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Properties

Label 9T1
9T1 1 2 1->2 3 2->3 4 3->4 5 4->5 6 5->6 7 6->7 8 7->8 9 8->9 9->1
Degree $9$
Order $9$
Cyclic yes
Abelian yes
Solvable yes
Primitive no
$p$-group yes
Group: $C_9$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(9, 1);
 

Group invariants

Abstract group:  $C_9$
Copy content magma:IdentifyGroup(G);
 
Order:  $9=3^{2}$
Copy content magma:Order(G);
 
Cyclic:  yes
Copy content magma:IsCyclic(G);
 
Abelian:  yes
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:  $1$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{9}$ $1$ $1$ $0$ $()$
3A1 $3^{3}$ $1$ $3$ $6$ $(1,4,7)(2,5,8)(3,6,9)$
3A-1 $3^{3}$ $1$ $3$ $6$ $(1,7,4)(2,8,5)(3,9,6)$
9A1 $9$ $1$ $9$ $8$ $(1,2,3,4,5,6,7,8,9)$
9A-1 $9$ $1$ $9$ $8$ $(1,9,8,7,6,5,4,3,2)$
9A2 $9$ $1$ $9$ $8$ $(1,3,5,7,9,2,4,6,8)$
9A-2 $9$ $1$ $9$ $8$ $(1,8,6,4,2,9,7,5,3)$
9A4 $9$ $1$ $9$ $8$ $(1,5,9,4,8,3,7,2,6)$
9A-4 $9$ $1$ $9$ $8$ $(1,6,2,7,3,8,4,9,5)$

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 3A1 3A-1 9A1 9A-1 9A2 9A-2 9A4 9A-4
Size 1 1 1 1 1 1 1 1 1
3 P 1A 3A-1 3A1 9A2 9A-2 9A4 9A-4 9A-1 9A1
Type
9.1.1a R 1 1 1 1 1 1 1 1 1
9.1.1b1 C 1 1 1 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3
9.1.1b2 C 1 1 1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1
9.1.1c1 C 1 ζ9−3 ζ93 ζ9−4 ζ92 ζ9−2 ζ9−1 ζ94 ζ9
9.1.1c2 C 1 ζ93 ζ9−3 ζ94 ζ9−2 ζ92 ζ9 ζ9−4 ζ9−1
9.1.1c3 C 1 ζ9−3 ζ93 ζ92 ζ9−1 ζ9 ζ9−4 ζ9−2 ζ94
9.1.1c4 C 1 ζ93 ζ9−3 ζ9−2 ζ9 ζ9−1 ζ94 ζ92 ζ9−4
9.1.1c5 C 1 ζ9−3 ζ93 ζ9−1 ζ9−4 ζ94 ζ92 ζ9 ζ9−2
9.1.1c6 C 1 ζ93 ζ9−3 ζ9 ζ94 ζ9−4 ζ9−2 ζ9−1 ζ92

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $t^{2} x^{9} + \left(-27 t^{6} - 7209 t^{4} - 2430 t^{2} - 81\right) x^{7} + \left(54 t^{8} + 14202 t^{6} - 52812 t^{4} - 19278 t^{2} - 648\right) x^{6} + \left(60264 t^{8} + 16104096 t^{6} + 9057096 t^{4} + 1405512 t^{2} + 40824\right) x^{5} + \left(-120528 t^{10} - 31869936 t^{8} + 72379008 t^{6} + 75384432 t^{4} + 17029440 t^{2} + 536544\right) x^{4} + \left(-48285072 t^{10} - 12913320672 t^{8} - 10008932832 t^{6} - 2361750048 t^{4} - 167545584 t^{2} - 3464208\right) x^{3} + \left(67254624 t^{12} + 17614272960 t^{10} - 85624304256 t^{8} - 76920814080 t^{6} - 24988277088 t^{4} - 3341386080 t^{2} - 94058496\right) x^{2} + \left(12478147200 t^{12} + 3340898758080 t^{10} + 3589241694336 t^{8} + 1102092215616 t^{6} + 56741767488 t^{4} - 14152024512 t^{2} - 559312128\right) x + \left(-2267481600 t^{14} - 456714184320 t^{12} + 39574425602304 t^{10} + 33326774047872 t^{8} + 9287723233152 t^{6} + 841757754240 t^{4} - 8833287168 t^{2} - 1088391168\right)$ Copy content Toggle raw display