Properties

Label 34T2
34T2 1 9 1->9 32 1->32 2 10 2->10 31 2->31 3 11 3->11 29 3->29 4 12 4->12 30 4->30 5 13 5->13 27 5->27 6 14 6->14 28 6->28 7 16 7->16 26 7->26 8 15 8->15 25 8->25 18 9->18 23 9->23 17 10->17 24 10->24 20 11->20 22 11->22 19 12->19 21 12->21 13->19 13->21 14->20 14->22 15->17 15->24 16->18 16->23 17->25 18->26 19->27 20->28 21->30 22->29 23->32 24->31 33 25->33 34 26->34 27->1 28->2 29->4 30->3 31->6 32->5 33->7 33->34 34->8
Degree $34$
Order $34$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{17}$

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Show commands: Magma

magma: G := TransitiveGroup(34, 2);
 

Group invariants

Abstract group:  $D_{17}$
magma: IdentifyGroup(G);
 
Order:  $34=2 \cdot 17$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $34$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $2$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $34$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,32)(2,31)(3,29)(4,30)(5,27)(6,28)(7,26)(8,25)(9,23)(10,24)(11,22)(12,21)(13,19)(14,20)(15,17)(16,18)(33,34)$, $(1,9,18,26,34,8,15,24,31,6,14,22,29,4,12,19,27)(2,10,17,25,33,7,16,23,32,5,13,21,30,3,11,20,28)$
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

Degree 2: $C_2$

Degree 17: $D_{17}$

Low degree siblings

17T2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{34}$ $1$ $1$ $0$ $()$
2A $2^{17}$ $17$ $2$ $17$ $( 1, 3)( 2, 4)( 5,34)( 6,33)( 7,31)( 8,32)( 9,30)(10,29)(11,27)(12,28)(13,26)(14,25)(15,23)(16,24)(17,22)(18,21)(19,20)$
17A1 $17^{2}$ $2$ $17$ $32$ $( 1, 4, 6, 8, 9,12,14,15,18,19,22,24,26,27,29,31,34)( 2, 3, 5, 7,10,11,13,16,17,20,21,23,25,28,30,32,33)$
17A2 $17^{2}$ $2$ $17$ $32$ $( 1, 6, 9,14,18,22,26,29,34, 4, 8,12,15,19,24,27,31)( 2, 5,10,13,17,21,25,30,33, 3, 7,11,16,20,23,28,32)$
17A3 $17^{2}$ $2$ $17$ $32$ $( 1, 8,14,19,26,31, 4, 9,15,22,27,34, 6,12,18,24,29)( 2, 7,13,20,25,32, 3,10,16,21,28,33, 5,11,17,23,30)$
17A4 $17^{2}$ $2$ $17$ $32$ $( 1, 9,18,26,34, 8,15,24,31, 6,14,22,29, 4,12,19,27)( 2,10,17,25,33, 7,16,23,32, 5,13,21,30, 3,11,20,28)$
17A5 $17^{2}$ $2$ $17$ $32$ $( 1,12,22,31, 8,18,27, 4,14,24,34, 9,19,29, 6,15,26)( 2,11,21,32, 7,17,28, 3,13,23,33,10,20,30, 5,16,25)$
17A6 $17^{2}$ $2$ $17$ $32$ $( 1,14,26, 4,15,27, 6,18,29, 8,19,31, 9,22,34,12,24)( 2,13,25, 3,16,28, 5,17,30, 7,20,32,10,21,33,11,23)$
17A7 $17^{2}$ $2$ $17$ $32$ $( 1,15,29, 9,24, 4,18,31,12,26, 6,19,34,14,27, 8,22)( 2,16,30,10,23, 3,17,32,11,25, 5,20,33,13,28, 7,21)$
17A8 $17^{2}$ $2$ $17$ $32$ $( 1,18,34,15,31,14,29,12,27, 9,26, 8,24, 6,22, 4,19)( 2,17,33,16,32,13,30,11,28,10,25, 7,23, 5,21, 3,20)$

Malle's constant $a(G)$:     $1/17$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 17A1 17A2 17A3 17A4 17A5 17A6 17A7 17A8
Size 1 17 2 2 2 2 2 2 2 2
2 P 1A 1A 17A2 17A4 17A6 17A8 17A7 17A5 17A3 17A1
17 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A
Type
34.1.1a R 1 1 1 1 1 1 1 1 1 1
34.1.1b R 1 1 1 1 1 1 1 1 1 1
34.1.2a1 R 2 0 ζ178+ζ178 ζ175+ζ175 ζ172+ζ172 ζ171+ζ17 ζ174+ζ174 ζ177+ζ177 ζ173+ζ173 ζ176+ζ176
34.1.2a2 R 2 0 ζ177+ζ177 ζ172+ζ172 ζ176+ζ176 ζ173+ζ173 ζ175+ζ175 ζ174+ζ174 ζ178+ζ178 ζ171+ζ17
34.1.2a3 R 2 0 ζ176+ζ176 ζ178+ζ178 ζ177+ζ177 ζ175+ζ175 ζ173+ζ173 ζ171+ζ17 ζ172+ζ172 ζ174+ζ174
34.1.2a4 R 2 0 ζ175+ζ175 ζ171+ζ17 ζ173+ζ173 ζ177+ζ177 ζ176+ζ176 ζ172+ζ172 ζ174+ζ174 ζ178+ζ178
34.1.2a5 R 2 0 ζ174+ζ174 ζ176+ζ176 ζ171+ζ17 ζ178+ζ178 ζ172+ζ172 ζ175+ζ175 ζ177+ζ177 ζ173+ζ173
34.1.2a6 R 2 0 ζ173+ζ173 ζ174+ζ174 ζ175+ζ175 ζ176+ζ176 ζ177+ζ177 ζ178+ζ178 ζ171+ζ17 ζ172+ζ172
34.1.2a7 R 2 0 ζ172+ζ172 ζ173+ζ173 ζ178+ζ178 ζ174+ζ174 ζ171+ζ17 ζ176+ζ176 ζ175+ζ175 ζ177+ζ177
34.1.2a8 R 2 0 ζ171+ζ17 ζ177+ζ177 ζ174+ζ174 ζ172+ζ172 ζ178+ζ178 ζ173+ζ173 ζ176+ζ176 ζ175+ζ175

magma: CharacterTable(G);
 

Regular extensions

Data not computed