↑

Properties

Label 34T32
34T32 1 15 1->15 20 1->20 2 12 2->12 27 2->27 3 9 3->9 34 3->34 4 6 4->6 24 4->24 5 5->3 31 5->31 17 6->17 21 6->21 7 14 7->14 28 7->28 8 11 8->11 18 8->18 9->8 25 9->25 10 10->5 32 10->32 11->2 22 11->22 16 12->16 29 12->29 13 19 13->19 14->10 26 14->26 15->7 33 15->33 16->4 23 16->23 17->1 30 17->30 18->4 18->32 19->9 19->26 20->14 21->2 21->31 22->7 22->25 23->12 23->19 24->17 24->30 25->5 25->24 26->10 26->18 27->15 27->29 28->3 28->23 29->8 29->34 30->13 30->28 31->1 31->22 32->6 32->33 33->11 33->27 34->16 34->21
Degree $34$
Order $9248$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{17}^2:D_{16}$

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(34, 32);
 
Copy content sage:G = TransitiveGroup(34, 32)
 
Copy content oscar:G = transitive_group(34, 32)
 

Group invariants

Abstract group:  $C_{17}^2:D_{16}$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Order:  $9248=2^{5} \cdot 17^{2}$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Cyclic:  no
Copy content comment:Determine if group is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content oscar:is_cyclic(G)
 
Abelian:  no
Copy content comment:Determine if group is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content oscar:is_abelian(G)
 
Solvable:  yes
Copy content comment:Determine if group is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content sage:G.is_solvable()
 
Copy content oscar:is_solvable(G)
 
Nilpotency class:   not nilpotent
Copy content comment:Nilpotency class
 
Copy content magma:NilpotencyClass(G);
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 

Group action invariants

Degree $n$:  $34$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Transitive number $t$:  $32$
Copy content comment:Transitive number
 
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Copy content sage:G.transitive_number()
 
Copy content oscar:transitive_group_identification(G)[2]
 
Parity:  $-1$
Copy content comment:Parity
 
Copy content magma:IsEven(G);
 
Copy content sage:all(g.SignPerm() == 1 for g in libgap(G).GeneratorsOfGroup())
 
Copy content oscar:is_even(G)
 
Primitive:  no
Copy content comment:Determine if group is primitive
 
Copy content magma:IsPrimitive(G);
 
Copy content sage:G.is_primitive()
 
Copy content oscar:is_primitive(G)
 
$\card{\Aut(F/K)}$:  $1$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(34).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(34), G)[1])
 
Generators:  $(1,20,14,26,10,32,6,21,2,27,15,33,11,22,7,28,3,34,16,23,12,29,8,18,4,24,17,30,13,19,9,25,5,31)$, $(1,15,7,14,10,5,3,9,8,11,2,12,16,4,6,17)(18,32,33,27,29,34,21,31,22,25,24,30,28,23,19,26)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$
$16$:  $D_{8}$
$32$:  $D_{16}$

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 $2^{17}$ $136$ $2$ $17$ $( 1,19)( 2,18)( 3,34)( 4,33)( 5,32)( 6,31)( 7,30)( 8,29)( 9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)$
2B $2^{17}$ $136$ $2$ $17$ $( 1,22)( 2,25)( 3,28)( 4,31)( 5,34)( 6,20)( 7,23)( 8,26)( 9,29)(10,32)(11,18)(12,21)(13,24)(14,27)(15,30)(16,33)(17,19)$
2C $2^{16},1^{2}$ $289$ $2$ $16$ $( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(30,34)(31,33)$
4A $4^{8},1^{2}$ $578$ $4$ $24$ $( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)(18,27,29,20)(19,31,28,33)(21,22,26,25)(23,30,24,34)$
8A1 $8^{4},1^{2}$ $578$ $8$ $28$ $( 2,10,14,16,17, 9, 5, 3)( 4,11, 6,12,15, 8,13, 7)(18,21,27,22,29,26,20,25)(19,23,31,30,28,24,33,34)$
8A3 $8^{4},1^{2}$ $578$ $8$ $28$ $( 2,16, 5,10,17, 3,14, 9)( 4,12,13,11,15, 7, 6, 8)(18,22,20,21,29,25,27,26)(19,30,33,23,28,34,31,24)$
16A1 $16^{2},1^{2}$ $578$ $16$ $30$ $( 2,15,10, 8,14,13,16, 7,17, 4, 9,11, 5, 6, 3,12)(18,31,21,30,27,28,22,24,29,33,26,34,20,19,25,23)$
16A3 $16^{2},1^{2}$ $578$ $16$ $30$ $( 2, 8,16, 4, 5,12,10,13,17,11, 3,15,14, 7, 9, 6)(18,30,22,33,20,23,21,28,29,34,25,31,27,24,26,19)$
16A5 $16^{2},1^{2}$ $578$ $16$ $30$ $( 2,13, 9,12,14, 4, 3, 8,17, 6,10, 7, 5,15,16,11)(18,28,26,23,27,33,25,30,29,19,21,24,20,31,22,34)$
16A7 $16^{2},1^{2}$ $578$ $16$ $30$ $( 2, 7, 3,13, 5, 8, 9,15,17,12,16, 6,14,11,10, 4)(18,24,25,28,20,30,26,31,29,23,22,19,27,34,21,33)$
17A1 $17^{2}$ $16$ $17$ $32$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$
17A2 $17^{2}$ $16$ $17$ $32$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$
17A3 $17^{2}$ $16$ $17$ $32$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$
17A4 $17^{2}$ $16$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$
17A5 $17^{2}$ $16$ $17$ $32$ $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$
17A6 $17^{2}$ $16$ $17$ $32$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$
17A7 $17^{2}$ $16$ $17$ $32$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$
17A8 $17^{2}$ $16$ $17$ $32$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$
17B1 $17^{2}$ $16$ $17$ $32$ $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$
17B2 $17^{2}$ $16$ $17$ $32$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$
17B3 $17^{2}$ $16$ $17$ $32$ $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$
17B4 $17^{2}$ $16$ $17$ $32$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$
17B5 $17^{2}$ $16$ $17$ $32$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$
17B6 $17^{2}$ $16$ $17$ $32$ $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$
17B7 $17^{2}$ $16$ $17$ $32$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$
17B8 $17^{2}$ $16$ $17$ $32$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$
17C $17,1^{17}$ $32$ $17$ $16$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)$
34A1 $34$ $272$ $34$ $33$ $( 1,33, 7,27,13,21, 2,32, 8,26,14,20, 3,31, 9,25,15,19, 4,30,10,24,16,18, 5,29,11,23,17,34, 6,28,12,22)$
34A3 $34$ $272$ $34$ $33$ $( 1,27, 2,26, 3,25, 4,24, 5,23, 6,22, 7,21, 8,20, 9,19,10,18,11,34,12,33,13,32,14,31,15,30,16,29,17,28)$
34A5 $34$ $272$ $34$ $33$ $( 1,21,14,25,10,29, 6,33, 2,20,15,24,11,28, 7,32, 3,19,16,23,12,27, 8,31, 4,18,17,22,13,26, 9,30, 5,34)$
34A7 $34$ $272$ $34$ $33$ $( 1,32, 9,24,17,33, 8,25,16,34, 7,26,15,18, 6,27,14,19, 5,28,13,20, 4,29,12,21, 3,30,11,22, 2,31,10,23)$
34A9 $34$ $272$ $34$ $33$ $( 1,26, 4,23, 7,20,10,34,13,31,16,28, 2,25, 5,22, 8,19,11,33,14,30,17,27, 3,24, 6,21, 9,18,12,32,15,29)$
34A11 $34$ $272$ $34$ $33$ $( 1,20,16,22,14,24,12,26,10,28, 8,30, 6,32, 4,34, 2,19,17,21,15,23,13,25,11,27, 9,29, 7,31, 5,33, 3,18)$
34A13 $34$ $272$ $34$ $33$ $( 1,31,11,21, 4,28,14,18, 7,25,17,32,10,22, 3,29,13,19, 6,26,16,33, 9,23, 2,30,12,20, 5,27,15,34, 8,24)$
34A15 $34$ $272$ $34$ $33$ $( 1,25, 6,20,11,32,16,27, 4,22, 9,34,14,29, 2,24, 7,19,12,31,17,26, 5,21,10,33,15,28, 3,23, 8,18,13,30)$
34B1 $34$ $272$ $34$ $33$ $( 1,33,14,21,10,26, 6,31, 2,19,15,24,11,29, 7,34, 3,22,16,27,12,32, 8,20, 4,25,17,30,13,18, 9,23, 5,28)$
34B3 $34$ $272$ $34$ $33$ $( 1,21, 6,19,11,34,16,32, 4,30, 9,28,14,26, 2,24, 7,22,12,20,17,18, 5,33,10,31,15,29, 3,27, 8,25,13,23)$
34B5 $34$ $272$ $34$ $33$ $( 1,26,15,34,12,25, 9,33, 6,24, 3,32,17,23,14,31,11,22, 8,30, 5,21, 2,29,16,20,13,28,10,19, 7,27, 4,18)$
34B7 $34$ $272$ $34$ $33$ $( 1,31, 7,32,13,33, 2,34, 8,18,14,19, 3,20, 9,21,15,22, 4,23,10,24,16,25, 5,26,11,27,17,28, 6,29,12,30)$
34B9 $34$ $272$ $34$ $33$ $( 1,19,16,30,14,24,12,18,10,29, 8,23, 6,34, 4,28, 2,22,17,33,15,27,13,21,11,32, 9,26, 7,20, 5,31, 3,25)$
34B11 $34$ $272$ $34$ $33$ $( 1,24, 8,28,15,32, 5,19,12,23, 2,27, 9,31,16,18, 6,22,13,26, 3,30,10,34,17,21, 7,25,14,29, 4,33,11,20)$
34B13 $34$ $272$ $34$ $33$ $( 1,29,17,26,16,23,15,20,14,34,13,31,12,28,11,25,10,22, 9,19, 8,33, 7,30, 6,27, 5,24, 4,21, 3,18, 2,32)$
34B15 $34$ $272$ $34$ $33$ $( 1,34, 9,24,17,31, 8,21,16,28, 7,18,15,25, 6,32,14,22, 5,29,13,19, 4,26,12,33, 3,23,11,30, 2,20,10,27)$

Malle's constant $a(G)$:     $1/16$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 

Character table

44 x 44 character table

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 

Regular extensions

Data not computed