ $3$ $(3,4,5)(6,8,7)$
$ 2, 2, 2, 1, 1, 1 $ $9$ $2$ $(3,6)(4,7)(5,8)$
$ 6, 1, 1, 1 $ $18$ $6$ $(3,6,4,7,5,8)$
$ 2, 2, 2, 1, 1, 1 $ $27$ $2$ $(2,9)(4,5)(7,8)$
$ 2, 2, 2, 2, 1 $ $27$ $2$ $(2,9)(3,6)(4,8)(5,7)$
$ 6, 2, 1 $ $54$ $6$ $(2,9)(3,6,4,8,5,7)$
$ 3, 3, 3 $ $2$ $3$ $(1,2,9)(3,4,5)(6,7,8)$
$ 3, 3, 3 $ $6$ $3$ $(1,2,9)(3,4,5)(6,8,7)$
$ 3, 2, 2, 2 $ $18$ $6$ $(1,2,9)(3,6)(4,7)(5,8)$
$ 6, 3 $ $18$ $6$ $(1,2,9)(3,6,4,7,5,8)$
$ 6, 3 $ $18$ $6$ $(1,2,9)(3,6,5,8,4,7)$
$ 3, 3, 3 $ $18$ $3$ $(1,3,6)(2,4,7)(5,8,9)$
$ 9 $ $36$ $9$ $(1,3,6,2,4,7,9,5,8)$
$ 6, 3 $ $54$ $6$ $(1,3,6)(2,5,7,9,4,8)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $324=2^{2} \cdot 3^{4}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  324.39
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 3D 3E 3F 6A 6B 6C 6D 6E 6F 9A
Size 1 9 27 27 2 6 6 6 6 18 18 18 18 18 54 54 36
2 P 1A 1A 1A 1A 3A 3C 3B 3E 3D 3F 3B 3D 3E 3A 3C 3F 9A
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2B 2C 3A
Type
324.39.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
324.39.1b R 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 1 1
324.39.1c R 1 −1 1 −1 1 1 1 1 1 1 −1 −1 −1 −1 1 −1 1
324.39.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 −1 −1 1
324.39.2a R 2 0 0 2 2 2 2 2 2 −1 0 0 0 0 0 −1 −1
324.39.2b R 2 2 0 0 2 −1 2 −1 −1 2 −1 −1 −1 2 0 0 −1
324.39.2c R 2 −2 0 0 2 −1 2 −1 −1 2 1 1 1 −2 0 0 −1
324.39.2d R 2 0 0 −2 2 2 2 2 2 −1 0 0 0 0 0 1 −1
324.39.4a R 4 0 0 0 4 −2 4 −2 −2 −2 0 0 0 0 0 0 1
324.39.6a R 6 0 2 0 6 0 −3 0 0 0 0 0 0 0 −1 0 0
324.39.6b R 6 2 0 0 −3 −3 0 3 0 0 −1 −1 2 −1 0 0 0
324.39.6c R 6 2 0 0 −3 0 0 −3 3 0 2 −1 −1 −1 0 0 0
324.39.6d R 6 2 0 0 −3 3 0 0 −3 0 −1 2 −1 −1 0 0 0
324.39.6e R 6 0 −2 0 6 0 −3 0 0 0 0 0 0 0 1 0 0
324.39.6f R 6 −2 0 0 −3 −3 0 3 0 0 1 1 −2 1 0 0 0
324.39.6g R 6 −2 0 0 −3 0 0 −3 3 0 −2 1 1 1 0 0 0
324.39.6h R 6 −2 0 0 −3 3 0 0 −3 0 1 −2 1 1 0 0 0

magma: CharacterTable(G);