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Properties

Label 17T10
17T10 1 2 1->2 1->2 3 2->3 4 3->4 5 4->5 6 5->6 7 6->7 8 7->8 9 8->9 10 9->10 11 10->11 12 11->12 13 12->13 14 13->14 15 14->15 16 15->16 17 16->17 17->1
Degree $17$
Order $3.557\times 10^{14}$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $S_{17}$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(17, 10);
 

Group invariants

Abstract group:  $S_{17}$
Copy content magma:IdentifyGroup(G);
 
Order:  $355687428096000=2^{15} \cdot 3^{6} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17$
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 $t$:  $10$
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:  $(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)$, $(1,2)$
Copy content magma:Generators(G);
 

Low degree resolvents

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

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

Conjugacy classes not computed

Copy content magma:ConjugacyClasses(G);
 

Character table

Character table not computed

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed