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Properties

Label 17T8
17T8 1 6 1->6 12 1->12 16 1->16 2 3 2->3 2->3 4 2->4 15 2->15 8 3->8 9 3->9 4->2 5 4->5 4->9 13 4->13 5->1 5->4 7 5->7 6->7 6->8 6->9 6->13 7->5 11 7->11 8->1 8->9 10 8->10 9->7 10->11 14 10->14 10->14 10->14 11->2 11->8 11->13 12->3 12->7 12->13 12->15 13->5 13->11 14->3 14->12 14->15 15->10 17 16->17
Degree $17$
Order $16320$
Cyclic no
Abelian no
Solvable no
Transitivity $3$
Primitive yes
$p$-group no
Group: $\PSL(2,16):C_4$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(17, 8);
 
Copy content sage:G = TransitiveGroup(17, 8)
 
Copy content oscar:G = transitive_group(17, 8)
 
Copy content gap:G := TransitiveGroup(17, 8);
 

Group invariants

Abstract group:  $\PSL(2,16):C_4$
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:  $16320=2^{6} \cdot 3 \cdot 5 \cdot 17$
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 $n$:  $17$
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 $t$:  $8$
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:  3
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(17).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(17), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(17), G));
 
Generators:  $(2,3)(4,9)(5,7)(6,8)(10,14)(11,13)(12,15)(16,17)$, $(1,12,7,5)(2,4,13,11)(3,8,10,14)(6,9)$, $(1,16)(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)$, $(1,6,13,5,4,2,15,10,14,12,3,9,7,11,8)$
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

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

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
$1^{17}$ $1$ $1$ $0$ $()$
$2^{6},1^{5}$ $68$ $2$ $6$ $( 1, 8)( 2,12)( 3,11)( 4, 6)( 7,17)(13,16)$
$5^{3},1^{2}$ $544$ $5$ $12$ $( 1,12,11,17, 6)( 2, 3, 7, 4, 8)( 5, 9,10,14,15)$
$10,5,2$ $1632$ $10$ $14$ $( 1, 7,12, 4,11, 8,17, 2, 6, 3)( 5,14, 9,15,10)(13,16)$
$3^{5},1^{2}$ $272$ $3$ $10$ $( 1,15, 8)( 2,12, 5)( 3,11, 9)( 4, 6,14)( 7,17,10)$
$15,1^{2}$ $1088$ $15$ $14$ $( 1, 2, 9,17, 4,15,12, 3,10, 6, 8, 5,11, 7,14)$
$4^{3},2,1^{3}$ $680$ $4$ $10$ $( 1,12, 6,17)( 2, 4, 7, 8)( 5,14,10,15)(13,16)$
$4^{3},2,1^{3}$ $680$ $4$ $10$ $( 1,17, 6,12)( 2, 8, 7, 4)( 5,15,10,14)(13,16)$
$2^{8},1$ $255$ $2$ $8$ $( 1,14)( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,16)$
$4^{4},1$ $1020$ $4$ $12$ $( 1,12,14, 3)( 2, 6,13, 9)( 4,16,11,15)( 5, 7,10, 8)$
$8^{2},1$ $2040$ $8$ $14$ $( 1,13,12, 9,14, 2, 3, 6)( 4,10,16, 8,11, 5,15, 7)$
$8^{2},1$ $2040$ $8$ $14$ $( 1, 6, 3, 2,14, 9,12,13)( 4, 7,15, 5,11, 8,16,10)$
$6^{2},3,1^{2}$ $1360$ $6$ $12$ $( 1,12,11,14,16,10)( 2,17, 3, 5, 8, 6)( 9,15,13)$
$12,3,2$ $1360$ $12$ $14$ $( 1,17,12, 3,11, 5,14, 8,16, 6,10, 2)( 4, 7)( 9,13,15)$
$12,3,2$ $1360$ $12$ $14$ $( 1, 8,12, 6,11, 2,14,17,16, 3,10, 5)( 4, 7)( 9,13,15)$
$17$ $960$ $17$ $16$ $( 1, 8, 2,14,16, 5,10,13,15,17, 6, 4, 3,12, 9, 7,11)$
$17$ $960$ $17$ $16$ $( 1,10, 3, 8,13,12, 2,15, 9,14,17, 7,16, 6,11, 5, 4)$

Malle's constant $a(G)$:     $1/6$

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

Character table not computed

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

Data not computed