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Properties

Label 34T26
34T26 1 12 1->12 34 1->34 2 10 2->10 24 2->24 3 8 3->8 31 3->31 4 6 4->6 21 4->21 5 5->4 28 5->28 6->2 18 6->18 7 17 7->17 25 7->25 15 8->15 32 8->32 9 13 9->13 22 9->22 11 10->11 29 10->29 11->9 19 11->19 12->7 26 12->26 13->5 33 13->33 14 14->3 23 14->23 15->1 30 15->30 16 20 16->20 17->14 27 17->27 18->8 18->21 19->23 20->14 20->25 21->17 21->27 22->3 22->29 23->6 23->31 24->9 24->33 25->12 25->18 26->15 26->20 27->1 27->22 28->4 28->24 29->7 29->26 30->10 30->28 31->13 31->30 32->16 33->2 33->34 34->5 34->19
Degree $34$
Order $4624$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{17}^2:\OD_{16}$

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Show commands: Magma

magma: G := TransitiveGroup(34, 26);
 

Group invariants

Abstract group:  $C_{17}^2:\OD_{16}$
magma: IdentifyGroup(G);
 
Order:  $4624=2^{4} \cdot 17^{2}$
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 $t$:  $26$
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:  $(1,12,7,17,14,3,8,15)(2,10,11,9,13,5,4,6)(18,21,27,22,29,26,20,25)(19,23,31,30,28,24,33,34)$, $(1,34,5,28,4,21,17,27)(2,24,9,22,3,31,13,33)(6,18,8,32,16,20,14,23)(7,25,12,26,15,30,10,29)(11,19)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$16$:  $C_8:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: None

Low degree siblings

34T25, 34T26

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 $2^{17}$ $34$ $2$ $17$ $( 1,22)( 2,25)( 3,28)( 4,31)( 5,34)( 6,20)( 7,23)( 8,26)( 9,29)(10,32)(11,18)(12,21)(13,24)(14,27)(15,30)(16,33)(17,19)$
2B $2^{16},1^{2}$ $289$ $2$ $16$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(30,34)(31,33)$
4A1 $4^{8},1^{2}$ $289$ $4$ $24$ $( 1, 4,16,13)( 2, 8,15, 9)( 3,12,14, 5)( 6, 7,11,10)(18,27,29,20)(19,31,28,33)(21,22,26,25)(23,30,24,34)$
4A-1 $4^{8},1^{2}$ $289$ $4$ $24$ $( 1,13,16, 4)( 2, 9,15, 8)( 3, 5,14,12)( 6,10,11, 7)(18,20,29,27)(19,33,28,31)(21,25,26,22)(23,34,24,30)$
4B $4^{8},2$ $578$ $4$ $25$ $( 1,33,14,19)( 2,28,13,24)( 3,23,12,29)( 4,18,11,34)( 5,30,10,22)( 6,25, 9,27)( 7,20, 8,32)(15,31,17,21)(16,26)$
8A1 $8^{4},2$ $578$ $8$ $29$ $( 1,25, 4,21,16,22,13,26)( 2,18, 8,27,15,29, 9,20)( 3,28,12,33,14,19, 5,31)( 6,24, 7,34,11,23,10,30)(17,32)$
8A-1 $8^{4},2$ $578$ $8$ $29$ $( 1,26,13,22,16,21, 4,25)( 2,20, 9,29,15,27, 8,18)( 3,31, 5,19,14,33,12,28)( 6,30,10,23,11,34, 7,24)(17,32)$
8B1 $8^{4},1^{2}$ $578$ $8$ $28$ $( 1, 5,13,12,10, 6,15,16)( 2, 7,17, 3, 9, 4,11, 8)(18,26,27,25,29,21,20,22)(19,24,31,34,28,23,33,30)$
8B-1 $8^{4},1^{2}$ $578$ $8$ $28$ $( 1,16,15, 6,10,12,13, 5)( 2, 8,11, 4, 9, 3,17, 7)(18,22,20,21,29,25,27,26)(19,30,33,23,28,34,31,24)$
17A1 $17^{2}$ $8$ $17$ $32$ $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$
17A2 $17^{2}$ $8$ $17$ $32$ $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$
17A3 $17^{2}$ $8$ $17$ $32$ $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$
17A6 $17^{2}$ $8$ $17$ $32$ $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$
17B1 $17,1^{17}$ $16$ $17$ $16$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)$
17B3 $17,1^{17}$ $16$ $17$ $16$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)$
17C1 $17^{2}$ $16$ $17$ $32$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$
17C3 $17^{2}$ $16$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$
17D1 $17^{2}$ $16$ $17$ $32$ $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$
17D2 $17^{2}$ $16$ $17$ $32$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$
17D3 $17^{2}$ $16$ $17$ $32$ $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$
17D6 $17^{2}$ $16$ $17$ $32$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$
17E1 $17^{2}$ $16$ $17$ $32$ $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$
17E2 $17^{2}$ $16$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$
17E3 $17^{2}$ $16$ $17$ $32$ $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$
17E6 $17^{2}$ $16$ $17$ $32$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$
17F1 $17^{2}$ $16$ $17$ $32$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$
17F2 $17^{2}$ $16$ $17$ $32$ $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$
17F3 $17^{2}$ $16$ $17$ $32$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$
17F6 $17^{2}$ $16$ $17$ $32$ $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$
34A1 $34$ $136$ $34$ $33$ $( 1,19,16,30,14,24,12,18,10,29, 8,23, 6,34, 4,28, 2,22,17,33,15,27,13,21,11,32, 9,26, 7,20, 5,31, 3,25)$
34A3 $34$ $136$ $34$ $33$ $( 1,30,12,29, 6,28,17,27,11,26, 5,25,16,24,10,23, 4,22,15,21, 9,20, 3,19,14,18, 8,34, 2,33,13,32, 7,31)$
34A7 $34$ $136$ $34$ $33$ $( 1,18, 4,27, 7,19,10,28,13,20,16,29, 2,21, 5,30, 8,22,11,31,14,23,17,32, 3,24, 6,33, 9,25,12,34,15,26)$
34A9 $34$ $136$ $34$ $33$ $( 1,29,17,26,16,23,15,20,14,34,13,31,12,28,11,25,10,22, 9,19, 8,33, 7,30, 6,27, 5,24, 4,21, 3,18, 2,32)$

Malle's constant $a(G)$:     $1/16$

magma: ConjugacyClasses(G);
 

Character table

34 x 34 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed