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Properties

Label 34T50
34T50 1 5 1->5 30 1->30 2 16 2->16 23 2->23 3 10 3->10 33 3->33 4 26 4->26 15 5->15 19 5->19 6 9 6->9 29 6->29 7 7->3 22 7->22 8 14 8->14 32 8->32 9->8 25 9->25 10->2 18 10->18 11 13 11->13 28 11->28 12 12->7 21 12->21 13->1 31 13->31 14->12 24 14->24 15->6 34 15->34 17 16->17 27 16->27 17->11 20 17->20 18->8 18->22 19->2 19->29 20->13 20->19 21->7 21->26 22->1 22->33 23->12 24->6 24->30 25->17 25->20 26->11 26->27 27->5 27->34 28->16 28->24 29->10 29->31 30->4 30->21 31->15 31->28 32->9 32->18 33->25 34->14 34->32
Degree $34$
Order $18496$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{17}^2.\OD_{16}$

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Copy content magma:G := TransitiveGroup(34, 50);
 

Group invariants

Abstract group:  $D_{17}^2.\OD_{16}$
Copy content magma:IdentifyGroup(G);
 
Order:  $18496=2^{6} \cdot 17^{2}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
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 $t$:  $50$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,30,4,26,11,28,16,27,5,19,2,23,12,21,7,22)(3,33)(6,29,10,18,8,32,9,25,17,20,13,31,15,34,14,24)$, $(1,5,15,6,9,8,14,12,7,3,10,2,16,17,11,13)(18,22,33,25,20,19,29,31,28,24,30,21,26,27,34,32)$
Copy content 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, $C_2^2$
$8$:  $D_{4}$ x 2, $C_8$ x 2, $C_4\times C_2$
$16$:  $C_8:C_2$, $C_2^2:C_4$, $C_8\times C_2$
$32$:  $C_2^2 : C_8$
$64$:  32T271

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: None

Low degree siblings

There are no siblings 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 $2^{8},1^{18}$ $34$ $2$ $8$ $(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(25,27)$
2B $2^{17}$ $68$ $2$ $17$ $( 1,28)( 2,24)( 3,20)( 4,33)( 5,29)( 6,25)( 7,21)( 8,34)( 9,30)(10,26)(11,22)(12,18)(13,31)(14,27)(15,23)(16,19)(17,32)$
2C $2^{16},1^{2}$ $289$ $2$ $16$ $( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(30,34)(31,33)$
4A1 $4^{8},1^{2}$ $289$ $4$ $24$ $( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)(18,27,29,20)(19,31,28,33)(21,22,26,25)(23,30,24,34)$
4A-1 $4^{8},1^{2}$ $289$ $4$ $24$ $( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)(18,20,29,27)(19,33,28,31)(21,25,26,22)(23,34,24,30)$
4B $4^{8},1^{2}$ $578$ $4$ $24$ $( 1, 3,12,10)( 2,16,11,14)( 4, 8, 9, 5)( 6,17, 7,13)(18,21,33,30)(19,25,32,26)(20,29,31,22)(23,24,28,27)$
4C $4^{8},2$ $1156$ $4$ $25$ $( 1,34,17,33)( 2,18,16,32)( 3,19,15,31)( 4,20,14,30)( 5,21,13,29)( 6,22,12,28)( 7,23,11,27)( 8,24,10,26)( 9,25)$
8A1 $8^{4},1^{2}$ $289$ $8$ $28$ $( 1, 9,13,15,16, 8, 4, 2)( 3,10, 5,11,14, 7,12, 6)(18,22,24,25,34,30,28,27)(19,31,20,23,33,21,32,29)$
8A-1 $8^{4},1^{2}$ $289$ $8$ $28$ $( 1, 2, 4, 8,16,15,13, 9)( 3, 6,12, 7,14,11, 5,10)(18,27,28,30,34,25,24,22)(19,29,32,21,33,23,20,31)$
8A3 $8^{4},1^{2}$ $289$ $8$ $28$ $( 1,15, 4, 9,16, 2,13, 8)( 3,11,12,10,14, 6, 5, 7)(18,25,28,22,34,27,24,30)(19,23,32,31,33,29,20,21)$
8A-3 $8^{4},1^{2}$ $289$ $8$ $28$ $( 1, 8,13, 2,16, 9, 4,15)( 3, 7, 5, 6,14,10,12,11)(18,30,24,27,34,22,28,25)(19,21,20,29,33,31,32,23)$
8B1 $8^{4},1^{2}$ $578$ $8$ $28$ $( 2, 3, 5, 9,17,16,14,10)( 4, 7,13, 8,15,12, 6,11)(18,26,27,25,29,21,20,22)(19,24,31,34,28,23,33,30)$
8B-1 $8^{4},1^{2}$ $578$ $8$ $28$ $( 2,10,14,16,17, 9, 5, 3)( 4,11, 6,12,15, 8,13, 7)(18,22,20,21,29,25,27,26)(19,30,33,23,28,34,31,24)$
8C1 $8^{4},2$ $1156$ $8$ $29$ $( 1,28, 3,27,11,23, 9,24)( 2,19, 7,25,10,32, 5,26)( 4,18,15,21, 8,33,14,30)( 6,34)(12,31,13,22,17,20,16,29)$
8C-1 $8^{4},2$ $1156$ $8$ $29$ $( 1,24, 9,23,11,27, 3,28)( 2,26, 5,32,10,25, 7,19)( 4,30,14,33, 8,21,15,18)( 6,34)(12,29,16,20,17,22,13,31)$
16A1 $16^{2},1^{2}$ $578$ $16$ $30$ $( 2,12, 3, 6, 5,11, 9, 4,17, 7,16,13,14, 8,10,15)(18,19,26,24,27,31,25,34,29,28,21,23,20,33,22,30)$
16A-1 $16^{2},1^{2}$ $578$ $16$ $30$ $( 2,15,10, 8,14,13,16, 7,17, 4, 9,11, 5, 6, 3,12)(18,30,22,33,20,23,21,28,29,34,25,31,27,24,26,19)$
16A3 $16^{2},1^{2}$ $578$ $16$ $30$ $( 2, 6, 9, 7,14,15, 3,11,17,13,10,12, 5, 4,16, 8)(18,24,25,28,20,30,26,31,29,23,22,19,27,34,21,33)$
16A-3 $16^{2},1^{2}$ $578$ $16$ $30$ $( 2, 8,16, 4, 5,12,10,13,17,11, 3,15,14, 7, 9, 6)(18,33,21,34,27,19,22,23,29,31,26,30,20,28,25,24)$
16A5 $16^{2},1^{2}$ $578$ $16$ $30$ $( 2,11,16,15, 5, 7,10, 6,17, 8, 3, 4,14,12, 9,13)(18,31,21,30,27,28,22,24,29,33,26,34,20,19,25,23)$
16A-5 $16^{2},1^{2}$ $578$ $16$ $30$ $( 2,13, 9,12,14, 4, 3, 8,17, 6,10, 7, 5,15,16,11)(18,23,25,19,20,34,26,33,29,24,22,28,27,30,21,31)$
16A7 $16^{2},1^{2}$ $578$ $16$ $30$ $( 2, 4,10,11,14, 6,16,12,17,15, 9, 8, 5,13, 3, 7)(18,34,22,31,20,24,21,19,29,30,25,33,27,23,26,28)$
16A-7 $16^{2},1^{2}$ $578$ $16$ $30$ $( 2, 7, 3,13, 5, 8, 9,15,17,12,16, 6,14,11,10, 4)(18,28,26,23,27,33,25,30,29,19,21,24,20,31,22,34)$
16B1 $16^{2},2$ $1156$ $16$ $31$ $( 1,33, 9,21,13,32,15,29,16,19, 8,31, 4,20, 2,23)( 3,30,10,28, 5,27,11,18,14,22, 7,24,12,25, 6,34)(17,26)$
16B-1 $16^{2},2$ $1156$ $16$ $31$ $( 1,23, 2,20, 4,31, 8,19,16,29,15,32,13,21, 9,33)( 3,34, 6,25,12,24, 7,22,14,18,11,27, 5,28,10,30)(17,26)$
16B3 $16^{2},2$ $1156$ $16$ $31$ $( 1,21,15,19, 4,23, 9,32,16,31, 2,33,13,29, 8,20)( 3,28,11,22,12,34,10,27,14,24, 6,30, 5,18, 7,25)(17,26)$
16B-3 $16^{2},2$ $1156$ $16$ $31$ $( 1,20, 8,29,13,33, 2,31,16,32, 9,23, 4,19,15,21)( 3,25, 7,18, 5,30, 6,24,14,27,10,34,12,22,11,28)(17,26)$
17A $17,1^{17}$ $32$ $17$ $16$ $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
17B1 $17^{2}$ $32$ $17$ $32$ $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$
17B3 $17^{2}$ $32$ $17$ $32$ $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$
17C $17^{2}$ $64$ $17$ $32$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$
17D1 $17^{2}$ $64$ $17$ $32$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$
17D3 $17^{2}$ $64$ $17$ $32$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$
34A $17,2^{8},1$ $544$ $34$ $24$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(25,27)$
34B1 $34$ $544$ $34$ $33$ $( 1,19,14,18,10,34, 6,33, 2,32,15,31,11,30, 7,29, 3,28,16,27,12,26, 8,25, 4,24,17,23,13,22, 9,21, 5,20)$
34B3 $34$ $544$ $34$ $33$ $( 1,18, 6,32,11,29,16,26, 4,23, 9,20,14,34, 2,31, 7,28,12,25,17,22, 5,19,10,33,15,30, 3,27, 8,24,13,21)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

37 x 37 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed