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Properties

Label 17T5
17T5 1 2 1->2 3 2->3 4 2->4 3->4 7 3->7 5 4->5 10 4->10 6 5->6 13 5->13 6->7 16 6->16 7->2 8 7->8 8->5 9 8->9 9->8 9->10 11 10->11 10->11 12 11->12 14 11->14 12->13 17 12->17 13->3 13->14 14->6 15 14->15 15->9 15->16 16->12 16->17 17->1 17->15
Degree $17$
Order $272$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $F_{17}$

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Copy content magma:G := TransitiveGroup(17, 5);
 

Group invariants

Abstract group:  $F_{17}$
Copy content magma:IdentifyGroup(G);
 
Order:  $272=2^{4} \cdot 17$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
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 $t$:  $5$
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:  $(2,4,10,11,14,6,16,12,17,15,9,8,5,13,3,7)$, $(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$8$:  $C_8$
$16$:  $C_{16}$

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
1A $1^{17}$ $1$ $1$ $0$ $()$
2A $2^{8},1$ $17$ $2$ $8$ $( 1,15)( 2,14)( 3,13)( 4,12)( 5,11)( 6,10)( 7, 9)(16,17)$
4A1 $4^{4},1$ $17$ $4$ $12$ $( 1,14,15, 2)( 3, 5,13,11)( 4, 9,12, 7)( 6,17,10,16)$
4A-1 $4^{4},1$ $17$ $4$ $12$ $( 1, 2,15,14)( 3,11,13, 5)( 4, 7,12, 9)( 6,16,10,17)$
8A1 $8^{2},1$ $17$ $8$ $14$ $( 1,11,14, 3,15, 5, 2,13)( 4,17, 9,10,12,16, 7, 6)$
8A-1 $8^{2},1$ $17$ $8$ $14$ $( 1,13, 2, 5,15, 3,14,11)( 4, 6, 7,16,12,10, 9,17)$
8A3 $8^{2},1$ $17$ $8$ $14$ $( 1, 3, 2,11,15,13,14, 5)( 4,10, 7,17,12, 6, 9,16)$
8A-3 $8^{2},1$ $17$ $8$ $14$ $( 1, 5,14,13,15,11, 2, 3)( 4,16, 9, 6,12,17, 7,10)$
16A1 $16,1$ $17$ $16$ $15$ $( 1,16,11, 7,14, 6, 3, 4,15,17, 5, 9, 2,10,13,12)$
16A-1 $16,1$ $17$ $16$ $15$ $( 1,12,13,10, 2, 9, 5,17,15, 4, 3, 6,14, 7,11,16)$
16A3 $16,1$ $17$ $16$ $15$ $( 1, 7, 3,17, 2,12,11, 6,15, 9,13,16,14, 4, 5,10)$
16A-3 $16,1$ $17$ $16$ $15$ $( 1,10, 5, 4,14,16,13, 9,15, 6,11,12, 2,17, 3, 7)$
16A5 $16,1$ $17$ $16$ $15$ $( 1, 6, 5,12,14,17,13, 7,15,10,11, 4, 2,16, 3, 9)$
16A-5 $16,1$ $17$ $16$ $15$ $( 1, 9, 3,16, 2, 4,11,10,15, 7,13,17,14,12, 5, 6)$
16A7 $16,1$ $17$ $16$ $15$ $( 1, 4,13, 6, 2, 7, 5,16,15,12, 3,10,14, 9,11,17)$
16A-7 $16,1$ $17$ $16$ $15$ $( 1,17,11, 9,14,10, 3,12,15,16, 5, 7, 2, 6,13, 4)$
17A $17$ $16$ $17$ $16$ $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 4A1 4A-1 8A1 8A-1 8A3 8A-3 16A1 16A-1 16A3 16A-3 16A5 16A-5 16A7 16A-7 17A
Size 1 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 16
2 P 1A 1A 2A 2A 4A1 4A-1 4A-1 4A1 8A1 8A-1 8A3 8A-3 8A-3 8A3 8A-1 8A1 17A
17 P 1A 2A 4A1 4A-1 8A1 8A-1 8A3 8A-3 16A1 16A-1 16A3 16A-3 16A5 16A-5 16A7 16A-7 1A
Type
272.50.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
272.50.1b R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1
272.50.1c1 C 1 1 1 1 −1 −1 −1 −1 −i −i i i −i i i −i 1
272.50.1c2 C 1 1 1 1 −1 −1 −1 −1 i i −i −i i −i −i i 1
272.50.1d1 C 1 1 −1 −1 −ζ82 −ζ82 ζ82 ζ82 ζ83 −ζ83 −ζ8 −ζ8 −ζ83 ζ8 ζ8 ζ83 1
272.50.1d2 C 1 1 −1 −1 ζ82 ζ82 −ζ82 −ζ82 −ζ8 ζ8 ζ83 ζ83 ζ8 −ζ83 −ζ83 −ζ8 1
272.50.1d3 C 1 1 −1 −1 −ζ82 −ζ82 ζ82 ζ82 −ζ83 ζ83 ζ8 ζ8 ζ83 −ζ8 −ζ8 −ζ83 1
272.50.1d4 C 1 1 −1 −1 ζ82 ζ82 −ζ82 −ζ82 ζ8 −ζ8 −ζ83 −ζ83 −ζ8 ζ83 ζ83 ζ8 1
272.50.1e1 C 1 −1 −ζ164 ζ164 ζ166 −ζ166 −ζ162 ζ162 −ζ163 −ζ167 −ζ165 ζ165 ζ167 −ζ16 ζ16 ζ163 1
272.50.1e2 C 1 −1 ζ164 −ζ164 −ζ162 ζ162 ζ166 −ζ166 ζ165 ζ16 ζ163 −ζ163 −ζ16 ζ167 −ζ167 −ζ165 1
272.50.1e3 C 1 −1 −ζ164 ζ164 ζ166 −ζ166 −ζ162 ζ162 ζ163 ζ167 ζ165 −ζ165 −ζ167 ζ16 −ζ16 −ζ163 1
272.50.1e4 C 1 −1 ζ164 −ζ164 −ζ162 ζ162 ζ166 −ζ166 −ζ165 −ζ16 −ζ163 ζ163 ζ16 −ζ167 ζ167 ζ165 1
272.50.1e5 C 1 −1 −ζ164 ζ164 −ζ166 ζ166 ζ162 −ζ162 ζ167 −ζ163 ζ16 −ζ16 ζ163 −ζ165 ζ165 −ζ167 1
272.50.1e6 C 1 −1 ζ164 −ζ164 ζ162 −ζ162 −ζ166 ζ166 −ζ16 ζ165 −ζ167 ζ167 −ζ165 ζ163 −ζ163 ζ16 1
272.50.1e7 C 1 −1 −ζ164 ζ164 −ζ166 ζ166 ζ162 −ζ162 −ζ167 ζ163 −ζ16 ζ16 −ζ163 ζ165 −ζ165 ζ167 1
272.50.1e8 C 1 −1 ζ164 −ζ164 ζ162 −ζ162 −ζ166 ζ166 ζ16 −ζ165 ζ167 −ζ167 ζ165 −ζ163 ζ163 −ζ16 1
272.50.16a R 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed