Properties

Label 18T377
18T377 1 5 1->5 8 1->8 2 6 2->6 16 2->16 3 3->2 3->16 4 4->6 10 4->10 15 5->15 5->15 6->3 13 6->13 7 17 7->17 18 7->18 8->4 11 8->11 9 9->2 9->10 10->7 10->18 11->1 14 11->14 12 12->11 12->13 13->7 13->12 14->3 14->4 15->9 15->9 16->1 16->8 17->5 17->14 18->12 18->17
Degree $18$
Order $2448$
Cyclic no
Abelian no
Solvable no
Transitivity $2$
Primitive yes
$p$-group no
Group: $\PSL(2,17)$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(18, 377);
 
Copy content sage:G = TransitiveGroup(18, 377)
 
Copy content oscar:G = transitive_group(18, 377)
 
Copy content gap:G := TransitiveGroup(18, 377);
 

Group invariants

Abstract group:  $\PSL(2,17)$
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:  $2448=2^{4} \cdot 3^{2} \cdot 17$
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 $n$:  $18$
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 $t$:  $377$
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);
 
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:  2
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(18).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(18), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(18), G));
 
Generators:  $(1,5,15,9,2,6,13,12,11)(3,16,8,4,10,7,18,17,14)$, $(1,8,11,14,4,6,3,2,16)(5,15,9,10,18,12,13,7,17)$
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 2: None

Degree 3: None

Degree 6: None

Degree 9: None

Low degree siblings

36T3474

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{18}$ $1$ $1$ $0$ $()$
2A $2^{8},1^{2}$ $153$ $2$ $8$ $( 1, 9)( 2,13)( 4,15)( 5,12)( 6,10)( 7,11)( 8,16)(17,18)$
3A $3^{6}$ $272$ $3$ $12$ $( 1, 4, 9)( 2, 5,14)( 3,15, 8)( 6,12,10)( 7,18,16)(11,17,13)$
4A $4^{4},1^{2}$ $306$ $4$ $12$ $( 1,16, 9, 8)( 2, 6,13,10)( 4,11,15, 7)( 5,18,12,17)$
8A1 $8^{2},1^{2}$ $306$ $8$ $14$ $( 1,17,16, 5, 9,18, 8,12)( 2,11, 6,15,13, 7,10, 4)$
8A3 $8^{2},1^{2}$ $306$ $8$ $14$ $( 1, 5, 8,17, 9,12,16,18)( 2,15,10,11,13, 4, 6, 7)$
9A1 $9^{2}$ $272$ $9$ $16$ $( 1, 2,18, 4, 5,16, 9,14, 7)( 3, 6,13,15,12,11, 8,10,17)$
9A2 $9^{2}$ $272$ $9$ $16$ $( 1,18, 5, 9, 7, 2, 4,16,14)( 3,13,12, 8,17, 6,15,11,10)$
9A4 $9^{2}$ $272$ $9$ $16$ $( 1, 5, 7, 4,14,18, 9, 2,16)( 3,12,17,15,10,13, 8, 6,11)$
17A1 $17,1$ $144$ $17$ $16$ $( 1, 7, 3,15,11,17, 5, 4, 8,16, 6,18,12, 2,10,14,13)$
17A3 $17,1$ $144$ $17$ $16$ $( 1,15, 5,16,12,14, 7,11, 4, 6, 2,13, 3,17, 8,18,10)$

Malle's constant $a(G)$:     $1/8$

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 3A 4A 8A1 8A3 9A1 9A2 9A4 17A1 17A3
Size 1 153 272 306 306 306 272 272 272 144 144
2 P 1A 1A 3A 2A 4A 4A 9A2 9A4 9A1 17A1 17A3
3 P 1A 2A 1A 4A 8A3 8A1 3A 3A 3A 17A3 17A1
17 P 1A 2A 3A 4A 8A1 8A3 9A1 9A2 9A4 1A 1A
Type
2448.a.1a R 1 1 1 1 1 1 1 1 1 1 1
2448.a.9a1 R 9 1 0 1 −1 −1 0 0 0 ζ17−7+ζ17−6+ζ17−5+ζ17−3+1+ζ173+ζ175+ζ176+ζ177 −ζ17−7−ζ17−6−ζ17−5−ζ17−3−ζ173−ζ175−ζ176−ζ177
2448.a.9a2 R 9 1 0 1 −1 −1 0 0 0 −ζ17−7−ζ17−6−ζ17−5−ζ17−3−ζ173−ζ175−ζ176−ζ177 ζ17−7+ζ17−6+ζ17−5+ζ17−3+1+ζ173+ζ175+ζ176+ζ177
2448.a.16a R 16 0 −2 0 0 0 1 1 1 −1 −1
2448.a.16b1 R 16 0 1 0 0 0 −ζ9−1−ζ9 −ζ9−2−ζ92 −ζ9−4−ζ94 −1 −1
2448.a.16b2 R 16 0 1 0 0 0 −ζ9−2−ζ92 −ζ9−4−ζ94 −ζ9−1−ζ9 −1 −1
2448.a.16b3 R 16 0 1 0 0 0 −ζ9−4−ζ94 −ζ9−1−ζ9 −ζ9−2−ζ92 −1 −1
2448.a.17a R 17 1 −1 1 1 1 −1 −1 −1 0 0
2448.a.18a R 18 2 0 −2 0 0 0 0 0 1 1
2448.a.18b1 R 18 −2 0 0 −ζ8−1−ζ8 ζ8−1+ζ8 0 0 0 1 1
2448.a.18b2 R 18 −2 0 0 ζ8−1+ζ8 −ζ8−1−ζ8 0 0 0 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

$f_{ 1 } =$ $\left(x^{3} - 7 x^{2} + 5 x - 2\right)^{6} - \left(x^{17} - 17 x^{15} + 34 x^{14} + 85 x^{13} - 408 x^{12} +289 x^{11} + 1190 x^{10} - 2907 x^{9} + 1462 x^{8} + 3281 x^{7} - 5780 x^{6} +3196 x^{5} + 238 x^{4} - 646 x^{3} - 68 x^{2} + 120 x - 16\right) \left(108/\left(17 t^{2}-1\right)\right) + \left(108/\left(17 t^{2}-1\right)\right)^{2}$ Copy content Toggle raw display