Properties

Label 34T30
Degree $34$
Order $8704$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^8.D_{17}$

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Show commands: Magma

magma: G := TransitiveGroup(34, 30);
 

Group invariants

Abstract group:  $C_2^8.D_{17}$
magma: IdentifyGroup(G);
 
Order:  $8704=2^{9} \cdot 17$
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 $t$:  $30$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,19)(2,20)(3,18)(4,17)(5,16)(6,15)(7,13,8,14)(9,11,10,12)(21,33,22,34)(23,32)(24,31)(25,29)(26,30)(27,28)$, $(1,13,2,14)(3,12)(4,11)(5,9,6,10)(15,34,16,33)(17,31,18,32)(19,29,20,30)(21,28,22,27)(23,25)(24,26)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$34$:  $D_{17}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 17: $D_{17}$

Low degree siblings

34T30 x 14, 34T31 x 15

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 $2^{6},1^{22}$ $17$ $2$ $6$ $( 1, 2)(17,18)(19,20)(23,24)(29,30)(33,34)$
2B $2^{10},1^{14}$ $17$ $2$ $10$ $( 1, 2)( 3, 4)( 5, 6)(11,12)(15,16)(21,22)(23,24)(25,26)(29,30)(31,32)$
2C $2^{6},1^{22}$ $17$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(15,16)(27,28)$
2D $2^{6},1^{22}$ $17$ $2$ $6$ $( 5, 6)(11,12)(13,14)(23,24)(25,26)(31,32)$
2E $2^{8},1^{18}$ $17$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 9,10)(23,24)(27,28)(29,30)(31,32)$
2F $2^{10},1^{14}$ $17$ $2$ $10$ $( 3, 4)( 5, 6)(11,12)(15,16)(17,18)(19,20)(21,22)(25,26)(31,32)(33,34)$
2G $2^{8},1^{18}$ $17$ $2$ $8$ $( 1, 2)( 3, 4)( 9,10)(11,12)(13,14)(25,26)(27,28)(29,30)$
2H $2^{10},1^{14}$ $17$ $2$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(13,14)(19,20)(25,26)(27,28)(29,30)(31,32)$
2I $2^{8},1^{18}$ $17$ $2$ $8$ $( 3, 4)( 5, 6)( 9,10)(17,18)(19,20)(27,28)(31,32)(33,34)$
2J $2^{12},1^{10}$ $17$ $2$ $12$ $( 3, 4)( 7, 8)(11,12)(13,14)(15,16)(17,18)(19,20)(25,26)(27,28)(29,30)(31,32)(33,34)$
2K $2^{6},1^{22}$ $17$ $2$ $6$ $( 5, 6)(11,12)(15,16)(19,20)(23,24)(29,30)$
2L $2^{10},1^{14}$ $17$ $2$ $10$ $( 3, 4)( 5, 6)( 7, 8)(15,16)(17,18)(19,20)(23,24)(27,28)(29,30)(33,34)$
2M $2^{8},1^{18}$ $17$ $2$ $8$ $( 5, 6)( 9,10)(13,14)(15,16)(21,22)(23,24)(27,28)(31,32)$
2N $2^{12},1^{10}$ $17$ $2$ $12$ $( 1, 2)( 9,10)(11,12)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(33,34)$
2O $2^{8},1^{18}$ $17$ $2$ $8$ $( 7, 8)(11,12)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2P $2^{16},1^{2}$ $272$ $2$ $16$ $( 1,13)( 2,14)( 3,12)( 4,11)( 5, 9)( 6,10)(15,34)(16,33)(17,31)(18,32)(19,29)(20,30)(21,28)(22,27)(23,26)(24,25)$
4A $4^{4},2^{8},1^{2}$ $272$ $4$ $20$ $( 1,13)( 2,14)( 3,12)( 4,11)( 5,10, 6, 9)(15,34)(16,33)(17,32)(18,31)(19,29,20,30)(21,28,22,27)(23,26,24,25)$
4B $4^{3},2^{11}$ $272$ $4$ $20$ $( 1,13, 2,14)( 3,12)( 4,11)( 5, 9)( 6,10)( 7, 8)(15,34)(16,33)(17,32)(18,31)(19,29)(20,30)(21,27,22,28)(23,26,24,25)$
4C $4^{5},2^{7}$ $272$ $4$ $22$ $( 1,13, 2,14)( 3,11, 4,12)( 5,10, 6, 9)( 7, 8)(15,34,16,33)(17,32)(18,31)(19,29)(20,30)(21,28,22,27)(23,25)(24,26)$
4D $4^{4},2^{8},1^{2}$ $272$ $4$ $20$ $( 1,13)( 2,14)( 3,11, 4,12)( 5,10, 6, 9)(15,34,16,33)(17,31)(18,32)(19,29)(20,30)(21,27)(22,28)(23,25,24,26)$
4E $4^{5},2^{7}$ $272$ $4$ $22$ $( 1,13)( 2,14)( 3,11, 4,12)( 5,10, 6, 9)( 7, 8)(15,33)(16,34)(17,32,18,31)(19,30)(20,29)(21,28,22,27)(23,25,24,26)$
4F $4^{4},2^{8},1^{2}$ $272$ $4$ $20$ $( 1,13, 2,14)( 3,11, 4,12)( 5,10, 6, 9)(15,33)(16,34)(17,31,18,32)(19,30)(20,29)(21,27)(22,28)(23,25)(24,26)$
4G $4^{6},2^{4},1^{2}$ $272$ $4$ $22$ $( 1,13, 2,14)( 3,11, 4,12)( 5, 9)( 6,10)(15,33)(16,34)(17,32,18,31)(19,30,20,29)(21,27,22,28)(23,25,24,26)$
4H $4^{4},2^{8},1^{2}$ $272$ $4$ $20$ $( 1,13)( 2,14)( 3,11, 4,12)( 5, 9)( 6,10)(15,34,16,33)(17,32)(18,31)(19,29,20,30)(21,27,22,28)(23,25)(24,26)$
4I $4^{3},2^{11}$ $272$ $4$ $20$ $( 1,13)( 2,14)( 3,11, 4,12)( 5, 9)( 6,10)( 7, 8)(15,33)(16,34)(17,31,18,32)(19,30,20,29)(21,28)(22,27)(23,25)(24,26)$
4J $4^{4},2^{8},1^{2}$ $272$ $4$ $20$ $( 1,13, 2,14)( 3,12)( 4,11)( 5, 9)( 6,10)(15,33,16,34)(17,31,18,32)(19,30)(20,29)(21,28)(22,27)(23,26,24,25)$
4K $4^{3},2^{11}$ $272$ $4$ $20$ $( 1,13, 2,14)( 3,12)( 4,11)( 5,10, 6, 9)( 7, 8)(15,34)(16,33)(17,31)(18,32)(19,29,20,30)(21,27)(22,28)(23,26)(24,25)$
4L $4^{6},2^{4},1^{2}$ $272$ $4$ $22$ $( 1,13, 2,14)( 3,12)( 4,11)( 5,10, 6, 9)(15,33,16,34)(17,32,18,31)(19,30,20,29)(21,28,22,27)(23,26)(24,25)$
4M $4^{5},2^{7}$ $272$ $4$ $22$ $( 1,13, 2,14)( 3,11, 4,12)( 5, 9)( 6,10)( 7, 8)(15,34,16,33)(17,31)(18,32)(19,29,20,30)(21,28)(22,27)(23,25,24,26)$
4N $4^{5},2^{7}$ $272$ $4$ $22$ $( 1,13)( 2,14)( 3,12)( 4,11)( 5,10, 6, 9)( 7, 8)(15,33,16,34)(17,31,18,32)(19,30,20,29)(21,27)(22,28)(23,26,24,25)$
4O $4^{3},2^{11}$ $272$ $4$ $20$ $( 1,13)( 2,14)( 3,12)( 4,11)( 5, 9)( 6,10)( 7, 8)(15,33,16,34)(17,32,18,31)(19,30)(20,29)(21,27,22,28)(23,26)(24,25)$
17A1 $17^{2}$ $512$ $17$ $32$ $( 1,24,11,33,22, 9,32,20, 8,29,17, 5,28,16, 3,26,14)( 2,23,12,34,21,10,31,19, 7,30,18, 6,27,15, 4,25,13)$
17A2 $17^{2}$ $512$ $17$ $32$ $( 1,11,22,32, 8,17,28, 3,14,24,33, 9,20,29, 5,16,26)( 2,12,21,31, 7,18,27, 4,13,23,34,10,19,30, 6,15,25)$
17A3 $17^{2}$ $512$ $17$ $32$ $( 1,33,32,29,28,26,24,22,20,17,16,14,11, 9, 8, 5, 3)( 2,34,31,30,27,25,23,21,19,18,15,13,12,10, 7, 6, 4)$
17A4 $17^{2}$ $512$ $17$ $32$ $( 1,22, 8,28,14,33,20, 5,26,11,32,17, 3,24, 9,29,16)( 2,21, 7,27,13,34,19, 6,25,12,31,18, 4,23,10,30,15)$
17A5 $17^{2}$ $512$ $17$ $32$ $( 1, 9,17,26,33, 8,16,24,32, 5,14,22,29, 3,11,20,28)( 2,10,18,25,34, 7,15,23,31, 6,13,21,30, 4,12,19,27)$
17A6 $17^{2}$ $512$ $17$ $32$ $( 1,32,28,24,20,16,11, 8, 3,33,29,26,22,17,14, 9, 5)( 2,31,27,23,19,15,12, 7, 4,34,30,25,21,18,13,10, 6)$
17A7 $17^{2}$ $512$ $17$ $32$ $( 1,20, 3,22, 5,24, 8,26, 9,28,11,29,14,32,16,33,17)( 2,19, 4,21, 6,23, 7,25,10,27,12,30,13,31,15,34,18)$
17A8 $17^{2}$ $512$ $17$ $32$ $( 1, 8,14,20,26,32, 3, 9,16,22,28,33, 5,11,17,24,29)( 2, 7,13,19,25,31, 4,10,15,21,27,34, 6,12,18,23,30)$

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(G);
 

Character table

40 x 40 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed