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Properties

Label 9T33
9T33 1 2 1->2 3 2->3 3->1 4 3->4 5 4->5 6 5->6 7 6->7 8 7->8 9 8->9 9->3
Degree $9$
Order $181440$
Cyclic no
Abelian no
Solvable no
Transitivity $7$
Primitive yes
$p$-group no
Group: $A_9$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(9, 33);
 
Copy content sage:G = TransitiveGroup(9, 33)
 
Copy content oscar:G = transitive_group(9, 33)
 
Copy content gap:G := TransitiveGroup(9, 33);
 

Group invariants

Abstract group:  $A_9$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Copy content oscar:small_group_identification(G)
 
Copy content gap:IdGroup(G);
 
Order:  $181440=2^{6} \cdot 3^{4} \cdot 5 \cdot 7$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Copy content gap:Order(G);
 
Cyclic:  no
Copy content comment:Determine if group is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content oscar:is_cyclic(G)
 
Copy content gap:IsCyclic(G);
 
Abelian:  no
Copy content comment:Determine if group is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content oscar:is_abelian(G)
 
Copy content gap:IsAbelian(G);
 
Solvable:  no
Copy content comment:Determine if group is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content sage:G.is_solvable()
 
Copy content oscar:is_solvable(G)
 
Copy content gap:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content comment:Nilpotency class
 
Copy content magma:NilpotencyClass(G);
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 
Copy content gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;
 

Group action invariants

Degree $n$:  $9$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Copy content gap:NrMovedPoints(G);
 
Transitive number $t$:  $33$
Copy content comment:Transitive number
 
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Copy content sage:G.transitive_number()
 
Copy content oscar:transitive_group_identification(G)[2]
 
Copy content gap:TransitiveIdentification(G);
 
CHM label:   $A9$
Parity:  $1$
Copy content comment:Parity
 
Copy content magma:IsEven(G);
 
Copy content sage:all(g.SignPerm() == 1 for g in libgap(G).GeneratorsOfGroup())
 
Copy content oscar:is_even(G)
 
Copy content gap:ForAll(GeneratorsOfGroup(G), g -> SignPerm(g) = 1);
 
Transitivity:  7
Primitive:  yes
Copy content comment:Determine if group is primitive
 
Copy content magma:IsPrimitive(G);
 
Copy content sage:G.is_primitive()
 
Copy content oscar:is_primitive(G)
 
Copy content gap:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(9).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(9), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(9), G));
 
Generators:  $(1,2,3)$, $(3,4,5,6,7,8,9)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 
Copy content gap:GeneratorsOfGroup(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Low degree siblings

36T23796

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 $2^{2},1^{5}$ $378$ $2$ $2$ $(1,6)(3,5)$
2B $2^{4},1$ $945$ $2$ $4$ $(1,2)(3,8)(4,6)(7,9)$
3A $3,1^{6}$ $168$ $3$ $2$ $(6,8,9)$
3B $3^{3}$ $2240$ $3$ $6$ $(1,9,5)(2,6,7)(3,4,8)$
3C $3^{2},1^{3}$ $3360$ $3$ $4$ $(1,3,4)(2,8,6)$
4A $4,2,1^{3}$ $7560$ $4$ $4$ $(1,7,9,8)(4,6)$
4B $4^{2},1$ $11340$ $4$ $6$ $(1,8,6,4)(2,5,3,9)$
5A $5,1^{4}$ $3024$ $5$ $4$ $(1,4,7,5,2)$
6A $3,2^{2},1^{2}$ $7560$ $6$ $4$ $(1,9)(2,5,3)(7,8)$
6B $6,2,1$ $30240$ $6$ $6$ $(1,6,3,2,4,8)(7,9)$
7A $7,1^{2}$ $25920$ $7$ $6$ $(2,6,9,5,8,4,7)$
9A $9$ $20160$ $9$ $8$ $(1,8,7,9,3,2,5,4,6)$
9B $9$ $20160$ $9$ $8$ $(1,3,6,5,4,8,9,2,7)$
10A $5,2^{2}$ $9072$ $10$ $6$ $(1,6)(2,9,7,8,4)(3,5)$
12A $4,3,2$ $15120$ $12$ $6$ $(1,8,9,7)(2,3,5)(4,6)$
15A1 $5,3,1$ $12096$ $15$ $6$ $(1,7,2,4,5)(6,9,8)$
15A-1 $5,3,1$ $12096$ $15$ $6$ $(1,5,4,2,7)(6,8,9)$

Malle's constant $a(G)$:     $1/2$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 
Copy content gap:ConjugacyClasses(G);
 

Character table

1A 2A 2B 3A 3B 3C 4A 4B 5A 6A 6B 7A 9A 9B 10A 12A 15A1 15A-1
Size 1 378 945 168 2240 3360 7560 11340 3024 7560 30240 25920 20160 20160 9072 15120 12096 12096
2 P 1A 1A 1A 3A 3B 3C 2A 2B 5A 3A 3C 7A 9A 9B 5A 6A 15A1 15A-1
3 P 1A 2A 2B 1A 1A 1A 4A 4B 5A 2A 2B 7A 3B 3B 10A 4A 5A 5A
5 P 1A 2A 2B 3A 3B 3C 4A 4B 1A 6A 6B 7A 9A 9B 2A 12A 3A 3A
7 P 1A 2A 2B 3A 3B 3C 4A 4B 5A 6A 6B 1A 9A 9B 10A 12A 15A-1 15A1
Type
181440.b.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
181440.b.8a R 8 4 0 5 −1 2 2 0 3 1 0 1 −1 −1 −1 −1 0 0
181440.b.21a1 C 21 1 −3 −3 3 0 −1 1 1 1 0 0 0 0 1 −1 1−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 −2+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157
181440.b.21a2 C 21 1 −3 −3 3 0 −1 1 1 1 0 0 0 0 1 −1 −2+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 1−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157
181440.b.27a R 27 7 3 9 0 0 1 −1 2 1 0 −1 0 0 2 1 −1 −1
181440.b.28a R 28 4 −4 10 1 1 0 0 3 −2 −1 0 1 1 −1 0 0 0
181440.b.35a R 35 −5 3 5 −1 2 −1 −1 0 1 0 0 −1 2 0 −1 0 0
181440.b.35b R 35 −5 3 5 −1 2 −1 −1 0 1 0 0 2 −1 0 −1 0 0
181440.b.42a R 42 6 2 0 −3 3 0 2 −3 0 −1 0 0 0 1 0 0 0
181440.b.48a R 48 8 0 6 3 0 0 0 −2 2 0 −1 0 0 −2 0 1 1
181440.b.56a R 56 −4 0 11 2 2 −2 0 1 −1 0 0 −1 −1 1 1 1 1
181440.b.84a R 84 4 4 −6 3 3 0 0 −1 −2 1 0 0 0 −1 0 −1 −1
181440.b.105a R 105 5 1 15 −3 −3 −1 1 0 −1 1 0 0 0 0 −1 0 0
181440.b.120a R 120 0 8 0 3 −3 0 0 0 0 −1 1 0 0 0 0 0 0
181440.b.162a R 162 6 −6 0 0 0 0 −2 −3 0 0 1 0 0 1 0 0 0
181440.b.168a R 168 4 0 −15 −3 0 −2 0 3 1 0 0 0 0 −1 1 0 0
181440.b.189a R 189 −11 −3 9 0 0 1 1 −1 1 0 0 0 0 −1 1 −1 −1
181440.b.216a R 216 −4 0 −9 0 0 2 0 1 −1 0 −1 0 0 1 −1 1 1

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 
Copy content gap:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{9} - 36 x^{5} + 900 x + t$ Copy content Toggle raw display