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Properties

Label 9T21
9T21 1 2 1->2 1->2 4 1->4 5 2->5 9 2->9 3 6 3->6 3->6 7 4->7 8 4->8 5->7 5->8 6->9 7->1 8->2 9->1 9->3
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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Copy content magma:G := TransitiveGroup(9, 21);
 

Group invariants

Abstract group:  $(C_3^3:C_3):C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $162=2 \cdot 3^{4}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $9$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $21$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $1/2.[3^{3}:2]S(3)$
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,2,9)$, $(1,2)(3,6)(4,8)(5,7)$, $(1,4,7)(2,5,8)(3,6,9)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$ x 4
$18$:  $C_3^2:C_2$
$54$:  $(C_3^2:C_3):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Low degree siblings

9T21 x 2, 18T88 x 3, 27T51 x 3, 27T52, 27T67

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{9}$ $1$ $1$ $0$ $()$
2A $2^{4},1$ $27$ $2$ $4$ $(1,5)(2,4)(3,9)(7,8)$
3A $3^{3}$ $2$ $3$ $6$ $(1,9,2)(3,5,4)(6,8,7)$
3B1 $3^{2},1^{3}$ $3$ $3$ $4$ $(1,9,2)(3,4,5)$
3B-1 $3^{2},1^{3}$ $3$ $3$ $4$ $(1,2,9)(3,5,4)$
3C $3^{2},1^{3}$ $6$ $3$ $4$ $(1,2,9)(6,7,8)$
3D $3,1^{6}$ $6$ $3$ $2$ $(3,5,4)$
3E $3^{3}$ $6$ $3$ $6$ $(1,9,2)(3,4,5)(6,8,7)$
3F $3^{3}$ $18$ $3$ $6$ $(1,4,7)(2,5,8)(3,6,9)$
6A1 $6,2,1$ $27$ $6$ $6$ $(1,4,9,5,2,3)(7,8)$
6A-1 $6,2,1$ $27$ $6$ $6$ $(1,3,2,5,9,4)(7,8)$
9A $9$ $18$ $9$ $8$ $(1,4,7,9,3,6,2,5,8)$
9B $9$ $18$ $9$ $8$ $(1,5,6,2,3,7,9,4,8)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 3A 3B1 3B-1 3C 3D 3E 3F 6A1 6A-1 9A 9B
Size 1 27 2 3 3 6 6 6 18 27 27 18 18
2 P 1A 1A 3A 3B-1 3B1 3C 3D 3E 3F 3B1 3B-1 9A 9B
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 2A 2A 3A 3A
Type
162.19.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
162.19.1b R 1 −1 1 1 1 1 1 1 1 −1 −1 1 1
162.19.2a R 2 0 2 2 2 −1 −1 −1 −1 0 0 −1 2
162.19.2b R 2 0 2 2 2 −1 −1 −1 −1 0 0 2 −1
162.19.2c R 2 0 2 2 2 −1 −1 −1 2 0 0 −1 −1
162.19.2d R 2 0 2 2 2 2 2 2 −1 0 0 −1 −1
162.19.3a1 C 3 1 3 3ζ3−1 3ζ3 0 0 0 0 ζ3 ζ3−1 0 0
162.19.3a2 C 3 1 3 3ζ3 3ζ3−1 0 0 0 0 ζ3−1 ζ3 0 0
162.19.3b1 C 3 −1 3 3ζ3−1 3ζ3 0 0 0 0 −ζ3 −ζ3−1 0 0
162.19.3b2 C 3 −1 3 3ζ3 3ζ3−1 0 0 0 0 −ζ3−1 −ζ3 0 0
162.19.6a R 6 0 −3 0 0 −3 0 3 0 0 0 0 0
162.19.6b R 6 0 −3 0 0 0 3 −3 0 0 0 0 0
162.19.6c R 6 0 −3 0 0 3 −3 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{9} - 9 x^{8} + \left(t + 22\right) x^{7} - 6 t x^{6} + \left(t - 32\right) x^{5} + \left(22 t - 2\right) x^{4} + \left(22 t + 10\right) x^{3} + \left(8 t - 4\right) x^{2} + \left(t - 5\right) x - 1$ Copy content Toggle raw display