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Properties

Label 34T6
Degree $34$
Order $136$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{17}:C_8$

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Show commands: Magma

magma: G := TransitiveGroup(34, 6);
 

Group invariants

Abstract group:  $C_{17}:C_8$
magma: IdentifyGroup(G);
 
Order:  $136=2^{3} \cdot 17$
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 $t$:  $6$
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:  $(1,28,24,21,4,12,15,17)(2,27,23,22,3,11,16,18)(5,30,7,14,34,9,31,25)(6,29,8,13,33,10,32,26)(19,20)$, $(1,20)(2,19)(3,17)(4,18)(5,16)(6,15)(7,13)(8,14)(9,11)(10,12)(21,34)(22,33)(23,31)(24,32)(25,30)(26,29)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$8$:  $C_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: $C_{17}:C_{8}$

Low degree siblings

17T4

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

Malle's constant $a(G)$:     $1/16$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 4A1 4A-1 8A1 8A-1 8A3 8A-3 17A1 17A3
Size 1 17 17 17 17 17 17 17 8 8
2 P 1A 1A 2A 2A 4A1 4A-1 4A-1 4A1 17A1 17A3
17 P 1A 2A 4A1 4A-1 8A1 8A-1 8A3 8A-3 1A 1A
Type
136.12.1a R 1 1 1 1 1 1 1 1 1 1
136.12.1b R 1 1 1 1 −1 −1 −1 −1 1 1
136.12.1c1 C 1 1 −1 −1 −i i i −i 1 1
136.12.1c2 C 1 1 −1 −1 i −i −i i 1 1
136.12.1d1 C 1 −1 −ζ82 ζ82 ζ83 −ζ8 ζ8 −ζ83 1 1
136.12.1d2 C 1 −1 ζ82 −ζ82 −ζ8 ζ83 −ζ83 ζ8 1 1
136.12.1d3 C 1 −1 −ζ82 ζ82 −ζ83 ζ8 −ζ8 ζ83 1 1
136.12.1d4 C 1 −1 ζ82 −ζ82 ζ8 −ζ83 ζ83 −ζ8 1 1
136.12.8a1 R 8 0 0 0 0 0 0 0 ζ17−7+ζ17−6+ζ17−5+ζ17−3+ζ173+ζ175+ζ176+ζ177 −ζ17−7−ζ17−6−ζ17−5−ζ17−3−1−ζ173−ζ175−ζ176−ζ177
136.12.8a2 R 8 0 0 0 0 0 0 0 −ζ17−7−ζ17−6−ζ17−5−ζ17−3−1−ζ173−ζ175−ζ176−ζ177 ζ17−7+ζ17−6+ζ17−5+ζ17−3+ζ173+ζ175+ζ176+ζ177

magma: CharacterTable(G);
 

Regular extensions

Data not computed