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Properties

Label 34T17
34T17 1 15 1->15 24 1->24 2 16 2->16 18 2->18 3 17 3->17 29 3->29 4 4->1 23 4->23 5 5->2 34 5->34 6 6->3 28 6->28 7 7->4 22 7->22 8 8->5 33 8->33 9 9->6 27 9->27 10 10->7 21 10->21 11 11->8 32 11->32 12 12->9 26 12->26 13 13->10 20 13->20 14 14->11 31 14->31 15->12 25 15->25 16->13 19 16->19 17->14 30 17->30 18->12 18->23 19->7 19->24 20->2 20->25 21->14 21->26 22->9 22->27 23->28 24->16 24->29 25->11 25->30 26->6 26->31 27->1 27->32 28->13 28->33 29->8 29->34 30->3 30->18 31->15 31->19 32->10 32->20 33->5 33->21 34->17 34->22
Degree $34$
Order $2312$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{17}^2:C_8$

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Show commands: Magma

magma: G := TransitiveGroup(34, 17);
 

Group invariants

Abstract group:  $C_{17}^2:C_8$
magma: IdentifyGroup(G);
 
Order:  $2312=2^{3} \cdot 17^{2}$
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 $t$:  $17$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,15,12,9,6,3,17,14,11,8,5,2,16,13,10,7,4)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$, $(1,24,16,19,7,22,9,27)(2,18,12,26,6,28,13,20)(3,29,8,33,5,34,17,30)(4,23)(10,21,14,31,15,25,11,32)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$8$:  $C_8$
$136$:  $C_{17}:C_{8}$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: None

Low degree siblings

34T17 x 7

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 $2^{16},1^{2}$ $289$ $2$ $16$ $( 1,13)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(14,17)(15,16)(18,20)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)$
4A1 $4^{8},1^{2}$ $289$ $4$ $24$ $( 1,17,13,14)( 2, 4,12,10)( 3, 8,11, 6)( 5,16, 9,15)(18,32,20,23)(21,27,34,28)(22,31,33,24)(25,26,30,29)$
4A-1 $4^{8},1^{2}$ $289$ $4$ $24$ $( 1,14,13,17)( 2,10,12, 4)( 3, 6,11, 8)( 5,15, 9,16)(18,23,20,32)(21,28,34,27)(22,24,33,31)(25,29,30,26)$
8A1 $8^{4},2$ $289$ $8$ $29$ $( 1,27,17,34,13,28,14,21)( 2,20, 4,23,12,18,10,32)( 3,30, 8,29,11,25, 6,26)( 5,33,16,24, 9,22,15,31)( 7,19)$
8A-1 $8^{4},2$ $289$ $8$ $29$ $( 1,21,14,28,13,34,17,27)( 2,32,10,18,12,23, 4,20)( 3,26, 6,25,11,29, 8,30)( 5,31,15,22, 9,24,16,33)( 7,19)$
8A3 $8^{4},2$ $289$ $8$ $29$ $( 1,34,14,27,13,21,17,28)( 2,23,10,20,12,32, 4,18)( 3,29, 6,30,11,26, 8,25)( 5,24,15,33, 9,31,16,22)( 7,19)$
8A-3 $8^{4},2$ $289$ $8$ $29$ $( 1,28,17,21,13,27,14,34)( 2,18, 4,32,12,20,10,23)( 3,25, 8,26,11,30, 6,29)( 5,22,16,31, 9,33,15,24)( 7,19)$
17A1 $17^{2}$ $8$ $17$ $32$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$
17A3 $17^{2}$ $8$ $17$ $32$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$
17B1 $17^{2}$ $8$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$
17B3 $17^{2}$ $8$ $17$ $32$ $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$
17C1 $17^{2}$ $8$ $17$ $32$ $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$
17C2 $17^{2}$ $8$ $17$ $32$ $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$
17C3 $17^{2}$ $8$ $17$ $32$ $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$
17C6 $17^{2}$ $8$ $17$ $32$ $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$
17D1 $17^{2}$ $8$ $17$ $32$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,20,22,24,26,28,30,32,34,19,21,23,25,27,29,31,33)$
17D2 $17^{2}$ $8$ $17$ $32$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$
17D3 $17^{2}$ $8$ $17$ $32$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$
17D6 $17^{2}$ $8$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$
17E1 $17^{2}$ $8$ $17$ $32$ $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$
17E2 $17^{2}$ $8$ $17$ $32$ $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,20,22,24,26,28,30,32,34,19,21,23,25,27,29,31,33)$
17E3 $17^{2}$ $8$ $17$ $32$ $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$
17E6 $17^{2}$ $8$ $17$ $32$ $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$
17F1 $17,1^{17}$ $8$ $17$ $16$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)$
17F2 $17,1^{17}$ $8$ $17$ $16$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)$
17F3 $17,1^{17}$ $8$ $17$ $16$ $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)$
17F6 $17,1^{17}$ $8$ $17$ $16$ $(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$
17G1 $17^{2}$ $8$ $17$ $32$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$
17G2 $17^{2}$ $8$ $17$ $32$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$
17G3 $17^{2}$ $8$ $17$ $32$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$
17G6 $17^{2}$ $8$ $17$ $32$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$
17H1 $17^{2}$ $8$ $17$ $32$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$
17H2 $17^{2}$ $8$ $17$ $32$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$
17H3 $17^{2}$ $8$ $17$ $32$ $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$
17H6 $17^{2}$ $8$ $17$ $32$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$
17I1 $17^{2}$ $8$ $17$ $32$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$
17I2 $17^{2}$ $8$ $17$ $32$ $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$
17I3 $17^{2}$ $8$ $17$ $32$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$
17I6 $17^{2}$ $8$ $17$ $32$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$
17J1 $17^{2}$ $8$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$
17J2 $17^{2}$ $8$ $17$ $32$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$
17J3 $17^{2}$ $8$ $17$ $32$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$
17J6 $17^{2}$ $8$ $17$ $32$ $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$

Malle's constant $a(G)$:     $1/16$

magma: ConjugacyClasses(G);
 

Character table

44 x 44 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed