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Properties

Label 34T46
Degree $34$
Order $17408$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^8.C_{17}:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(34, 46);
 

Group invariants

Abstract group:  $C_2^8.C_{17}:C_4$
magma: IdentifyGroup(G);
 
Order:  $17408=2^{10} \cdot 17$
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 $t$:  $46$
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:  $(1,16,28,13)(2,15,27,14)(3,7,26,21,4,8,25,22)(5,33,23,29)(6,34,24,30)(9,18,19,11,10,17,20,12)$, $(1,24)(2,23)(3,22)(4,21)(5,20,6,19)(7,17,8,18)(9,16,10,15)(11,14,12,13)(25,34)(26,33)(27,32,28,31)(29,30)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$68$:  $C_{17}:C_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 17: $C_{17}:C_{4}$

Low degree siblings

34T46 x 2, 34T47 x 3

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 $2^{8},1^{18}$ $17$ $2$ $8$ $( 5, 6)( 7, 8)( 9,10)(13,14)(25,26)(29,30)(31,32)(33,34)$
2B $2^{12},1^{10}$ $17$ $2$ $12$ $( 5, 6)( 7, 8)(13,14)(15,16)(17,18)(19,20)(21,22)(25,26)(27,28)(29,30)(31,32)(33,34)$
2C $2^{12},1^{10}$ $17$ $2$ $12$ $( 3, 4)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(21,22)(25,26)(27,28)(31,32)(33,34)$
2D $2^{8},1^{18}$ $34$ $2$ $8$ $( 1, 2)( 7, 8)( 9,10)(17,18)(19,20)(25,26)(29,30)(31,32)$
2E $2^{10},1^{14}$ $34$ $2$ $10$ $( 3, 4)( 5, 6)( 9,10)(11,12)(13,14)(15,16)(27,28)(29,30)(31,32)(33,34)$
2F $2^{6},1^{22}$ $34$ $2$ $6$ $( 5, 6)( 7, 8)(17,18)(21,22)(25,26)(29,30)$
2G $2^{8},1^{18}$ $34$ $2$ $8$ $( 1, 2)( 5, 6)( 9,10)(13,14)(15,16)(21,22)(27,28)(33,34)$
2H $2^{6},1^{22}$ $34$ $2$ $6$ $( 9,10)(15,16)(17,18)(19,20)(21,22)(27,28)$
2I $2^{10},1^{14}$ $34$ $2$ $10$ $( 1, 2)( 7, 8)( 9,10)(13,14)(15,16)(17,18)(25,26)(27,28)(29,30)(33,34)$
2J $2^{16},1^{2}$ $272$ $2$ $16$ $( 1,28)( 2,27)( 3,26)( 4,25)( 5,23)( 6,24)( 7,22)( 8,21)( 9,19)(10,20)(11,17)(12,18)(13,15)(14,16)(29,33)(30,34)$
4A $4^{6},2^{4},1^{2}$ $272$ $4$ $22$ $( 1,27, 2,28)( 3,26, 4,25)( 5,23, 6,24)( 7,21, 8,22)( 9,19)(10,20)(11,18)(12,17)(13,16,14,15)(29,33,30,34)$
4B $4^{6},2^{4},1^{2}$ $272$ $4$ $22$ $( 1,27, 2,28)( 3,26)( 4,25)( 5,23, 6,24)( 7,22)( 8,21)( 9,19,10,20)(11,18,12,17)(13,15,14,16)(29,33,30,34)$
4C $4^{4},2^{8},1^{2}$ $272$ $4$ $20$ $( 1,28)( 2,27)( 3,26, 4,25)( 5,23)( 6,24)( 7,21, 8,22)( 9,19,10,20)(11,17,12,18)(13,16)(14,15)(29,33)(30,34)$
4D $4^{5},2^{7}$ $544$ $4$ $22$ $( 1,28, 2,27)( 3,26, 4,25)( 5,24)( 6,23)( 7,22, 8,21)( 9,20)(10,19)(11,17,12,18)(13,15)(14,16)(29,34,30,33)(31,32)$
4E $4^{4},2^{8},1^{2}$ $544$ $4$ $20$ $( 1,28)( 2,27)( 3,26)( 4,25)( 5,23, 6,24)( 7,21, 8,22)( 9,20,10,19)(11,17)(12,18)(13,16)(14,15)(29,33,30,34)$
4F $4^{3},2^{11}$ $544$ $4$ $20$ $( 1,27)( 2,28)( 3,26)( 4,25)( 5,24, 6,23)( 7,21)( 8,22)( 9,20)(10,19)(11,18,12,17)(13,16,14,15)(29,34)(30,33)(31,32)$
4G $4^{3},2^{11}$ $544$ $4$ $20$ $( 1,28, 2,27)( 3,26)( 4,25)( 5,24, 6,23)( 7,22, 8,21)( 9,19)(10,20)(11,17)(12,18)(13,15)(14,16)(29,34)(30,33)(31,32)$
4H $4^{4},2^{8},1^{2}$ $544$ $4$ $20$ $( 1,27, 2,28)( 3,26, 4,25)( 5,23)( 6,24)( 7,22)( 8,21)( 9,20,10,19)(11,18)(12,17)(13,15,14,16)(29,33)(30,34)$
4I $4^{5},2^{7}$ $544$ $4$ $22$ $( 1,28, 2,27)( 3,26, 4,25)( 5,24, 6,23)( 7,21)( 8,22)( 9,19,10,20)(11,17,12,18)(13,16)(14,15)(29,34)(30,33)(31,32)$
4J1 $4^{8},1^{2}$ $1088$ $4$ $24$ $( 1,15,27,14)( 2,16,28,13)( 3, 7,25,22)( 4, 8,26,21)( 5,33,23,29)( 6,34,24,30)( 9,17,20,11)(10,18,19,12)$
4J-1 $4^{8},1^{2}$ $1088$ $4$ $24$ $( 1,14,28,16)( 2,13,27,15)( 3,21,25, 7)( 4,22,26, 8)( 5,29,23,33)( 6,30,24,34)( 9,12,20,17)(10,11,19,18)$
8A1 $8^{2},4^{4},1^{2}$ $1088$ $8$ $26$ $( 1,16,28,13)( 2,15,27,14)( 3, 7,26,21, 4, 8,25,22)( 5,33,23,29)( 6,34,24,30)( 9,18,19,11,10,17,20,12)$
8A-1 $8^{3},4^{2},2$ $1088$ $8$ $28$ $( 1,16,27,13, 2,15,28,14)( 3, 7,25,21)( 4, 8,26,22)( 5,34,23,29, 6,33,24,30)( 9,18,20,11,10,17,19,12)(31,32)$
8B1 $8^{3},4^{2},2$ $1088$ $8$ $28$ $( 1,14,27,16, 2,13,28,15)( 3,21,26, 7, 4,22,25, 8)( 5,30,23,33, 6,29,24,34)( 9,12,20,18)(10,11,19,17)(31,32)$
8B-1 $8^{3},4^{2},2$ $1088$ $8$ $28$ $( 1,13,27,16, 2,14,28,15)( 3,21,26, 7)( 4,22,25, 8)( 5,30,23,33, 6,29,24,34)( 9,11,20,17,10,12,19,18)(31,32)$
8C1 $8^{2},4^{4},1^{2}$ $1088$ $8$ $26$ $( 1,13,28,16)( 2,14,27,15)( 3,21,25, 7, 4,22,26, 8)( 5,29,23,33)( 6,30,24,34)( 9,11,20,18,10,12,19,17)$
8C-1 $8^{3},4^{2},2$ $1088$ $8$ $28$ $( 1,15,28,14, 2,16,27,13)( 3, 7,26,22, 4, 8,25,21)( 5,34,23,29, 6,33,24,30)( 9,17,19,11)(10,18,20,12)(31,32)$
17A1 $17^{2}$ $1024$ $17$ $32$ $( 1,16,29, 9,23, 4,18,32,12,26, 5,19,34,13,28, 7,22)( 2,15,30,10,24, 3,17,31,11,25, 6,20,33,14,27, 8,21)$
17A2 $17^{2}$ $1024$ $17$ $32$ $( 1,29,23,18,12, 5,34,28,22,16, 9, 4,32,26,19,13, 7)( 2,30,24,17,11, 6,33,27,21,15,10, 3,31,25,20,14, 8)$
17A3 $17^{2}$ $1024$ $17$ $32$ $( 1, 9,18,26,34, 7,16,23,32, 5,13,22,29, 4,12,19,28)( 2,10,17,25,33, 8,15,24,31, 6,14,21,30, 3,11,20,27)$
17A6 $17^{2}$ $1024$ $17$ $32$ $( 1,18,34,16,32,13,29,12,28, 9,26, 7,23, 5,22, 4,19)( 2,17,33,15,31,14,30,11,27,10,25, 8,24, 6,21, 3,20)$

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(G);
 

Character table

32 x 32 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed