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Properties

Label 17T5
Degree $17$
Order $272$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $F_{17}$

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Show commands: Magma

magma: G := TransitiveGroup(17, 5);
 

Group action invariants

Degree $n$:  $17$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $5$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $F_{17}$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (2,4,10,11,14,6,16,12,17,15,9,8,5,13,3,7), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)
magma: Generators(G);
 

Low degree resolvents

|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

Prime degree - 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 TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 8, 8, 1 $ $17$ $8$ $( 2, 3, 5, 9,17,16,14,10)( 4, 7,13, 8,15,12, 6,11)$
$ 16, 1 $ $17$ $16$ $( 2, 4,10,11,14, 6,16,12,17,15, 9, 8, 5,13, 3, 7)$
$ 4, 4, 4, 4, 1 $ $17$ $4$ $( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)$
$ 16, 1 $ $17$ $16$ $( 2, 6, 9, 7,14,15, 3,11,17,13,10,12, 5, 4,16, 8)$
$ 16, 1 $ $17$ $16$ $( 2, 7, 3,13, 5, 8, 9,15,17,12,16, 6,14,11,10, 4)$
$ 16, 1 $ $17$ $16$ $( 2, 8,16, 4, 5,12,10,13,17,11, 3,15,14, 7, 9, 6)$
$ 8, 8, 1 $ $17$ $8$ $( 2, 9,14, 3,17,10, 5,16)( 4, 8, 6, 7,15,11,13,12)$
$ 8, 8, 1 $ $17$ $8$ $( 2,10,14,16,17, 9, 5, 3)( 4,11, 6,12,15, 8,13, 7)$
$ 16, 1 $ $17$ $16$ $( 2,11,16,15, 5, 7,10, 6,17, 8, 3, 4,14,12, 9,13)$
$ 16, 1 $ $17$ $16$ $( 2,12, 3, 6, 5,11, 9, 4,17, 7,16,13,14, 8,10,15)$
$ 16, 1 $ $17$ $16$ $( 2,13, 9,12,14, 4, 3, 8,17, 6,10, 7, 5,15,16,11)$
$ 4, 4, 4, 4, 1 $ $17$ $4$ $( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)$
$ 16, 1 $ $17$ $16$ $( 2,15,10, 8,14,13,16, 7,17, 4, 9,11, 5, 6, 3,12)$
$ 8, 8, 1 $ $17$ $8$ $( 2,16, 5,10,17, 3,14, 9)( 4,12,13,11,15, 7, 6, 8)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $17$ $2$ $( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)$
$ 17 $ $16$ $17$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $272=2^{4} \cdot 17$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  272.50
magma: IdentifyGroup(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 4A1 4A-1 4A-1 8A1 8A-3 8A-1 8A-1 8A-3 8A3 8A3 8A1 17A
17 P 1A 2A 4A1 4A-1 8A1 8A-3 8A-1 8A3 16A1 16A-3 16A7 16A-1 16A5 16A-5 16A3 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 −ζ8 ζ8 −ζ83 −ζ83 ζ8 −ζ8 ζ83 1
272.50.1d2 C 1 1 −1 −1 ζ82 −ζ82 −ζ82 ζ82 −ζ8 ζ83 −ζ83 ζ8 ζ8 −ζ83 ζ83 −ζ8 1
272.50.1d3 C 1 1 −1 −1 −ζ82 ζ82 ζ82 −ζ82 −ζ83 ζ8 −ζ8 ζ83 ζ83 −ζ8 ζ8 −ζ83 1
272.50.1d4 C 1 1 −1 −1 ζ82 −ζ82 −ζ82 ζ82 ζ8 −ζ83 ζ83 −ζ8 −ζ8 ζ83 −ζ83 ζ8 1
272.50.1e1 C 1 −1 −ζ164 ζ164 ζ166 −ζ162 ζ162 −ζ166 −ζ163 ζ165 ζ16 −ζ167 ζ167 −ζ16 −ζ165 ζ163 1
272.50.1e2 C 1 −1 ζ164 −ζ164 −ζ162 ζ166 −ζ166 ζ162 ζ165 −ζ163 −ζ167 ζ16 −ζ16 ζ167 ζ163 −ζ165 1
272.50.1e3 C 1 −1 −ζ164 ζ164 ζ166 −ζ162 ζ162 −ζ166 ζ163 −ζ165 −ζ16 ζ167 −ζ167 ζ16 ζ165 −ζ163 1
272.50.1e4 C 1 −1 ζ164 −ζ164 −ζ162 ζ166 −ζ166 ζ162 −ζ165 ζ163 ζ167 −ζ16 ζ16 −ζ167 −ζ163 ζ165 1
272.50.1e5 C 1 −1 −ζ164 ζ164 −ζ166 ζ162 −ζ162 ζ166 ζ167 −ζ16 ζ165 −ζ163 ζ163 −ζ165 ζ16 −ζ167 1
272.50.1e6 C 1 −1 ζ164 −ζ164 ζ162 −ζ166 ζ166 −ζ162 −ζ16 ζ167 −ζ163 ζ165 −ζ165 ζ163 −ζ167 ζ16 1
272.50.1e7 C 1 −1 −ζ164 ζ164 −ζ166 ζ162 −ζ162 ζ166 −ζ167 ζ16 −ζ165 ζ163 −ζ163 ζ165 −ζ16 ζ167 1
272.50.1e8 C 1 −1 ζ164 −ζ164 ζ162 −ζ166 ζ166 −ζ162 ζ16 −ζ167 ζ163 −ζ165 ζ165 −ζ163 ζ167 −ζ16 1
272.50.16a R 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1

magma: CharacterTable(G);