↑

Properties

Label 34T3
34T3 1 5 1->5 24 1->24 2 6 2->6 23 2->23 3 8 3->8 21 3->21 4 7 4->7 22 4->22 10 5->10 19 5->19 9 6->9 20 6->20 12 7->12 17 7->17 11 8->11 18 8->18 13 9->13 16 9->16 14 10->14 15 10->15 11->13 11->16 12->14 12->15 13->18 14->17 15->19 16->20 17->22 18->21 19->24 20->23 26 21->26 25 22->25 27 23->27 28 24->28 29 25->29 34 25->34 30 26->30 33 26->33 32 27->32 27->32 31 28->31 28->31 29->30 29->33 30->34 31->1 32->2 33->3 34->4
Degree $34$
Order $68$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{34}$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(34, 3);
 

Group invariants

Abstract group:  $D_{34}$
magma: IdentifyGroup(G);
 
Order:  $68=2^{2} \cdot 17$
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 $t$:  $3$
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:  $(1,5,10,14,17,22,25,29,33,3,8,11,16,20,23,27,32,2,6,9,13,18,21,26,30,34,4,7,12,15,19,24,28,31)$, $(1,24)(2,23)(3,21)(4,22)(5,19)(6,20)(7,17)(8,18)(9,16)(10,15)(11,13)(12,14)(25,34)(26,33)(27,32)(28,31)(29,30)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$34$:  $D_{17}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: $D_{17}$

Low degree siblings

34T3

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 $2^{16},1^{2}$ $17$ $2$ $16$ $( 3,34)( 4,33)( 5,31)( 6,32)( 7,29)( 8,30)( 9,27)(10,28)(11,26)(12,25)(13,23)(14,24)(15,22)(16,21)(17,19)(18,20)$
2C $2^{17}$ $17$ $2$ $17$ $( 1,31)( 2,32)( 3,30)( 4,29)( 5,28)( 6,27)( 7,25)( 8,26)( 9,23)(10,24)(11,21)(12,22)(13,20)(14,19)(15,17)(16,18)(33,34)$
17A1 $17^{2}$ $2$ $17$ $32$ $( 1,28,19,12, 4,30,21,13, 6,32,23,16, 8,33,25,17,10)( 2,27,20,11, 3,29,22,14, 5,31,24,15, 7,34,26,18, 9)$
17A2 $17^{2}$ $2$ $17$ $32$ $( 1,19, 4,21, 6,23, 8,25,10,28,12,30,13,32,16,33,17)( 2,20, 3,22, 5,24, 7,26, 9,27,11,29,14,31,15,34,18)$
17A3 $17^{2}$ $2$ $17$ $32$ $( 1,12,21,32, 8,17,28, 4,13,23,33,10,19,30, 6,16,25)( 2,11,22,31, 7,18,27, 3,14,24,34, 9,20,29, 5,15,26)$
17A4 $17^{2}$ $2$ $17$ $32$ $( 1, 4, 6, 8,10,12,13,16,17,19,21,23,25,28,30,32,33)( 2, 3, 5, 7, 9,11,14,15,18,20,22,24,26,27,29,31,34)$
17A5 $17^{2}$ $2$ $17$ $32$ $( 1,30,23,17,12, 6,33,28,21,16,10, 4,32,25,19,13, 8)( 2,29,24,18,11, 5,34,27,22,15, 9, 3,31,26,20,14, 7)$
17A6 $17^{2}$ $2$ $17$ $32$ $( 1,21, 8,28,13,33,19, 6,25,12,32,17, 4,23,10,30,16)( 2,22, 7,27,14,34,20, 5,26,11,31,18, 3,24, 9,29,15)$
17A7 $17^{2}$ $2$ $17$ $32$ $( 1,13,25, 4,16,28, 6,17,30, 8,19,32,10,21,33,12,23)( 2,14,26, 3,15,27, 5,18,29, 7,20,31, 9,22,34,11,24)$
17A8 $17^{2}$ $2$ $17$ $32$ $( 1, 6,10,13,17,21,25,30,33, 4, 8,12,16,19,23,28,32)( 2, 5, 9,14,18,22,26,29,34, 3, 7,11,15,20,24,27,31)$
34A1 $34$ $2$ $34$ $33$ $( 1,31,28,24,19,15,12, 7, 4,34,30,26,21,18,13, 9, 6, 2,32,27,23,20,16,11, 8, 3,33,29,25,22,17,14,10, 5)$
34A3 $34$ $2$ $34$ $33$ $( 1,24,12,34,21, 9,32,20, 8,29,17, 5,28,15, 4,26,13, 2,23,11,33,22,10,31,19, 7,30,18, 6,27,16, 3,25,14)$
34A5 $34$ $2$ $34$ $33$ $( 1,15,30, 9,23, 3,17,31,12,26, 6,20,33,14,28, 7,21, 2,16,29,10,24, 4,18,32,11,25, 5,19,34,13,27, 8,22)$
34A7 $34$ $2$ $34$ $33$ $( 1, 7,13,20,25,31, 4, 9,16,22,28,34, 6,11,17,24,30, 2, 8,14,19,26,32, 3,10,15,21,27,33, 5,12,18,23,29)$
34A9 $34$ $2$ $34$ $33$ $( 1,34,32,29,28,26,23,22,19,18,16,14,12, 9, 8, 5, 4, 2,33,31,30,27,25,24,21,20,17,15,13,11,10, 7, 6, 3)$
34A11 $34$ $2$ $34$ $33$ $( 1,26,16, 5,30,20,10,34,23,14, 4,27,17, 7,32,22,12, 2,25,15, 6,29,19, 9,33,24,13, 3,28,18, 8,31,21,11)$
34A13 $34$ $2$ $34$ $33$ $( 1,18,33,15,32,14,30,11,28, 9,25, 7,23, 5,21, 3,19, 2,17,34,16,31,13,29,12,27,10,26, 8,24, 6,22, 4,20)$
34A15 $34$ $2$ $34$ $33$ $( 1, 9,17,26,33, 7,16,24,32, 5,13,22,30, 3,12,20,28, 2,10,18,25,34, 8,15,23,31, 6,14,21,29, 4,11,19,27)$

Malle's constant $a(G)$:     $1/16$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 17A1 17A2 17A3 17A4 17A5 17A6 17A7 17A8 34A1 34A3 34A5 34A7 34A9 34A11 34A13 34A15
Size 1 1 17 17 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 17A2 17A4 17A6 17A8 17A7 17A5 17A3 17A1 17A1 17A3 17A5 17A7 17A8 17A6 17A4 17A2
17 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 2A 2A
Type
68.4.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
68.4.1b R 1 −1 −1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
68.4.1c R 1 −1 1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
68.4.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
68.4.2a1 R 2 2 0 0 ζ17−8+ζ178 ζ17−3+ζ173 ζ17−6+ζ176 ζ17−5+ζ175 ζ17−1+ζ17 ζ17−7+ζ177 ζ17−4+ζ174 ζ17−2+ζ172 ζ17−5+ζ175 ζ17−6+ζ176 ζ17−3+ζ173 ζ17−8+ζ178 ζ17−2+ζ172 ζ17−4+ζ174 ζ17−7+ζ177 ζ17−1+ζ17
68.4.2a2 R 2 2 0 0 ζ17−7+ζ177 ζ17−8+ζ178 ζ17−1+ζ17 ζ17−2+ζ172 ζ17−3+ζ173 ζ17−4+ζ174 ζ17−5+ζ175 ζ17−6+ζ176 ζ17−2+ζ172 ζ17−1+ζ17 ζ17−8+ζ178 ζ17−7+ζ177 ζ17−6+ζ176 ζ17−5+ζ175 ζ17−4+ζ174 ζ17−3+ζ173
68.4.2a3 R 2 2 0 0 ζ17−6+ζ176 ζ17−2+ζ172 ζ17−4+ζ174 ζ17−8+ζ178 ζ17−5+ζ175 ζ17−1+ζ17 ζ17−3+ζ173 ζ17−7+ζ177 ζ17−8+ζ178 ζ17−4+ζ174 ζ17−2+ζ172 ζ17−6+ζ176 ζ17−7+ζ177 ζ17−3+ζ173 ζ17−1+ζ17 ζ17−5+ζ175
68.4.2a4 R 2 2 0 0 ζ17−5+ζ175 ζ17−4+ζ174 ζ17−8+ζ178 ζ17−1+ζ17 ζ17−7+ζ177 ζ17−2+ζ172 ζ17−6+ζ176 ζ17−3+ζ173 ζ17−1+ζ17 ζ17−8+ζ178 ζ17−4+ζ174 ζ17−5+ζ175 ζ17−3+ζ173 ζ17−6+ζ176 ζ17−2+ζ172 ζ17−7+ζ177
68.4.2a5 R 2 2 0 0 ζ17−4+ζ174 ζ17−7+ζ177 ζ17−3+ζ173 ζ17−6+ζ176 ζ17−8+ζ178 ζ17−5+ζ175 ζ17−2+ζ172 ζ17−1+ζ17 ζ17−6+ζ176 ζ17−3+ζ173 ζ17−7+ζ177 ζ17−4+ζ174 ζ17−1+ζ17 ζ17−2+ζ172 ζ17−5+ζ175 ζ17−8+ζ178
68.4.2a6 R 2 2 0 0 ζ17−3+ζ173 ζ17−1+ζ17 ζ17−2+ζ172 ζ17−4+ζ174 ζ17−6+ζ176 ζ17−8+ζ178 ζ17−7+ζ177 ζ17−5+ζ175 ζ17−4+ζ174 ζ17−2+ζ172 ζ17−1+ζ17 ζ17−3+ζ173 ζ17−5+ζ175 ζ17−7+ζ177 ζ17−8+ζ178 ζ17−6+ζ176
68.4.2a7 R 2 2 0 0 ζ17−2+ζ172 ζ17−5+ζ175 ζ17−7+ζ177 ζ17−3+ζ173 ζ17−4+ζ174 ζ17−6+ζ176 ζ17−1+ζ17 ζ17−8+ζ178 ζ17−3+ζ173 ζ17−7+ζ177 ζ17−5+ζ175 ζ17−2+ζ172 ζ17−8+ζ178 ζ17−1+ζ17 ζ17−6+ζ176 ζ17−4+ζ174
68.4.2a8 R 2 2 0 0 ζ17−1+ζ17 ζ17−6+ζ176 ζ17−5+ζ175 ζ17−7+ζ177 ζ17−2+ζ172 ζ17−3+ζ173 ζ17−8+ζ178 ζ17−4+ζ174 ζ17−7+ζ177 ζ17−5+ζ175 ζ17−6+ζ176 ζ17−1+ζ17 ζ17−4+ζ174 ζ17−8+ζ178 ζ17−3+ζ173 ζ17−2+ζ172
68.4.2b1 R 2 −2 0 0 ζ17−8+ζ178 ζ17−3+ζ173 ζ17−6+ζ176 ζ17−5+ζ175 ζ17−1+ζ17 ζ17−7+ζ177 ζ17−4+ζ174 ζ17−2+ζ172 −ζ17−5−ζ175 −ζ17−6−ζ176 −ζ17−3−ζ173 −ζ17−8−ζ178 −ζ17−2−ζ172 −ζ17−4−ζ174 −ζ17−7−ζ177 −ζ17−1−ζ17
68.4.2b2 R 2 −2 0 0 ζ17−7+ζ177 ζ17−8+ζ178 ζ17−1+ζ17 ζ17−2+ζ172 ζ17−3+ζ173 ζ17−4+ζ174 ζ17−5+ζ175 ζ17−6+ζ176 −ζ17−2−ζ172 −ζ17−1−ζ17 −ζ17−8−ζ178 −ζ17−7−ζ177 −ζ17−6−ζ176 −ζ17−5−ζ175 −ζ17−4−ζ174 −ζ17−3−ζ173
68.4.2b3 R 2 −2 0 0 ζ17−6+ζ176 ζ17−2+ζ172 ζ17−4+ζ174 ζ17−8+ζ178 ζ17−5+ζ175 ζ17−1+ζ17 ζ17−3+ζ173 ζ17−7+ζ177 −ζ17−8−ζ178 −ζ17−4−ζ174 −ζ17−2−ζ172 −ζ17−6−ζ176 −ζ17−7−ζ177 −ζ17−3−ζ173 −ζ17−1−ζ17 −ζ17−5−ζ175
68.4.2b4 R 2 −2 0 0 ζ17−5+ζ175 ζ17−4+ζ174 ζ17−8+ζ178 ζ17−1+ζ17 ζ17−7+ζ177 ζ17−2+ζ172 ζ17−6+ζ176 ζ17−3+ζ173 −ζ17−1−ζ17 −ζ17−8−ζ178 −ζ17−4−ζ174 −ζ17−5−ζ175 −ζ17−3−ζ173 −ζ17−6−ζ176 −ζ17−2−ζ172 −ζ17−7−ζ177
68.4.2b5 R 2 −2 0 0 ζ17−4+ζ174 ζ17−7+ζ177 ζ17−3+ζ173 ζ17−6+ζ176 ζ17−8+ζ178 ζ17−5+ζ175 ζ17−2+ζ172 ζ17−1+ζ17 −ζ17−6−ζ176 −ζ17−3−ζ173 −ζ17−7−ζ177 −ζ17−4−ζ174 −ζ17−1−ζ17 −ζ17−2−ζ172 −ζ17−5−ζ175 −ζ17−8−ζ178
68.4.2b6 R 2 −2 0 0 ζ17−3+ζ173 ζ17−1+ζ17 ζ17−2+ζ172 ζ17−4+ζ174 ζ17−6+ζ176 ζ17−8+ζ178 ζ17−7+ζ177 ζ17−5+ζ175 −ζ17−4−ζ174 −ζ17−2−ζ172 −ζ17−1−ζ17 −ζ17−3−ζ173 −ζ17−5−ζ175 −ζ17−7−ζ177 −ζ17−8−ζ178 −ζ17−6−ζ176
68.4.2b7 R 2 −2 0 0 ζ17−2+ζ172 ζ17−5+ζ175 ζ17−7+ζ177 ζ17−3+ζ173 ζ17−4+ζ174 ζ17−6+ζ176 ζ17−1+ζ17 ζ17−8+ζ178 −ζ17−3−ζ173 −ζ17−7−ζ177 −ζ17−5−ζ175 −ζ17−2−ζ172 −ζ17−8−ζ178 −ζ17−1−ζ17 −ζ17−6−ζ176 −ζ17−4−ζ174
68.4.2b8 R 2 −2 0 0 ζ17−1+ζ17 ζ17−6+ζ176 ζ17−5+ζ175 ζ17−7+ζ177 ζ17−2+ζ172 ζ17−3+ζ173 ζ17−8+ζ178 ζ17−4+ζ174 −ζ17−7−ζ177 −ζ17−5−ζ175 −ζ17−6−ζ176 −ζ17−1−ζ17 −ζ17−4−ζ174 −ζ17−8−ζ178 −ζ17−3−ζ173 −ζ17−2−ζ172

magma: CharacterTable(G);
 

Regular extensions

Data not computed