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Properties

Label 34T14
34T14 1 17 1->17 25 1->25 2 4 2->4 20 2->20 3 8 3->8 32 3->32 12 4->12 27 4->27 5 16 5->16 22 5->22 6 6->3 34 6->34 7 29 7->29 11 8->11 24 8->24 9 15 9->15 19 9->19 10 10->2 31 10->31 11->6 26 11->26 12->10 21 12->21 13 14 13->14 33 13->33 14->1 28 14->28 15->5 23 15->23 16->9 18 16->18 17->13 30 17->30 18->1 18->20 19->8 19->33 20->15 20->29 21->5 21->25 22->12 22->21 23->2 23->34 24->9 24->30 25->16 25->26 26->6 26->22 27->13 27->18 28->3 28->31 29->10 29->27 30->23 31->7 31->19 32->14 33->4 33->28 34->11 34->24
Degree $34$
Order $2312$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{17}^2:Q_8$

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Show commands: Magma

magma: G := TransitiveGroup(34, 14);
 

Group invariants

Abstract group:  $C_{17}^2:Q_8$
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 $t$:  $14$
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:  $(1,25,16,18)(2,20,15,23)(3,32,14,28)(4,27,13,33)(5,22,12,21)(6,34,11,26)(7,29,10,31)(8,24,9,19)(17,30)$, $(1,17,13,14)(2,4,12,10)(3,8,11,6)(5,16,9,15)(18,20,29,27)(19,33,28,31)(21,25,26,22)(23,34,24,30)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $Q_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: None

Low degree siblings

34T14 x 2

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 $2^{16},1^{2}$ $289$ $2$ $16$ $( 1, 2)( 3,17)( 4,16)( 5,15)( 6,14)( 7,13)( 8,12)( 9,11)(18,19)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)$
4A $4^{8},2$ $578$ $4$ $25$ $( 1,33, 2,21)( 3,26,17,28)( 4,31,16,23)( 5,19,15,18)( 6,24,14,30)( 7,29,13,25)( 8,34,12,20)( 9,22,11,32)(10,27)$
4B $4^{8},2$ $578$ $4$ $25$ $( 1,24,15,33)( 2,21,14,19)( 3,18,13,22)( 4,32,12,25)( 5,29,11,28)( 6,26,10,31)( 7,23, 9,34)( 8,20)(16,30,17,27)$
4C $4^{8},1^{2}$ $578$ $4$ $24$ $( 1, 8,14, 7)( 2, 4,13,11)( 3,17,12,15)( 5, 9,10, 6)(19,22,34,31)(20,26,33,27)(21,30,32,23)(24,25,29,28)$
17A1 $17,1^{17}$ $8$ $17$ $16$ $(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$
17A2 $17,1^{17}$ $8$ $17$ $16$ $(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$
17A3 $17,1^{17}$ $8$ $17$ $16$ $(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$
17A6 $17,1^{17}$ $8$ $17$ $16$ $(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$
17B1 $17^{2}$ $8$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$
17B2 $17^{2}$ $8$ $17$ $32$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$
17B3 $17^{2}$ $8$ $17$ $32$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$
17B6 $17^{2}$ $8$ $17$ $32$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$
17C1 $17^{2}$ $8$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$
17C2 $17^{2}$ $8$ $17$ $32$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$
17C3 $17^{2}$ $8$ $17$ $32$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$
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 $17^{2}$ $8$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$
17D2 $17^{2}$ $8$ $17$ $32$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$
17D3 $17^{2}$ $8$ $17$ $32$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$
17D4 $17^{2}$ $8$ $17$ $32$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$
17D5 $17^{2}$ $8$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$
17D6 $17^{2}$ $8$ $17$ $32$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,20,22,24,26,28,30,32,34,19,21,23,25,27,29,31,33)$
17D7 $17^{2}$ $8$ $17$ $32$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$
17D8 $17^{2}$ $8$ $17$ $32$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$
17E1 $17^{2}$ $8$ $17$ $32$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$
17E2 $17^{2}$ $8$ $17$ $32$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$
17E3 $17^{2}$ $8$ $17$ $32$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$
17E4 $17^{2}$ $8$ $17$ $32$ $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$
17E5 $17^{2}$ $8$ $17$ $32$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,20,22,24,26,28,30,32,34,19,21,23,25,27,29,31,33)$
17E6 $17^{2}$ $8$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$
17E7 $17^{2}$ $8$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$
17E8 $17^{2}$ $8$ $17$ $32$ $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$
17F1 $17^{2}$ $8$ $17$ $32$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$
17F2 $17^{2}$ $8$ $17$ $32$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$
17F3 $17^{2}$ $8$ $17$ $32$ $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$
17F4 $17^{2}$ $8$ $17$ $32$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$
17F5 $17^{2}$ $8$ $17$ $32$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$
17F6 $17^{2}$ $8$ $17$ $32$ $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$
17F7 $17^{2}$ $8$ $17$ $32$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$
17F8 $17^{2}$ $8$ $17$ $32$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$

Malle's constant $a(G)$:     $1/16$

magma: ConjugacyClasses(G);
 

Character table

41 x 41 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed