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Properties

Label 34T8
Degree $34$
Order $272$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_{17}$

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Copy content magma:G := TransitiveGroup(34, 8);
 

Group invariants

Abstract group:  $F_{17}$
Copy content magma:IdentifyGroup(G);
 
Order:  $272=2^{4} \cdot 17$
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 $t$:  $8$
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:  $(1,16,20,25,17,6,22,12,14,33,30,23,31,10,27,4)(2,15,19,26,18,5,21,11,13,34,29,24,32,9,28,3)(7,8)$, $(1,31,30,11,20,24,26,9)(2,32,29,12,19,23,25,10)(3,15,22,8,17,5,34,14)(4,16,21,7,18,6,33,13)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$8$:  $C_8$
$16$:  $C_{16}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: $F_{17}$

Low degree siblings

17T5

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 $2^{16},1^{2}$ $17$ $2$ $16$ $( 1, 9)( 2,10)( 3, 8)( 4, 7)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,27)(18,28)(19,25)(20,26)(21,23)(22,24)$
4A1 $4^{8},1^{2}$ $17$ $4$ $24$ $( 1,24, 9,22)( 2,23,10,21)( 3,31, 8,14)( 4,32, 7,13)(11,30,34,15)(12,29,33,16)(17,20,27,26)(18,19,28,25)$
4A-1 $4^{8},1^{2}$ $17$ $4$ $24$ $( 1,22, 9,24)( 2,21,10,23)( 3,14, 8,31)( 4,13, 7,32)(11,15,34,30)(12,16,33,29)(17,26,27,20)(18,25,28,19)$
8A1 $8^{4},1^{2}$ $17$ $8$ $28$ $( 1,31,24, 8, 9,14,22, 3)( 2,32,23, 7,10,13,21, 4)(11,17,30,20,34,27,15,26)(12,18,29,19,33,28,16,25)$
8A-1 $8^{4},1^{2}$ $17$ $8$ $28$ $( 1, 3,22,14, 9, 8,24,31)( 2, 4,21,13,10, 7,23,32)(11,26,15,27,34,20,30,17)(12,25,16,28,33,19,29,18)$
8A3 $8^{4},1^{2}$ $17$ $8$ $28$ $( 1, 8,22,31, 9, 3,24,14)( 2, 7,21,32,10, 4,23,13)(11,20,15,17,34,26,30,27)(12,19,16,18,33,25,29,28)$
8A-3 $8^{4},1^{2}$ $17$ $8$ $28$ $( 1,14,24, 3, 9,31,22, 8)( 2,13,23, 4,10,32,21, 7)(11,27,30,26,34,17,15,20)(12,28,29,25,33,18,16,19)$
16A1 $16^{2},2$ $17$ $16$ $31$ $( 1,29,31,19,24,33, 8,28, 9,16,14,25,22,12, 3,18)( 2,30,32,20,23,34, 7,27,10,15,13,26,21,11, 4,17)( 5, 6)$
16A-1 $16^{2},2$ $17$ $16$ $31$ $( 1,18, 3,12,22,25,14,16, 9,28, 8,33,24,19,31,29)( 2,17, 4,11,21,26,13,15,10,27, 7,34,23,20,32,30)( 5, 6)$
16A3 $16^{2},2$ $17$ $16$ $31$ $( 1,19, 8,16,22,18,31,33, 9,25, 3,29,24,28,14,12)( 2,20, 7,15,21,17,32,34,10,26, 4,30,23,27,13,11)( 5, 6)$
16A-3 $16^{2},2$ $17$ $16$ $31$ $( 1,12,14,28,24,29, 3,25, 9,33,31,18,22,16, 8,19)( 2,11,13,27,23,30, 4,26,10,34,32,17,21,15, 7,20)( 5, 6)$
16A5 $16^{2},2$ $17$ $16$ $31$ $( 1,33,14,18,24,16, 3,19, 9,12,31,28,22,29, 8,25)( 2,34,13,17,23,15, 4,20,10,11,32,27,21,30, 7,26)( 5, 6)$
16A-5 $16^{2},2$ $17$ $16$ $31$ $( 1,25, 8,29,22,28,31,12, 9,19, 3,16,24,18,14,33)( 2,26, 7,30,21,27,32,11,10,20, 4,15,23,17,13,34)( 5, 6)$
16A7 $16^{2},2$ $17$ $16$ $31$ $( 1,28, 3,33,22,19,14,29, 9,18, 8,12,24,25,31,16)( 2,27, 4,34,21,20,13,30,10,17, 7,11,23,26,32,15)( 5, 6)$
16A-7 $16^{2},2$ $17$ $16$ $31$ $( 1,16,31,25,24,12, 8,18, 9,29,14,19,22,33, 3,28)( 2,15,32,26,23,11, 7,17,10,30,13,20,21,34, 4,27)( 5, 6)$
17A $17^{2}$ $16$ $17$ $32$ $( 1, 8,14,20,26,31, 3, 9,15,22,27,34, 5,11,17,24,30)( 2, 7,13,19,25,32, 4,10,16,21,28,33, 6,12,18,23,29)$

Malle's constant $a(G)$:     $1/16$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 4A1 4A-1 8A1 8A-1 8A3 8A-3 16A1 16A-1 16A3 16A-3 16A5 16A-5 16A7 16A-7 17A
Size 1 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 16
2 P 1A 1A 2A 2A 4A1 4A-1 4A-1 4A1 8A1 8A-1 8A3 8A-3 8A-3 8A3 8A-1 8A1 17A
17 P 1A 2A 4A1 4A-1 8A1 8A-1 8A3 8A-3 16A1 16A-1 16A3 16A-3 16A5 16A-5 16A7 16A-7 1A
Type
272.50.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
272.50.1b R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1
272.50.1c1 C 1 1 1 1 −1 −1 −1 −1 −i −i i i −i i i −i 1
272.50.1c2 C 1 1 1 1 −1 −1 −1 −1 i i −i −i i −i −i i 1
272.50.1d1 C 1 1 −1 −1 −ζ82 −ζ82 ζ82 ζ82 ζ83 −ζ83 −ζ8 −ζ8 −ζ83 ζ8 ζ8 ζ83 1
272.50.1d2 C 1 1 −1 −1 ζ82 ζ82 −ζ82 −ζ82 −ζ8 ζ8 ζ83 ζ83 ζ8 −ζ83 −ζ83 −ζ8 1
272.50.1d3 C 1 1 −1 −1 −ζ82 −ζ82 ζ82 ζ82 −ζ83 ζ83 ζ8 ζ8 ζ83 −ζ8 −ζ8 −ζ83 1
272.50.1d4 C 1 1 −1 −1 ζ82 ζ82 −ζ82 −ζ82 ζ8 −ζ8 −ζ83 −ζ83 −ζ8 ζ83 ζ83 ζ8 1
272.50.1e1 C 1 −1 −ζ164 ζ164 ζ166 −ζ166 −ζ162 ζ162 −ζ163 −ζ167 −ζ165 ζ165 ζ167 −ζ16 ζ16 ζ163 1
272.50.1e2 C 1 −1 ζ164 −ζ164 −ζ162 ζ162 ζ166 −ζ166 ζ165 ζ16 ζ163 −ζ163 −ζ16 ζ167 −ζ167 −ζ165 1
272.50.1e3 C 1 −1 −ζ164 ζ164 ζ166 −ζ166 −ζ162 ζ162 ζ163 ζ167 ζ165 −ζ165 −ζ167 ζ16 −ζ16 −ζ163 1
272.50.1e4 C 1 −1 ζ164 −ζ164 −ζ162 ζ162 ζ166 −ζ166 −ζ165 −ζ16 −ζ163 ζ163 ζ16 −ζ167 ζ167 ζ165 1
272.50.1e5 C 1 −1 −ζ164 ζ164 −ζ166 ζ166 ζ162 −ζ162 ζ167 −ζ163 ζ16 −ζ16 ζ163 −ζ165 ζ165 −ζ167 1
272.50.1e6 C 1 −1 ζ164 −ζ164 ζ162 −ζ162 −ζ166 ζ166 −ζ16 ζ165 −ζ167 ζ167 −ζ165 ζ163 −ζ163 ζ16 1
272.50.1e7 C 1 −1 −ζ164 ζ164 −ζ166 ζ166 ζ162 −ζ162 −ζ167 ζ163 −ζ16 ζ16 −ζ163 ζ165 −ζ165 ζ167 1
272.50.1e8 C 1 −1 ζ164 −ζ164 ζ162 −ζ162 −ζ166 ζ166 ζ16 −ζ165 ζ167 −ζ167 ζ165 −ζ163 ζ163 −ζ16 1
272.50.16a R 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed