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Properties

Label 9T22
Degree $9$
Order $162$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $(C_3^3:C_3):C_2$

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Show commands: Magma

magma: G := TransitiveGroup(9, 22);
 

Group action invariants

Degree $n$:  $9$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $22$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $(C_3^3:C_3):C_2$
CHM label:   $[3^{3}:2]3$
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), (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)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$
$54$:  $C_3^2 : C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Low degree siblings

9T22 x 2, 18T85 x 3, 27T53 x 3, 27T62, 27T63

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 $ $27$ $2$ $(2,9)(4,5)(7,8)$
$ 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, 3, 3 $ $9$ $3$ $(1,3,6)(2,4,7)(5,8,9)$
$ 9 $ $18$ $9$ $(1,3,6,2,4,7,9,5,8)$
$ 6, 3 $ $27$ $6$ $(1,3,6)(2,5,7,9,4,8)$
$ 3, 3, 3 $ $9$ $3$ $(1,6,3)(2,7,4)(5,9,8)$
$ 9 $ $18$ $9$ $(1,6,4,2,7,5,9,8,3)$
$ 6, 3 $ $27$ $6$ $(1,6,3)(2,8,4,9,7,5)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $162=2 \cdot 3^{4}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  162.11
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 3B 3C 3D 3E 3F1 3F-1 6A1 6A-1 9A1 9A-1
Size 1 27 2 6 6 6 6 9 9 27 27 18 18
2 P 1A 1A 3A 3D 3B 3C 3E 3F-1 3F1 3F1 3F-1 9A-1 9A1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 2A 2A 3A 3A
Type
162.11.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
162.11.1b R 1 −1 1 1 1 1 1 1 1 −1 −1 1 1
162.11.1c1 C 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
162.11.1c2 C 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
162.11.1d1 C 1 −1 1 1 1 1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3
162.11.1d2 C 1 −1 1 1 1 1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1
162.11.2a R 2 0 2 −1 −1 −1 2 2 2 0 0 −1 −1
162.11.2b1 C 2 0 2 −1 −1 −1 2 2ζ3−1 2ζ3 0 0 −ζ3−1 −ζ3
162.11.2b2 C 2 0 2 −1 −1 −1 2 2ζ3 2ζ3−1 0 0 −ζ3 −ζ3−1
162.11.6a R 6 0 6 0 0 0 −3 0 0 0 0 0 0
162.11.6b R 6 0 −3 −3 0 3 0 0 0 0 0 0 0
162.11.6c R 6 0 −3 0 3 −3 0 0 0 0 0 0 0
162.11.6d R 6 0 −3 3 −3 0 0 0 0 0 0 0 0

magma: CharacterTable(G);