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Properties

Label 17T6
17T6 1 6 1->6 16 1->16 2 3 2->3 2->3 15 2->15 9 3->9 4 4->2 5 4->5 4->9 5->4 7 5->7 6->7 8 6->8 13 6->13 11 7->11 8->1 8->9 9->7 10 10->11 14 10->14 10->14 11->8 11->13 12 12->3 12->13 12->15 13->5 14->12 14->15 15->10 17 16->17
Degree $17$
Order $4080$
Cyclic no
Abelian no
Solvable no
Transitivity $3$
Primitive yes
$p$-group no
Group: $\PSL(2,16)$

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, 6);
 
Copy content sage:G = TransitiveGroup(17, 6)
 
Copy content oscar:G = transitive_group(17, 6)
 
Copy content gap:G := TransitiveGroup(17, 6);
 

Group invariants

Abstract group:  $\PSL(2,16)$
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:  $4080=2^{4} \cdot 3 \cdot 5 \cdot 17$
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 $t$:  $6$
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:  $(2,3)(4,9)(5,7)(6,8)(10,14)(11,13)(12,15)(16,17)$, $(1,16)(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)$, $(1,6,13,5,4,2,15,10,14,12,3,9,7,11,8)$
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

none

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
$1^{17}$ $1$ $1$ $0$ $()$
$3^{5},1^{2}$ $272$ $3$ $10$ $( 1, 2, 5)( 3, 9,17)( 6,13,16)( 7,12,10)( 8,11,15)$
$5^{3},1^{2}$ $272$ $5$ $12$ $( 1,17,11, 6,12)( 2, 3,15,13,10)( 5, 9, 8,16, 7)$
$5^{3},1^{2}$ $272$ $5$ $12$ $( 1, 6,17,12,11)( 2,13, 3,10,15)( 5,16, 9, 7, 8)$
$15,1^{2}$ $272$ $15$ $14$ $( 1, 8,10,17,16, 2,11, 7, 3, 6, 5,15,12, 9,13)$
$15,1^{2}$ $272$ $15$ $14$ $( 1,15, 7,17,13, 5,11,10, 9, 6, 2, 8,12, 3,16)$
$15,1^{2}$ $272$ $15$ $14$ $( 1, 7,13,11, 9, 2,12,16,15,17, 5,10, 6, 8, 3)$
$15,1^{2}$ $272$ $15$ $14$ $( 1,10,16,11, 3, 5,12,13, 8,17, 2, 7, 6,15, 9)$
$17$ $240$ $17$ $16$ $( 1,16,10,13, 2,14, 5,17, 3, 8, 4,11,12, 6, 7,15, 9)$
$17$ $240$ $17$ $16$ $( 1, 2, 3,12, 9,13,17,11,15,10, 5, 4, 7,16,14, 8, 6)$
$17$ $240$ $17$ $16$ $( 1,10, 2, 5, 3, 4,12, 7, 9,16,13,14,17, 8,11, 6,15)$
$17$ $240$ $17$ $16$ $( 1, 3, 9,17,15, 5, 7,14, 6, 2,12,13,11,10, 4,16, 8)$
$17$ $240$ $17$ $16$ $( 1, 5,12,16,17, 6,10, 3, 7,13, 8,15, 2, 4, 9,14,11)$
$17$ $240$ $17$ $16$ $( 1,17, 7, 2,11,16, 3,15,14,12,10, 8, 9, 5, 6,13, 4)$
$17$ $240$ $17$ $16$ $( 1,12,17,10, 7, 8, 2, 9,11, 5,16, 6, 3,13,15, 4,14)$
$17$ $240$ $17$ $16$ $( 1, 7,11, 3,14,10, 9, 6, 4,17, 2,16,15,12, 8, 5,13)$
$2^{8},1$ $255$ $2$ $8$ $( 1, 9)( 2, 7)( 3,15)( 4,16)( 6,12)( 8,14)(10,11)(13,17)$

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 5A1 5A2 15A1 15A2 15A4 15A7 17A1 17A2 17A3 17A4 17A5 17A6 17A7 17A8
Size 1 255 272 272 272 272 272 272 272 240 240 240 240 240 240 240 240
2 P 1A 1A 3A 5A2 5A1 15A2 15A4 15A7 15A1 17A2 17A4 17A6 17A8 17A7 17A5 17A3 17A1
3 P 1A 2A 1A 5A2 5A1 5A1 5A2 5A1 5A2 17A3 17A6 17A8 17A5 17A2 17A1 17A4 17A7
5 P 1A 2A 3A 1A 1A 3A 3A 3A 3A 17A5 17A7 17A2 17A3 17A8 17A4 17A1 17A6
17 P 1A 2A 3A 5A2 5A1 15A2 15A4 15A7 15A1 1A 1A 1A 1A 1A 1A 1A 1A
Type
4080.a.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
4080.a.15a1 R 15 −1 0 0 0 0 0 0 0 −ζ17−1−ζ17 −ζ17−2−ζ172 −ζ17−3−ζ173 −ζ17−4−ζ174 −ζ17−5−ζ175 −ζ17−6−ζ176 −ζ17−7−ζ177 −ζ17−8−ζ178
4080.a.15a2 R 15 −1 0 0 0 0 0 0 0 −ζ17−2−ζ172 −ζ17−4−ζ174 −ζ17−6−ζ176 −ζ17−8−ζ178 −ζ17−7−ζ177 −ζ17−5−ζ175 −ζ17−3−ζ173 −ζ17−1−ζ17
4080.a.15a3 R 15 −1 0 0 0 0 0 0 0 −ζ17−3−ζ173 −ζ17−6−ζ176 −ζ17−8−ζ178 −ζ17−5−ζ175 −ζ17−2−ζ172 −ζ17−1−ζ17 −ζ17−4−ζ174 −ζ17−7−ζ177
4080.a.15a4 R 15 −1 0 0 0 0 0 0 0 −ζ17−4−ζ174 −ζ17−8−ζ178 −ζ17−5−ζ175 −ζ17−1−ζ17 −ζ17−3−ζ173 −ζ17−7−ζ177 −ζ17−6−ζ176 −ζ17−2−ζ172
4080.a.15a5 R 15 −1 0 0 0 0 0 0 0 −ζ17−5−ζ175 −ζ17−7−ζ177 −ζ17−2−ζ172 −ζ17−3−ζ173 −ζ17−8−ζ178 −ζ17−4−ζ174 −ζ17−1−ζ17 −ζ17−6−ζ176
4080.a.15a6 R 15 −1 0 0 0 0 0 0 0 −ζ17−6−ζ176 −ζ17−5−ζ175 −ζ17−1−ζ17 −ζ17−7−ζ177 −ζ17−4−ζ174 −ζ17−2−ζ172 −ζ17−8−ζ178 −ζ17−3−ζ173
4080.a.15a7 R 15 −1 0 0 0 0 0 0 0 −ζ17−7−ζ177 −ζ17−3−ζ173 −ζ17−4−ζ174 −ζ17−6−ζ176 −ζ17−1−ζ17 −ζ17−8−ζ178 −ζ17−2−ζ172 −ζ17−5−ζ175
4080.a.15a8 R 15 −1 0 0 0 0 0 0 0 −ζ17−8−ζ178 −ζ17−1−ζ17 −ζ17−7−ζ177 −ζ17−2−ζ172 −ζ17−6−ζ176 −ζ17−3−ζ173 −ζ17−5−ζ175 −ζ17−4−ζ174
4080.a.16a R 16 0 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
4080.a.17a R 17 1 −1 2 2 −1 −1 −1 −1 0 0 0 0 0 0 0 0
4080.a.17b1 R 17 1 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 0 0 0 0 0 0 0 0
4080.a.17b2 R 17 1 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 0 0 0 0 0 0 0 0
4080.a.17c1 R 17 1 −1 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 0 0 0 0 0 0 0 0
4080.a.17c2 R 17 1 −1 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 0 0 0 0 0 0 0 0
4080.a.17c3 R 17 1 −1 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 0 0 0 0 0 0 0 0
4080.a.17c4 R 17 1 −1 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 0 0 0 0 0 0 0 0

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