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Properties

Label 34T7
Degree $34$
Order $272$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{34}:C_8$

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Show commands: Magma

magma: G := TransitiveGroup(34, 7);
 

Group invariants

Abstract group:  $C_{34}:C_8$
magma: IdentifyGroup(G);
 
Order:  $272=2^{4} \cdot 17$
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 $t$:  $7$
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:  $(1,34,17,25,21,24,5,31)(2,33,18,26,22,23,6,32)(3,16,10,29,20,8,13,27)(4,15,9,30,19,7,14,28)$, $(1,7,29,19,5,33,11,22)(2,8,30,20,6,34,12,21)(3,4)(9,25,28,24,32,16,14,17)(10,26,27,23,31,15,13,18)$
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, $C_4\times C_2$
$16$:  $C_8\times C_2$
$136$:  $C_{17}:C_{8}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

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

Low degree siblings

34T7

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

Malle's constant $a(G)$:     $1/16$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 8A1 8A-1 8A3 8A-3 8B1 8B-1 8B3 8B-3 17A1 17A3 34A1 34A3
Size 1 1 17 17 17 17 17 17 17 17 17 17 17 17 17 17 8 8 8 8
2 P 1A 1A 1A 1A 2B 2B 2B 2B 4A1 4A1 4A-1 4A1 4A-1 4A-1 4A-1 4A1 17A1 17A3 17A1 17A3
17 P 1A 2A 2B 2C 4A1 4B1 4B-1 4A-1 8A1 8A-3 8B3 8B1 8A-1 8A3 8B-1 8B-3 1A 1A 2A 2A
Type
272.51.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
272.51.1b R 1 −1 1 −1 1 1 −1 −1 −1 −1 −1 −1 1 1 1 1 1 1 −1 −1
272.51.1c R 1 −1 1 −1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1
272.51.1d R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1
272.51.1e1 C 1 −1 1 −1 −1 −1 1 1 −i i i −i i −i −i i 1 1 −1 −1
272.51.1e2 C 1 −1 1 −1 −1 −1 1 1 i −i −i i −i i i −i 1 1 −1 −1
272.51.1f1 C 1 1 1 1 −1 −1 −1 −1 −i i i −i −i i i −i 1 1 1 1
272.51.1f2 C 1 1 1 1 −1 −1 −1 −1 i −i −i i i −i −i i 1 1 1 1
272.51.1g1 C 1 −1 −1 1 −ζ82 ζ82 ζ82 −ζ82 ζ83 −ζ8 ζ8 −ζ83 ζ83 −ζ8 ζ8 −ζ83 1 1 −1 −1
272.51.1g2 C 1 −1 −1 1 ζ82 −ζ82 −ζ82 ζ82 −ζ8 ζ83 −ζ83 ζ8 −ζ8 ζ83 −ζ83 ζ8 1 1 −1 −1
272.51.1g3 C 1 −1 −1 1 −ζ82 ζ82 ζ82 −ζ82 −ζ83 ζ8 −ζ8 ζ83 −ζ83 ζ8 −ζ8 ζ83 1 1 −1 −1
272.51.1g4 C 1 −1 −1 1 ζ82 −ζ82 −ζ82 ζ82 ζ8 −ζ83 ζ83 −ζ8 ζ8 −ζ83 ζ83 −ζ8 1 1 −1 −1
272.51.1h1 C 1 1 −1 −1 −ζ82 ζ82 −ζ82 ζ82 ζ83 −ζ8 ζ8 −ζ83 −ζ83 ζ8 −ζ8 ζ83 1 1 1 1
272.51.1h2 C 1 1 −1 −1 ζ82 −ζ82 ζ82 −ζ82 −ζ8 ζ83 −ζ83 ζ8 ζ8 −ζ83 ζ83 −ζ8 1 1 1 1
272.51.1h3 C 1 1 −1 −1 −ζ82 ζ82 −ζ82 ζ82 −ζ83 ζ8 −ζ8 ζ83 ζ83 −ζ8 ζ8 −ζ83 1 1 1 1
272.51.1h4 C 1 1 −1 −1 ζ82 −ζ82 ζ82 −ζ82 ζ8 −ζ83 ζ83 −ζ8 −ζ8 ζ83 −ζ83 ζ8 1 1 1 1
272.51.8a1 R 8 8 0 0 0 0 0 0 0 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 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
272.51.8a2 R 8 8 0 0 0 0 0 0 0 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 −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
272.51.8b1 R 8 −8 0 0 0 0 0 0 0 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 −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
272.51.8b2 R 8 −8 0 0 0 0 0 0 0 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 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