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Properties

Label 34T21
34T21 1 17 1->17 25 1->25 2 9 2->9 30 2->30 3 3->1 18 3->18 4 10 4->10 23 4->23 5 5->2 28 5->28 6 11 6->11 33 6->33 7 7->3 21 7->21 8 12 8->12 26 8->26 9->4 31 9->31 13 10->13 19 10->19 11->5 24 11->24 14 12->14 29 12->29 13->6 34 13->34 15 14->15 22 14->22 15->7 27 15->27 16 32 16->32 17->8 20 17->20 18->9 18->21 19->16 19->30 20->6 20->22 21->13 21->31 22->3 22->23 23->10 23->32 24->17 25->7 25->33 26->14 26->25 27->4 27->34 28->11 28->26 29->1 29->18 30->8 30->27 31->15 31->19 32->5 32->28 33->12 33->20 34->2 34->29
Degree $34$
Order $4624$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{17}^2.C_4$

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Show commands: Magma

magma: G := TransitiveGroup(34, 21);
 

Group invariants

Abstract group:  $D_{17}^2.C_4$
magma: IdentifyGroup(G);
 
Order:  $4624=2^{4} \cdot 17^{2}$
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 $t$:  $21$
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:  $(1,17,8,12,14,15,7,3)(2,9,4,10,13,6,11,5)(18,21,31,19,30,27,34,29)(20,22,23,32,28,26,25,33)$, $(1,25,7,21,13,34,2,30,8,26,14,22,3,18,9,31,15,27,4,23,10,19,16,32,5,28,11,24,17,20,6,33,12,29)$
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, $C_4\times C_2$
$16$:  $C_8\times C_2$
$136$:  $C_{17}:C_{8}$ x 2
$272$:  34T7 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: None

Low degree siblings

34T21 x 7

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 $2^{17}$ $17$ $2$ $17$ $( 1,24)( 2,19)( 3,31)( 4,26)( 5,21)( 6,33)( 7,28)( 8,23)( 9,18)(10,30)(11,25)(12,20)(13,32)(14,27)(15,22)(16,34)(17,29)$
2B $2^{17}$ $17$ $2$ $17$ $( 1,27)( 2,32)( 3,20)( 4,25)( 5,30)( 6,18)( 7,23)( 8,28)( 9,33)(10,21)(11,26)(12,31)(13,19)(14,24)(15,29)(16,34)(17,22)$
2C $2^{16},1^{2}$ $289$ $2$ $16$ $( 1,14)( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,17)(18,28)(19,27)(20,26)(21,25)(22,24)(29,34)(30,33)(31,32)$
4A1 $4^{8},1^{2}$ $289$ $4$ $24$ $( 1, 8,14, 7)( 2, 4,13,11)( 3,17,12,15)( 5, 9,10, 6)(18,26,28,20)(19,22,27,24)(21,31,25,32)(29,33,34,30)$
4A-1 $4^{8},1^{2}$ $289$ $4$ $24$ $( 1, 7,14, 8)( 2,11,13, 4)( 3,15,12,17)( 5, 6,10, 9)(18,20,28,26)(19,24,27,22)(21,32,25,31)(29,30,34,33)$
4B1 $4^{8},2$ $289$ $4$ $25$ $( 1,18,15,26)( 2,21,14,23)( 3,24,13,20)( 4,27,12,34)( 5,30,11,31)( 6,33,10,28)( 7,19, 9,25)( 8,22)(16,29,17,32)$
4B-1 $4^{8},2$ $289$ $4$ $25$ $( 1,26,15,18)( 2,23,14,21)( 3,20,13,24)( 4,34,12,27)( 5,31,11,30)( 6,28,10,33)( 7,25, 9,19)( 8,22)(16,32,17,29)$
8A1 $8^{4},1^{2}$ $289$ $8$ $28$ $( 1,17, 8,12,14,15, 7, 3)( 2, 9, 4,10,13, 6,11, 5)(18,29,26,33,28,34,20,30)(19,21,22,31,27,25,24,32)$
8A-1 $8^{4},1^{2}$ $289$ $8$ $28$ $( 1, 3, 7,15,14,12, 8,17)( 2, 5,11, 6,13,10, 4, 9)(18,30,20,34,28,33,26,29)(19,32,24,25,27,31,22,21)$
8A3 $8^{4},1^{2}$ $289$ $8$ $28$ $( 1,12, 7,17,14, 3, 8,15)( 2,10,11, 9,13, 5, 4, 6)(18,33,20,29,28,30,26,34)(19,31,24,21,27,32,22,25)$
8A-3 $8^{4},1^{2}$ $289$ $8$ $28$ $( 1,15, 8, 3,14,17, 7,12)( 2, 6, 4, 5,13, 9,11,10)(18,34,26,30,28,29,20,33)(19,25,22,32,27,21,24,31)$
8B1 $8^{4},2$ $289$ $8$ $29$ $( 1,24, 5,18, 4,28,17,34)( 2,31, 9,29, 3,21,13,23)( 6,25, 8,22,16,27,14,30)( 7,32,12,33,15,20,10,19)(11,26)$
8B-1 $8^{4},2$ $289$ $8$ $29$ $( 1,34,17,28, 4,18, 5,24)( 2,23,13,21, 3,29, 9,31)( 6,30,14,27,16,22, 8,25)( 7,19,10,20,15,33,12,32)(11,26)$
8B3 $8^{4},2$ $289$ $8$ $29$ $( 1,18,17,24, 4,34, 5,28)( 2,29,13,31, 3,23, 9,21)( 6,22,14,25,16,30, 8,27)( 7,33,10,32,15,19,12,20)(11,26)$
8B-3 $8^{4},2$ $289$ $8$ $29$ $( 1,28, 5,34, 4,24,17,18)( 2,21, 9,23, 3,31,13,29)( 6,27, 8,30,16,25,14,22)( 7,20,12,19,15,32,10,33)(11,26)$
17A1 $17^{2}$ $8$ $17$ $32$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,20,22,24,26,28,30,32,34,19,21,23,25,27,29,31,33)$
17A3 $17^{2}$ $8$ $17$ $32$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$
17B1 $17^{2}$ $8$ $17$ $32$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$
17B3 $17^{2}$ $8$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$
17C1 $17^{2}$ $16$ $17$ $32$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$
17C3 $17^{2}$ $16$ $17$ $32$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$
17D1 $17^{2}$ $16$ $17$ $32$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$
17D3 $17^{2}$ $16$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$
17E1 $17^{2}$ $16$ $17$ $32$ $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$
17E3 $17^{2}$ $16$ $17$ $32$ $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$
17F1 $17^{2}$ $16$ $17$ $32$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$
17F3 $17^{2}$ $16$ $17$ $32$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$
17G1 $17^{2}$ $16$ $17$ $32$ $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$
17G3 $17^{2}$ $16$ $17$ $32$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$
17H1 $17,1^{17}$ $16$ $17$ $16$ $(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$
17H3 $17,1^{17}$ $16$ $17$ $16$ $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)$
17I1 $17^{2}$ $16$ $17$ $32$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$
17I3 $17^{2}$ $16$ $17$ $32$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$
17J1 $17^{2}$ $16$ $17$ $32$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$
17J3 $17^{2}$ $16$ $17$ $32$ $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$
34A1 $34$ $136$ $34$ $33$ $( 1,25, 4,27, 7,29,10,31,13,33,16,18, 2,20, 5,22, 8,24,11,26,14,28,17,30, 3,32, 6,34, 9,19,12,21,15,23)$
34A3 $34$ $136$ $34$ $33$ $( 1,27,10,33, 2,22,11,28, 3,34,12,23, 4,29,13,18, 5,24,14,30, 6,19,15,25, 7,31,16,20, 8,26,17,32, 9,21)$
34B1 $34$ $136$ $34$ $33$ $( 1,30, 9,19,17,25, 8,31,16,20, 7,26,15,32, 6,21,14,27, 5,33,13,22, 4,28,12,34, 3,23,11,29, 2,18,10,24)$
34B3 $34$ $136$ $34$ $33$ $( 1,19, 8,20,15,21, 5,22,12,23, 2,24, 9,25,16,26, 6,27,13,28, 3,29,10,30,17,31, 7,32,14,33, 4,34,11,18)$

Malle's constant $a(G)$:     $1/16$

magma: ConjugacyClasses(G);
 

Character table

40 x 40 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed