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Properties

Label 34T9
34T9 1 4 1->4 20 1->20 2 3 2->3 19 2->19 23 3->23 34 3->34 24 4->24 33 4->33 5 9 5->9 13 5->13 6 10 6->10 14 6->14 7 28 7->28 30 7->30 8 27 8->27 29 8->29 9->8 15 9->15 10->7 16 10->16 11 11->1 21 11->21 12 12->2 22 12->22 13->1 13->21 14->2 14->22 15->7 15->16 16->8 17 17->28 17->30 18 18->27 18->29 19->10 19->14 20->9 20->13 21->24 21->33 22->23 22->34 23->4 23->20 24->3 24->19 25 25->6 25->18 26 26->5 26->17 27->26 32 27->32 28->25 31 28->31 29->11 29->12 30->11 30->12 31->25 32->26 33->5 33->17 34->6 34->18
Degree $34$
Order $544$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times F_{17}$

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Copy content magma:G := TransitiveGroup(34, 9);
 

Group invariants

Abstract group:  $C_2\times F_{17}$
Copy content magma:IdentifyGroup(G);
 
Order:  $544=2^{5} \cdot 17$
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 $n$:  $34$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $9$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,20,9,8,27,32,26,17,30,12,22,23,4,33,5,13)(2,19,10,7,28,31,25,18,29,11,21,24,3,34,6,14)(15,16)$, $(1,4,24,19,14,22,34,18,27,26,5,9,15,7,30,11)(2,3,23,20,13,21,33,17,28,25,6,10,16,8,29,12)$
Copy content 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, $C_2^2$
$8$:  $C_8$ x 2, $C_4\times C_2$
$16$:  $C_{16}$ x 2, $C_8\times C_2$
$32$:  32T32
$272$:  $F_{17}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: $F_{17}$

Low degree siblings

34T9

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 $2^{17}$ $1$ $2$ $17$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)$
2B $2^{17}$ $17$ $2$ $17$ $( 1,20)( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,13)( 8,14)( 9,12)(10,11)(21,34)(22,33)(23,31)(24,32)(25,30)(26,29)(27,28)$
2C $2^{16},1^{2}$ $17$ $2$ $16$ $( 1,26)( 2,25)( 3,23)( 4,24)( 5,22)( 6,21)( 7,19)( 8,20)( 9,18)(10,17)(11,15)(12,16)(27,34)(28,33)(29,32)(30,31)$
4A1 $4^{8},2$ $17$ $4$ $25$ $( 1,28,26,33)( 2,27,25,34)( 3,19,23, 7)( 4,20,24, 8)( 5,12,22,16)( 6,11,21,15)( 9,29,18,32)(10,30,17,31)(13,14)$
4A-1 $4^{8},2$ $17$ $4$ $25$ $( 1,17,14,32)( 2,18,13,31)( 3,26,12,24)( 4,25,11,23)( 5,33, 9,16)( 6,34,10,15)( 7, 8)(19,21,30,28)(20,22,29,27)$
4B1 $4^{8},1^{2}$ $17$ $4$ $24$ $( 1, 5,24,19)( 2, 6,23,20)( 3,32,21,28)( 4,31,22,27)( 7,15,18, 9)( 8,16,17,10)(11,34,14,26)(12,33,13,25)$
4B-1 $4^{8},1^{2}$ $17$ $4$ $24$ $( 1,19,24, 5)( 2,20,23, 6)( 3,28,21,32)( 4,27,22,31)( 7, 9,18,15)( 8,10,17,16)(11,26,14,34)(12,25,13,33)$
8A1 $8^{4},2$ $17$ $8$ $29$ $( 1,16, 5,17,24,10,19, 8)( 2,15, 6,18,23, 9,20, 7)( 3,34,32,14,21,26,28,11)( 4,33,31,13,22,25,27,12)(29,30)$
8A-1 $8^{4},2$ $17$ $8$ $29$ $( 1,12,31, 3,15, 6,19,13)( 2,11,32, 4,16, 5,20,14)( 7,23,22,17, 9,28,30,33)( 8,24,21,18,10,27,29,34)(25,26)$
8A3 $8^{4},2$ $17$ $8$ $29$ $( 1,33, 4,29,11,13, 9,17)( 2,34, 3,30,12,14,10,18)( 5,25,19,32, 7,21,27,16)( 6,26,20,31, 8,22,28,15)(23,24)$
8A-3 $8^{4},2$ $17$ $8$ $29$ $( 1,29,15, 6,27,33,14,23)( 2,30,16, 5,28,34,13,24)( 3,11, 8, 9,25,18,21,19)( 4,12, 7,10,26,17,22,20)(31,32)$
8B1 $8^{4},1^{2}$ $17$ $8$ $28$ $( 1,26, 4, 9,30, 5,27,22)( 2,25, 3,10,29, 6,28,21)( 7,11,14,31,24,19,18,34)( 8,12,13,32,23,20,17,33)$
8B-1 $8^{4},1^{2}$ $17$ $8$ $28$ $( 1,22,27, 5,30, 9, 4,26)( 2,21,28, 6,29,10, 3,25)( 7,34,18,19,24,31,14,11)( 8,33,17,20,23,32,13,12)$
8B3 $8^{4},1^{2}$ $17$ $8$ $28$ $( 1, 9,27,26,30,22, 4, 5)( 2,10,28,25,29,21, 3, 6)( 7,31,18,11,24,34,14,19)( 8,32,17,12,23,33,13,20)$
8B-3 $8^{4},1^{2}$ $17$ $8$ $28$ $( 1, 5, 4,22,30,26,27, 9)( 2, 6, 3,21,29,25,28,10)( 7,19,14,34,24,11,18,31)( 8,20,13,33,23,12,17,32)$
16A1 $16^{2},2$ $17$ $16$ $31$ $( 1, 8,26,12, 4,13, 9,32,30,23, 5,20,27,17,22,33)( 2, 7,25,11, 3,14,10,31,29,24, 6,19,28,18,21,34)(15,16)$
16A-1 $16^{2},2$ $17$ $16$ $31$ $( 1,33,22,17,27,20, 5,23,30,32, 9,13, 4,12,26, 8)( 2,34,21,18,28,19, 6,24,29,31,10,14, 3,11,25, 7)(15,16)$
16A3 $16^{2},2$ $17$ $16$ $31$ $( 1,12, 9,23,27,33,26,13,30,20,22, 8, 4,32, 5,17)( 2,11,10,24,28,34,25,14,29,19,21, 7, 3,31, 6,18)(15,16)$
16A-3 $16^{2},2$ $17$ $16$ $31$ $( 1,17, 5,32, 4, 8,22,20,30,13,26,33,27,23, 9,12)( 2,18, 6,31, 3, 7,21,19,29,14,25,34,28,24,10,11)(15,16)$
16A5 $16^{2},2$ $17$ $16$ $31$ $( 1,13, 5,33, 4,23,22,12,30,17,26,32,27, 8, 9,20)( 2,14, 6,34, 3,24,21,11,29,18,25,31,28, 7,10,19)(15,16)$
16A-5 $16^{2},2$ $17$ $16$ $31$ $( 1,20, 9, 8,27,32,26,17,30,12,22,23, 4,33, 5,13)( 2,19,10, 7,28,31,25,18,29,11,21,24, 3,34, 6,14)(15,16)$
16A7 $16^{2},2$ $17$ $16$ $31$ $( 1,32,22,13,27,12, 5, 8,30,33, 9,17, 4,20,26,23)( 2,31,21,14,28,11, 6, 7,29,34,10,18, 3,19,25,24)(15,16)$
16A-7 $16^{2},2$ $17$ $16$ $31$ $( 1,23,26,20, 4,17, 9,33,30, 8, 5,12,27,13,22,32)( 2,24,25,19, 3,18,10,34,29, 7, 6,11,28,14,21,31)(15,16)$
16B1 $16^{2},1^{2}$ $17$ $16$ $30$ $( 1,31,19,18,11,27, 7,15, 5, 9,22,24,30,14,34,26)( 2,32,20,17,12,28, 8,16, 6,10,21,23,29,13,33,25)$
16B-1 $16^{2},1^{2}$ $17$ $16$ $30$ $( 1,24,19,30,22, 7,26,31,34,11,15, 5,14,27, 9, 4)( 2,23,20,29,21, 8,25,32,33,12,16, 6,13,28,10, 3)$
16B3 $16^{2},1^{2}$ $17$ $16$ $30$ $( 3,21,32,29,10,13,20,12,33,16, 6, 8,28,23,17,25)( 4,22,31,30, 9,14,19,11,34,15, 5, 7,27,24,18,26)$
16B-3 $16^{2},1^{2}$ $17$ $16$ $30$ $( 1, 7,11,26,24,34,18,30, 4,31,27,14,15, 5,22, 9)( 2, 8,12,25,23,33,17,29, 3,32,28,13,16, 6,21,10)$
16B5 $16^{2},1^{2}$ $17$ $16$ $30$ $( 1, 4,14,30, 7,34,27,31,18,15, 5,24,11,19,26,22)( 2, 3,13,29, 8,33,28,32,17,16, 6,23,12,20,25,21)$
16B-5 $16^{2},1^{2}$ $17$ $16$ $30$ $( 1, 9,31,15, 5, 4,24,27,22,14,26, 7,18,19,34,30)( 2,10,32,16, 6, 3,23,28,21,13,25, 8,17,20,33,29)$
16B7 $16^{2},1^{2}$ $17$ $16$ $30$ $( 1,22, 4, 9, 7,19,15, 5,31,11,30,24,26,14,18,27)( 2,21, 3,10, 8,20,16, 6,32,12,29,23,25,13,17,28)$
16B-7 $16^{2},1^{2}$ $17$ $16$ $30$ $( 1,14,11,18,34,19,27, 4, 7,30,31,26, 9,24,15, 5)( 2,13,12,17,33,20,28, 3, 8,29,32,25,10,23,16, 6)$
17A $17^{2}$ $16$ $17$ $32$ $( 1,15,30, 9,24, 4,18,31,11,26, 5,19,34,14,27, 7,22)( 2,16,29,10,23, 3,17,32,12,25, 6,20,33,13,28, 8,21)$
34A $34$ $16$ $34$ $33$ $( 1,25,15, 6,30,20, 9,33,24,13, 4,28,18, 8,31,21,11, 2,26,16, 5,29,19,10,34,23,14, 3,27,17, 7,32,22,12)$

Malle's constant $a(G)$:     $1/16$

Copy content magma:ConjugacyClasses(G);
 

Character table

34 x 34 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed