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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
Primitive yes
$p$-group no
Group: $\PSL(2,16)$

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Copy content magma:G := TransitiveGroup(17, 6);
 

Group invariants

Abstract group:  $\PSL(2,16)$
Copy content magma:IdentifyGroup(G);
 
Order:  $4080=2^{4} \cdot 3 \cdot 5 \cdot 17$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $17$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $6$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  yes
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), 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 magma:Generators(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
$1^{17}$ $1$ $1$ $0$ $()$
$17$ $240$ $17$ $16$ $( 1, 4, 6, 3, 7,14, 8,11,16,13,17,15, 2, 9,10,12, 5)$
$17$ $240$ $17$ $16$ $( 1, 7,16, 2, 5, 3,11,15,12, 6, 8,17,10, 4,14,13, 9)$
$17$ $240$ $17$ $16$ $( 1, 6, 7, 8,16,17, 2,10, 5, 4, 3,14,11,13,15, 9,12)$
$17$ $240$ $17$ $16$ $( 1,16, 5,11,12, 8,10,14, 9, 7, 2, 3,15, 6,17, 4,13)$
$17$ $240$ $17$ $16$ $( 1, 8, 2, 4,11, 9, 6,16,10, 3,13,12, 7,17, 5,14,15)$
$17$ $240$ $17$ $16$ $( 1,11,10, 7,15, 4,16,12,14, 2, 6,13, 5, 8, 9, 3,17)$
$17$ $240$ $17$ $16$ $( 1, 2,11, 6,10,13, 7, 5,15, 8, 4, 9,16, 3,12,17,14)$
$17$ $240$ $17$ $16$ $( 1,10,15,16,14, 6, 5, 9,17,11, 7, 4,12, 2,13, 8, 3)$
$5^{3},1^{2}$ $272$ $5$ $12$ $( 1,15, 3,14, 6)( 2, 7,12, 9,17)( 4,10,13, 5, 8)$
$5^{3},1^{2}$ $272$ $5$ $12$ $( 1,14,15, 6, 3)( 2, 9, 7,17,12)( 4, 5,10, 8,13)$
$3^{5},1^{2}$ $272$ $3$ $10$ $( 1, 4, 7)( 2, 6, 8)( 3,13, 9)( 5,17,14)(10,12,15)$
$15,1^{2}$ $272$ $15$ $14$ $( 1,17,10, 6, 9, 4,14,12, 8, 3, 7, 5,15, 2,13)$
$15,1^{2}$ $272$ $15$ $14$ $( 1, 5,12, 6,13, 7,14,10, 2, 3, 4,17,15, 8, 9)$
$15,1^{2}$ $272$ $15$ $14$ $( 1,12,13,14, 2, 4,15, 9, 5, 6, 7,10, 3,17, 8)$
$15,1^{2}$ $272$ $15$ $14$ $( 1,10, 9,14, 8, 7,15,13,17, 6, 4,12, 3, 5, 2)$
$2^{8},1$ $255$ $2$ $8$ $( 1, 9)( 2,17)( 3,12)( 4, 8)( 5,11)( 7,16)(10,14)(13,15)$

Malle's constant $a(G)$:     $1/8$

Copy content magma: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 −ζ17−6−ζ176 −ζ17−3−ζ173 −ζ17−4−ζ174 −ζ17−5−ζ175 −ζ17−8−ζ178 −ζ17−7−ζ177 −ζ17−2−ζ172
4080.a.15a2 R 15 −1 0 0 0 0 0 0 0 −ζ17−2−ζ172 −ζ17−5−ζ175 −ζ17−6−ζ176 −ζ17−8−ζ178 −ζ17−7−ζ177 −ζ17−1−ζ17 −ζ17−3−ζ173 −ζ17−4−ζ174
4080.a.15a3 R 15 −1 0 0 0 0 0 0 0 −ζ17−3−ζ173 −ζ17−1−ζ17 −ζ17−8−ζ178 −ζ17−5−ζ175 −ζ17−2−ζ172 −ζ17−7−ζ177 −ζ17−4−ζ174 −ζ17−6−ζ176
4080.a.15a4 R 15 −1 0 0 0 0 0 0 0 −ζ17−4−ζ174 −ζ17−7−ζ177 −ζ17−5−ζ175 −ζ17−1−ζ17 −ζ17−3−ζ173 −ζ17−2−ζ172 −ζ17−6−ζ176 −ζ17−8−ζ178
4080.a.15a5 R 15 −1 0 0 0 0 0 0 0 −ζ17−5−ζ175 −ζ17−4−ζ174 −ζ17−2−ζ172 −ζ17−3−ζ173 −ζ17−8−ζ178 −ζ17−6−ζ176 −ζ17−1−ζ17 −ζ17−7−ζ177
4080.a.15a6 R 15 −1 0 0 0 0 0 0 0 −ζ17−6−ζ176 −ζ17−2−ζ172 −ζ17−1−ζ17 −ζ17−7−ζ177 −ζ17−4−ζ174 −ζ17−3−ζ173 −ζ17−8−ζ178 −ζ17−5−ζ175
4080.a.15a7 R 15 −1 0 0 0 0 0 0 0 −ζ17−7−ζ177 −ζ17−8−ζ178 −ζ17−4−ζ174 −ζ17−6−ζ176 −ζ17−1−ζ17 −ζ17−5−ζ175 −ζ17−2−ζ172 −ζ17−3−ζ173
4080.a.15a8 R 15 −1 0 0 0 0 0 0 0 −ζ17−8−ζ178 −ζ17−3−ζ173 −ζ17−7−ζ177 −ζ17−2−ζ172 −ζ17−6−ζ176 −ζ17−4−ζ174 −ζ17−5−ζ175 −ζ17−1−ζ17
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 ζ5−1+ζ5 ζ5−2+ζ52 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 ζ5−2+ζ52 ζ5−1+ζ5 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 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−1+ζ15 0 0 0 0 0 0 0 0
4080.a.17c2 R 17 1 −1 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−2+ζ152 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−4+ζ154 0 0 0 0 0 0 0 0
4080.a.17c3 R 17 1 −1 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−4+ζ154 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−7+ζ157 0 0 0 0 0 0 0 0
4080.a.17c4 R 17 1 −1 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−1+ζ15 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−2+ζ152 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed