↑

Properties

Label 34T36
34T36 1 12 1->12 34 1->34 2 4 2->4 31 2->31 3 13 3->13 28 3->28 5 4->5 25 4->25 14 5->14 22 5->22 6 19 6->19 7 15 7->15 33 7->33 8 8->7 30 8->30 9 16 9->16 27 9->27 10 10->8 24 10->24 11 17 11->17 21 11->21 12->9 18 12->18 13->1 32 13->32 14->10 29 14->29 15->2 26 15->26 16->11 23 16->23 17->3 20 17->20 18->11 18->19 19->14 19->27 20->18 21->3 21->26 22->6 22->34 23->9 23->25 24->12 24->33 25->15 25->24 26->1 26->32 27->4 27->23 28->7 28->31 29->10 29->22 30->13 31->16 31->21 32->2 32->29 33->5 33->20 34->8 34->28
Degree $34$
Order $9248$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{17}^2:\OD_{32}$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(34, 36);
 

Group invariants

Abstract group:  $C_{17}^2:\OD_{32}$
magma: IdentifyGroup(G);
 
Order:  $9248=2^{5} \cdot 17^{2}$
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 $t$:  $36$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,34,8,30,13,32,2,31,16,23,9,27,4,25,15,26)(3,28,7,33,5,22,6,19,14,29,10,24,12,18,11,21)(17,20)$, $(1,12,9,16,11,17,3,13)(2,4,5,14,10,8,7,15)(18,19,27,23,25,24,33,20)(21,26,32,29,22,34,28,31)$
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, $C_4\times C_2$
$16$:  $C_8\times C_2$
$32$:  $C_{16} : C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: None

Low degree siblings

34T34 x 2

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 $2^{8},1^{18}$ $34$ $2$ $8$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,17)(10,16)(11,15)(12,14)$
2B $2^{16},1^{2}$ $289$ $2$ $16$ $( 1, 6)( 2, 5)( 3, 4)( 7,17)( 8,16)( 9,15)(10,14)(11,13)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(32,34)$
4A1 $4^{8},1^{2}$ $289$ $4$ $24$ $( 1, 2, 6, 5)( 3,10, 4,14)( 7, 9,17,15)( 8,13,16,11)(18,24,31,25)(19,28,30,21)(20,32,29,34)(22,23,27,26)$
4A-1 $4^{8},1^{2}$ $289$ $4$ $24$ $( 1, 5, 6, 2)( 3,14, 4,10)( 7,15,17, 9)( 8,11,16,13)(18,25,31,24)(19,21,30,28)(20,34,29,32)(22,26,27,23)$
4B $4^{8},1^{2}$ $578$ $4$ $24$ $( 1,17, 4, 5)( 2,13, 3, 9)( 6,14,16, 8)( 7,10,15,12)(18,32,20,23)(21,27,34,28)(22,31,33,24)(25,26,30,29)$
8A1 $8^{4},1^{2}$ $289$ $8$ $28$ $( 1,17, 2,15, 6, 7, 5, 9)( 3,13,10,16, 4,11,14, 8)(18,29,24,34,31,20,25,32)(19,27,28,26,30,22,21,23)$
8A-1 $8^{4},1^{2}$ $289$ $8$ $28$ $( 1, 9, 5, 7, 6,15, 2,17)( 3, 8,14,11, 4,16,10,13)(18,32,25,20,31,34,24,29)(19,23,21,22,30,26,28,27)$
8A3 $8^{4},1^{2}$ $289$ $8$ $28$ $( 1,15, 5,17, 6, 9, 2, 7)( 3,16,14,13, 4, 8,10,11)(18,34,25,29,31,32,24,20)(19,26,21,27,30,23,28,22)$
8A-3 $8^{4},1^{2}$ $289$ $8$ $28$ $( 1, 7, 2, 9, 6,17, 5,15)( 3,11,10, 8, 4,13,14,16)(18,20,24,32,31,29,25,34)(19,22,28,23,30,27,21,26)$
8B1 $8^{4},1^{2}$ $578$ $8$ $28$ $( 1,11,16,10, 7,14, 9,15)( 2, 3,12, 8, 6, 5,13,17)(18,31,33,32,24,28,26,27)(19,22,29,34,23,20,30,25)$
8B-1 $8^{4},1^{2}$ $578$ $8$ $28$ $( 1,15, 9,14, 7,10,16,11)( 2,17,13, 5, 6, 8,12, 3)(18,27,26,28,24,32,33,31)(19,25,30,20,23,34,29,22)$
16A1 $16^{2},2$ $578$ $16$ $31$ $( 1,18,17,29, 2,24,15,34, 6,31, 7,20, 5,25, 9,32)( 3,30,13,22,10,21,16,23, 4,19,11,27,14,28, 8,26)(12,33)$
16A-1 $16^{2},2$ $578$ $16$ $31$ $( 1,32, 9,25, 5,20, 7,31, 6,34,15,24, 2,29,17,18)( 3,26, 8,28,14,27,11,19, 4,23,16,21,10,22,13,30)(12,33)$
16A3 $16^{2},2$ $578$ $16$ $31$ $( 1,29,15,31, 5,32,17,24, 6,20, 9,18, 2,34, 7,25)( 3,22,16,19,14,26,13,21, 4,27, 8,30,10,23,11,28)(12,33)$
16A-3 $16^{2},2$ $578$ $16$ $31$ $( 1,25, 7,34, 2,18, 9,20, 6,24,17,32, 5,31,15,29)( 3,28,11,23,10,30, 8,27, 4,21,13,26,14,19,16,22)(12,33)$
16B1 $16^{2},2$ $578$ $16$ $31$ $( 1,28,12,29, 9,21,16,34,11,32,17,31, 3,22,13,26)( 2,25, 4,19, 5,33,14,23,10,18, 8,24, 7,27,15,20)( 6,30)$
16B-1 $16^{2},2$ $578$ $16$ $31$ $( 1,26,13,22, 3,31,17,32,11,34,16,21, 9,29,12,28)( 2,20,15,27, 7,24, 8,18,10,23,14,33, 5,19, 4,25)( 6,30)$
16B3 $16^{2},2$ $578$ $16$ $31$ $( 1,29,16,32, 3,26,12,21,11,31,13,28, 9,34,17,22)( 2,19,14,18, 7,20, 4,33,10,24,15,25, 5,23, 8,27)( 6,30)$
16B-3 $16^{2},2$ $578$ $16$ $31$ $( 1,22,17,34, 9,28,13,31,11,21,12,26, 3,32,16,29)( 2,27, 8,23, 5,25,15,24,10,33, 4,20, 7,18,14,19)( 6,30)$
17A1 $17,1^{17}$ $16$ $17$ $16$ $(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$
17A3 $17,1^{17}$ $16$ $17$ $16$ $(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$
17B $17^{2}$ $32$ $17$ $32$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$
17C $17^{2}$ $32$ $17$ $32$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$
17D1 $17^{2}$ $32$ $17$ $32$ $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$
17D3 $17^{2}$ $32$ $17$ $32$ $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$
17E1 $17^{2}$ $32$ $17$ $32$ $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,20,22,24,26,28,30,32,34,19,21,23,25,27,29,31,33)$
17E3 $17^{2}$ $32$ $17$ $32$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$
17F1 $17^{2}$ $32$ $17$ $32$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$
17F3 $17^{2}$ $32$ $17$ $32$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$
34A1 $17,2^{8},1$ $272$ $34$ $24$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,17)(10,16)(11,15)(12,14)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$
34A3 $17,2^{8},1$ $272$ $34$ $24$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,17)(10,16)(11,15)(12,14)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$

Malle's constant $a(G)$:     $1/8$

magma: ConjugacyClasses(G);
 

Character table

32 x 32 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed