Properties

Label 34T44
34T44 1 30 1->30 1->30 2 29 2->29 2->29 3 4 3->4 34 3->34 33 4->33 5 7 5->7 21 5->21 6 8 6->8 22 6->22 10 7->10 24 7->24 9 8->9 23 8->23 9->3 9->8 10->4 10->7 11 11->21 25 11->25 12 12->22 26 12->26 13 13->5 14 13->14 14->6 15 15->12 27 15->27 16 16->11 28 16->28 17 17->26 17->33 18 18->25 18->34 19 19->13 19->17 20 20->14 20->18 21->15 21->24 22->16 22->23 31 23->31 23->31 32 24->32 24->32 25->5 25->19 26->6 26->20 27->11 27->15 28->12 28->16 29->9 29->27 30->10 30->28 31->2 31->17 32->1 32->18 33->2 33->20 34->1 34->19
Degree $34$
Order $16320$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $\SL(2,16).C_4$

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Copy content magma:G := TransitiveGroup(34, 44);
 

Group invariants

Abstract group:  $\SL(2,16).C_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $16320=2^{6} \cdot 3 \cdot 5 \cdot 17$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $34$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $44$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,30,28,16,11,21,24,32)(2,29,27,15,12,22,23,31)(3,34,19,17,26,6,8,9)(4,33,20,18,25,5,7,10)(13,14)$, $(1,30,10,7,24,32,18,34)(2,29,9,8,23,31,17,33)(3,4)(5,21,15,27,11,25,19,13)(6,22,16,28,12,26,20,14)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: $\PSL(2,16):C_4$

Low degree siblings

17T8

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
$1^{34}$ $1$ $1$ $0$ $()$
$2^{16},1^{2}$ $255$ $2$ $16$ $( 1,15)( 2,16)( 3,33)( 4,34)( 5,29)( 6,30)( 9,32)(10,31)(11,18)(12,17)(13,20)(14,19)(21,25)(22,26)(23,27)(24,28)$
$4^{8},1^{2}$ $1020$ $4$ $24$ $( 1, 3,15,33)( 2, 4,16,34)( 5,24,29,28)( 6,23,30,27)( 9,13,32,20)(10,14,31,19)(11,22,18,26)(12,21,17,25)$
$8^{4},2$ $2040$ $8$ $29$ $( 1,30, 3,27,15, 6,33,23)( 2,29, 4,28,16, 5,34,24)( 7, 8)( 9,22,13,18,32,26,20,11)(10,21,14,17,31,25,19,12)$
$8^{4},2$ $2040$ $8$ $29$ $( 1,23,33, 6,15,27, 3,30)( 2,24,34, 5,16,28, 4,29)( 7, 8)( 9,11,20,26,32,18,13,22)(10,12,19,25,31,17,14,21)$
$2^{12},1^{10}$ $68$ $2$ $12$ $( 3,14)( 4,13)( 5,15)( 6,16)( 9,21)(10,22)(11,26)(12,25)(17,23)(18,24)(27,32)(28,31)$
$5^{6},1^{4}$ $544$ $5$ $24$ $( 1,33, 8,19,29)( 2,34, 7,20,30)( 5,22,31,26,18)( 6,21,32,25,17)( 9,27,12,23,16)(10,28,11,24,15)$
$10^{2},5^{2},2^{2}$ $1632$ $10$ $28$ $( 1,19,33,29, 8)( 2,20,34,30, 7)( 3,14)( 4,13)( 5,11,22,24,31,15,26,10,18,28)( 6,12,21,23,32,16,25, 9,17,27)$
$3^{10},1^{4}$ $272$ $3$ $20$ $( 1,11,26)( 2,12,25)( 5, 8,15)( 6, 7,16)( 9,21,20)(10,22,19)(17,34,23)(18,33,24)(27,32,30)(28,31,29)$
$15^{2},1^{4}$ $1088$ $15$ $28$ $( 1, 5,28,33,22,11, 8,31,24,19,26,15,29,18,10)( 2, 6,27,34,21,12, 7,32,23,20,25,16,30,17, 9)$
$4^{6},2^{5}$ $680$ $4$ $23$ $( 1,16,33,13)( 2,15,34,14)( 3,30, 5,20)( 4,29, 6,19)( 7, 8)( 9,22,17,24)(10,21,18,23)(11,32)(12,31)(25,26)(27,28)$
$4^{6},2^{5}$ $680$ $4$ $23$ $( 1,13,33,16)( 2,14,34,15)( 3,20, 5,30)( 4,19, 6,29)( 7, 8)( 9,24,17,22)(10,23,18,21)(11,32)(12,31)(25,26)(27,28)$
$6^{4},3^{2},1^{4}$ $1360$ $6$ $24$ $( 1,29,10,33,19,18)( 2,30, 9,34,20,17)( 3,24,15, 5,22,14)( 4,23,16, 6,21,13)( 7,27,25)( 8,28,26)$
$12^{2},6,2^{2}$ $1360$ $12$ $29$ $( 1,23,29,16,10, 6,33,21,19,13,18, 4)( 2,24,30,15, 9, 5,34,22,20,14,17, 3)( 7,26,27, 8,25,28)(11,32)(12,31)$
$12^{2},6,2^{2}$ $1360$ $12$ $29$ $( 1,21,29,13,10, 4,33,23,19,16,18, 6)( 2,22,30,14, 9, 3,34,24,20,15,17, 5)( 7,26,27, 8,25,28)(11,32)(12,31)$
$17^{2}$ $960$ $17$ $32$ $( 1, 5, 3,28,26,11,29,19,33,24,18,31,22,15, 8,10,14)( 2, 6, 4,27,25,12,30,20,34,23,17,32,21,16, 7, 9,13)$
$17^{2}$ $960$ $17$ $32$ $( 1,29,22, 5,19,15, 3,33, 8,28,24,10,26,18,14,11,31)( 2,30,21, 6,20,16, 4,34, 7,27,23, 9,25,17,13,12,32)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

Character table not computed

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed