Properties

Label 34T5
Degree $34$
Order $136$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{34}:C_4$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(34, 5);
 

Group invariants

Abstract group:  $C_{34}:C_4$
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 $t$:  $5$
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:  $(1,9,8,33)(2,10,7,34)(3,17,6,26)(4,18,5,25)(11,16,31,27)(12,15,32,28)(13,23,29,19)(14,24,30,20)$, $(1,26,20,29)(2,25,19,30)(3,33,17,21)(4,34,18,22)(5,7,16,13)(6,8,15,14)(9,23,11,32)(10,24,12,31)(27,28)$
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, $C_2^2$
$8$:  $C_4\times C_2$
$68$:  $C_{17}:C_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

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

Low degree siblings

34T5

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 $2^{16},1^{2}$ $17$ $2$ $16$ $( 3,34)( 4,33)( 5,31)( 6,32)( 7,29)( 8,30)( 9,27)(10,28)(11,25)(12,26)(13,23)(14,24)(15,22)(16,21)(17,19)(18,20)$
2C $2^{17}$ $17$ $2$ $17$ $( 1, 6)( 2, 5)( 3, 4)( 7,33)( 8,34)( 9,32)(10,31)(11,29)(12,30)(13,27)(14,28)(15,25)(16,26)(17,24)(18,23)(19,21)(20,22)$
4A1 $4^{8},1^{2}$ $17$ $4$ $24$ $( 3,10,34,28)( 4, 9,33,27)( 5,18,31,20)( 6,17,32,19)( 7,26,29,12)( 8,25,30,11)(13,15,23,22)(14,16,24,21)$
4A-1 $4^{8},1^{2}$ $17$ $4$ $24$ $( 3,28,34,10)( 4,27,33, 9)( 5,20,31,18)( 6,19,32,17)( 7,12,29,26)( 8,11,30,25)(13,22,23,15)(14,21,24,16)$
4B1 $4^{8},2$ $17$ $4$ $25$ $( 1,19,24, 6)( 2,20,23, 5)( 3,27,22,31)( 4,28,21,32)( 7, 9,17,16)( 8,10,18,15)(11,26,14,34)(12,25,13,33)(29,30)$
4B-1 $4^{8},2$ $17$ $4$ $25$ $( 1,17,21, 6)( 2,18,22, 5)( 3, 9,19,14)( 4,10,20,13)( 7,27,15,30)( 8,28,16,29)(11,12)(23,31,34,25)(24,32,33,26)$
17A1 $17^{2}$ $4$ $17$ $32$ $( 1,11,21,31, 8,18,27, 4,14,24,33, 9,20,30, 5,16,25)( 2,12,22,32, 7,17,28, 3,13,23,34,10,19,29, 6,15,26)$
17A2 $17^{2}$ $4$ $17$ $32$ $( 1,21, 8,27,14,33,20, 5,25,11,31,18, 4,24, 9,30,16)( 2,22, 7,28,13,34,19, 6,26,12,32,17, 3,23,10,29,15)$
17A3 $17^{2}$ $4$ $17$ $32$ $( 1,27,20,11, 4,30,21,14, 5,31,24,16, 8,33,25,18, 9)( 2,28,19,12, 3,29,22,13, 6,32,23,15, 7,34,26,17,10)$
17A6 $17^{2}$ $4$ $17$ $32$ $( 1,20, 4,21, 5,24, 8,25, 9,27,11,30,14,31,16,33,18)( 2,19, 3,22, 6,23, 7,26,10,28,12,29,13,32,15,34,17)$
34A1 $34$ $4$ $34$ $33$ $( 1, 7,14,19,25,32, 4,10,16,22,27,34, 5,12,18,23,30, 2, 8,13,20,26,31, 3, 9,15,21,28,33, 6,11,17,24,29)$
34A3 $34$ $4$ $34$ $33$ $( 1,34,31,29,27,26,24,22,20,17,16,13,11,10, 8, 6, 4, 2,33,32,30,28,25,23,21,19,18,15,14,12, 9, 7, 5, 3)$
34A7 $34$ $4$ $34$ $33$ $( 1,32,27,23,20,15,11, 7, 4,34,30,26,21,17,14,10, 5, 2,31,28,24,19,16,12, 8, 3,33,29,25,22,18,13, 9, 6)$
34A9 $34$ $4$ $34$ $33$ $( 1,23,11,34,21,10,31,19, 8,29,18, 6,27,15, 4,26,14, 2,24,12,33,22, 9,32,20, 7,30,17, 5,28,16, 3,25,13)$

Malle's constant $a(G)$:     $1/16$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 17A1 17A2 17A3 17A6 34A1 34A3 34A7 34A9
Size 1 1 17 17 17 17 17 17 4 4 4 4 4 4 4 4
2 P 1A 1A 1A 1A 2B 2B 2B 2B 17A6 17A3 17A2 17A1 17A6 17A2 17A1 17A3
17 P 1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 1A 1A 1A 1A 2A 2A 2A 2A
Type
136.13.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
136.13.1b R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
136.13.1c R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
136.13.1d R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
136.13.1e1 C 1 1 1 1 i i i i 1 1 1 1 1 1 1 1
136.13.1e2 C 1 1 1 1 i i i i 1 1 1 1 1 1 1 1
136.13.1f1 C 1 1 1 1 i i i i 1 1 1 1 1 1 1 1
136.13.1f2 C 1 1 1 1 i i i i 1 1 1 1 1 1 1 1
136.13.4a1 R 4 4 0 0 0 0 0 0 ζ177+ζ176+ζ176+ζ177 ζ175+ζ173+ζ173+ζ175 ζ174+ζ171+ζ17+ζ174 ζ178+ζ172+ζ172+ζ178 ζ175+ζ173+ζ173+ζ175 ζ178+ζ172+ζ172+ζ178 ζ174+ζ171+ζ17+ζ174 ζ177+ζ176+ζ176+ζ177
136.13.4a2 R 4 4 0 0 0 0 0 0 ζ178+ζ172+ζ172+ζ178 ζ174+ζ171+ζ17+ζ174 ζ177+ζ176+ζ176+ζ177 ζ175+ζ173+ζ173+ζ175 ζ174+ζ171+ζ17+ζ174 ζ175+ζ173+ζ173+ζ175 ζ177+ζ176+ζ176+ζ177 ζ178+ζ172+ζ172+ζ178
136.13.4a3 R 4 4 0 0 0 0 0 0 ζ175+ζ173+ζ173+ζ175 ζ177+ζ176+ζ176+ζ177 ζ178+ζ172+ζ172+ζ178 ζ174+ζ171+ζ17+ζ174 ζ177+ζ176+ζ176+ζ177 ζ174+ζ171+ζ17+ζ174 ζ178+ζ172+ζ172+ζ178 ζ175+ζ173+ζ173+ζ175
136.13.4a4 R 4 4 0 0 0 0 0 0 ζ174+ζ171+ζ17+ζ174 ζ178+ζ172+ζ172+ζ178 ζ175+ζ173+ζ173+ζ175 ζ177+ζ176+ζ176+ζ177 ζ178+ζ172+ζ172+ζ178 ζ177+ζ176+ζ176+ζ177 ζ175+ζ173+ζ173+ζ175 ζ174+ζ171+ζ17+ζ174
136.13.4b1 R 4 4 0 0 0 0 0 0 ζ177+ζ176+ζ176+ζ177 ζ175+ζ173+ζ173+ζ175 ζ174+ζ171+ζ17+ζ174 ζ178+ζ172+ζ172+ζ178 ζ175ζ173ζ173ζ175 ζ178ζ172ζ172ζ178 ζ174ζ171ζ17ζ174 ζ177ζ176ζ176ζ177
136.13.4b2 R 4 4 0 0 0 0 0 0 ζ178+ζ172+ζ172+ζ178 ζ174+ζ171+ζ17+ζ174 ζ177+ζ176+ζ176+ζ177 ζ175+ζ173+ζ173+ζ175 ζ174ζ171ζ17ζ174 ζ175ζ173ζ173ζ175 ζ177ζ176ζ176ζ177 ζ178ζ172ζ172ζ178
136.13.4b3 R 4 4 0 0 0 0 0 0 ζ175+ζ173+ζ173+ζ175 ζ177+ζ176+ζ176+ζ177 ζ178+ζ172+ζ172+ζ178 ζ174+ζ171+ζ17+ζ174 ζ177ζ176ζ176ζ177 ζ174ζ171ζ17ζ174 ζ178ζ172ζ172ζ178 ζ175ζ173ζ173ζ175
136.13.4b4 R 4 4 0 0 0 0 0 0 ζ174+ζ171+ζ17+ζ174 ζ178+ζ172+ζ172+ζ178 ζ175+ζ173+ζ173+ζ175 ζ177+ζ176+ζ176+ζ177 ζ178ζ172ζ172ζ178 ζ177ζ176ζ176ζ177 ζ175ζ173ζ173ζ175 ζ174ζ171ζ17ζ174

magma: CharacterTable(G);
 

Regular extensions

Data not computed