Properties

Label 34T31
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, 31);
 

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 $t$:  $31$
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:  $(1,28)(2,27)(3,25)(4,26)(5,23)(6,24)(7,22)(8,21)(9,20)(10,19)(11,17)(12,18)(13,15)(14,16)(29,34)(30,33)(31,32)$, $(1,30,24,18,12,6,34,28,22,15,9,4,32,25,20,14,8)(2,29,23,17,11,5,33,27,21,16,10,3,31,26,19,13,7)$
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 15, 34T31 x 14

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

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