Properties

Label 34T16
34T16 1 31 1->31 32 1->32 2 27 2->27 28 2->28 3 22 3->22 25 3->25 4 4->22 34 4->34 5 19 5->19 29 5->29 6 24 6->24 33 6->33 7 7->19 30 7->30 8 8->27 8->31 9 9->24 26 9->26 10 21 10->21 10->21 11 18 11->18 11->33 12 12->28 12->32 13 23 13->23 13->29 14 14->18 14->26 15 15->23 15->30 16 20 16->20 16->25 17 17->20 17->34 18->8 18->16 19->1 20->11 20->11 21->4 21->17 22->6 22->14 23->7 23->12 24->1 24->17 25->7 25->10 26->3 26->13 27->2 27->13 28->6 28->8 29->14 29->16 30->3 30->9 31->2 31->9 32->12 32->15 33->4 33->5 34->10 34->15
Degree $34$
Order $2312$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{17}^2.C_2$

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Show commands: Magma

magma: G := TransitiveGroup(34, 16);
 

Group invariants

Abstract group:  $D_{17}^2.C_2$
magma: IdentifyGroup(G);
 
Order:  $2312=2^{3} \cdot 17^{2}$
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 $t$:  $16$
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:  $(1,31,9,24)(2,28,8,27)(3,25,7,30)(4,22,6,33)(5,19)(10,21,17,34)(11,18,16,20)(12,32,15,23)(13,29,14,26)$, $(1,32,12,28,6,24,17,20,11,33,5,29,16,25,10,21,4,34,15,30,9,26,3,22,14,18,8,31,2,27,13,23,7,19)$
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, $C_2^2$
$8$:  $C_4\times C_2$
$68$:  $C_{17}:C_{4}$ x 2
$136$:  34T5 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: None

Low degree siblings

34T16 x 7

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Conjugacy classes not computed

magma: ConjugacyClasses(G);
 

Character table

56 x 56 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed