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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
Primitive yes
$p$-group no
Group: $\PSL(2,16):C_4$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(17, 8);
 

Group invariants

Abstract group:  $\PSL(2,16):C_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $16320=2^{6} \cdot 3 \cdot 5 \cdot 17$
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 $t$:  $8$
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:  $(2,3)(4,9)(5,7)(6,8)(10,14)(11,13)(12,15)(16,17)$, $(1,12,7,5)(2,4,13,11)(3,8,10,14)(6,9)$, $(1,16)(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)$, $(1,6,13,5,4,2,15,10,14,12,3,9,7,11,8)$
Copy content magma:Generators(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
$1^{17}$ $1$ $1$ $0$ $()$
$2^{8},1$ $255$ $2$ $8$ $( 1, 7)( 2, 6)( 3,16)( 4,10)( 5, 8)( 9,11)(12,17)(14,15)$
$4^{4},1$ $1020$ $4$ $12$ $( 1, 5, 7, 8)( 2,10, 6, 4)( 3,12,16,17)( 9,14,11,15)$
$8^{2},1$ $2040$ $8$ $14$ $( 1,15, 5, 9, 7,14, 8,11)( 2,12,10,16, 6,17, 4, 3)$
$8^{2},1$ $2040$ $8$ $14$ $( 1,11, 8,14, 7, 9, 5,15)( 2, 3, 4,17, 6,16,10,12)$
$2^{6},1^{5}$ $68$ $2$ $6$ $( 1, 3)( 4,12)( 5,11)( 6,15)( 7,10)(16,17)$
$4^{3},2,1^{3}$ $680$ $4$ $10$ $( 1, 6, 3,15)( 4,17,12,16)( 5,10,11, 7)( 9,13)$
$4^{3},2,1^{3}$ $680$ $4$ $10$ $( 1,15, 3, 6)( 4,16,12,17)( 5, 7,11,10)( 9,13)$
$3^{5},1^{2}$ $272$ $3$ $10$ $( 1, 5,17)( 2, 8,14)( 3,11,16)( 4,15, 7)( 6,10,12)$
$6^{2},3,1^{2}$ $1360$ $6$ $12$ $( 1,16, 5, 3,17,11)( 2,14, 8)( 4,10,15,12, 7, 6)$
$5^{3},1^{2}$ $544$ $5$ $12$ $( 2,14, 8, 9,13)( 4,17, 6, 5, 7)(10,12,16,15,11)$
$10,5,2$ $1632$ $10$ $14$ $( 1, 3)( 2, 7,14, 4, 8,17, 9, 6,13, 5)(10,15,12,11,16)$
$17$ $960$ $17$ $16$ $( 1, 4, 7,15, 8, 5,11, 9, 6,17, 3,12,14,16,13,10, 2)$
$17$ $960$ $17$ $16$ $( 1,11,14, 4, 9,16, 7, 6,13,15,17,10, 8, 3, 2, 5,12)$
$12,3,2$ $1360$ $12$ $14$ $( 1,12,15)( 2, 9,14,17,10, 8, 3, 6,11,16,13, 4)( 5, 7)$
$12,3,2$ $1360$ $12$ $14$ $( 1,12,15)( 2, 6,14,16,10, 4, 3, 9,11,17,13, 8)( 5, 7)$
$15,1^{2}$ $1088$ $15$ $14$ $( 1, 4,10, 5, 7,15,13, 2,12, 9, 3,16,17, 8,11)$

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

Character table not computed

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed