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Properties

Label 17T1
Degree $17$
Order $17$
Cyclic yes
Abelian yes
Solvable yes
Primitive yes
$p$-group yes
Group: $C_{17}$

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Show commands: Magma

magma: G := TransitiveGroup(17, 1);
 

Group action invariants

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

Low degree resolvents

none

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
$1^{17}$ $1$ $1$ $()$
$17$ $1$ $17$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)$
$17$ $1$ $17$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)$
$17$ $1$ $17$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)$
$17$ $1$ $17$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)$
$17$ $1$ $17$ $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)$
$17$ $1$ $17$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)$
$17$ $1$ $17$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)$
$17$ $1$ $17$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)$
$17$ $1$ $17$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)$
$17$ $1$ $17$ $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)$
$17$ $1$ $17$ $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)$
$17$ $1$ $17$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)$
$17$ $1$ $17$ $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)$
$17$ $1$ $17$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)$
$17$ $1$ $17$ $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)$
$17$ $1$ $17$ $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $17$ (is prime)
magma: Order(G);
 
Cyclic:  yes
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  17.1
magma: IdentifyGroup(G);
 
Character table:

1A 17A1 17A-1 17A2 17A-2 17A3 17A-3 17A4 17A-4 17A5 17A-5 17A6 17A-6 17A7 17A-7 17A8 17A-8
Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
17 P 1A 17A5 17A-2 17A7 17A-1 17A6 17A3 17A-8 17A-4 17A-7 17A-5 17A-6 17A-3 17A2 17A4 17A1 17A8
Type
17.1.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
17.1.1b1 C 1 ζ17−8 ζ178 ζ17 ζ17−1 ζ17−7 ζ177 ζ172 ζ17−2 ζ17−6 ζ176 ζ173 ζ17−3 ζ17−5 ζ175 ζ174 ζ17−4
17.1.1b2 C 1 ζ178 ζ17−8 ζ17−1 ζ17 ζ177 ζ17−7 ζ17−2 ζ172 ζ176 ζ17−6 ζ17−3 ζ173 ζ175 ζ17−5 ζ17−4 ζ174
17.1.1b3 C 1 ζ17−7 ζ177 ζ173 ζ17−3 ζ17−4 ζ174 ζ176 ζ17−6 ζ17−1 ζ17 ζ17−8 ζ178 ζ172 ζ17−2 ζ17−5 ζ175
17.1.1b4 C 1 ζ177 ζ17−7 ζ17−3 ζ173 ζ174 ζ17−4 ζ17−6 ζ176 ζ17 ζ17−1 ζ178 ζ17−8 ζ17−2 ζ172 ζ175 ζ17−5
17.1.1b5 C 1 ζ17−6 ζ176 ζ175 ζ17−5 ζ17−1 ζ17 ζ17−7 ζ177 ζ174 ζ17−4 ζ17−2 ζ172 ζ17−8 ζ178 ζ173 ζ17−3
17.1.1b6 C 1 ζ176 ζ17−6 ζ17−5 ζ175 ζ17 ζ17−1 ζ177 ζ17−7 ζ17−4 ζ174 ζ172 ζ17−2 ζ178 ζ17−8 ζ17−3 ζ173
17.1.1b7 C 1 ζ17−5 ζ175 ζ177 ζ17−7 ζ172 ζ17−2 ζ17−3 ζ173 ζ17−8 ζ178 ζ174 ζ17−4 ζ17−1 ζ17 ζ17−6 ζ176
17.1.1b8 C 1 ζ175 ζ17−5 ζ17−7 ζ177 ζ17−2 ζ172 ζ173 ζ17−3 ζ178 ζ17−8 ζ17−4 ζ174 ζ17 ζ17−1 ζ176 ζ17−6
17.1.1b9 C 1 ζ17−4 ζ174 ζ17−8 ζ178 ζ175 ζ17−5 ζ17 ζ17−1 ζ17−3 ζ173 ζ17−7 ζ177 ζ176 ζ17−6 ζ172 ζ17−2
17.1.1b10 C 1 ζ174 ζ17−4 ζ178 ζ17−8 ζ17−5 ζ175 ζ17−1 ζ17 ζ173 ζ17−3 ζ177 ζ17−7 ζ17−6 ζ176 ζ17−2 ζ172
17.1.1b11 C 1 ζ17−3 ζ173 ζ17−6 ζ176 ζ178 ζ17−8 ζ175 ζ17−5 ζ172 ζ17−2 ζ17−1 ζ17 ζ17−4 ζ174 ζ17−7 ζ177
17.1.1b12 C 1 ζ173 ζ17−3 ζ176 ζ17−6 ζ17−8 ζ178 ζ17−5 ζ175 ζ17−2 ζ172 ζ17 ζ17−1 ζ174 ζ17−4 ζ177 ζ17−7
17.1.1b13 C 1 ζ17−2 ζ172 ζ17−4 ζ174 ζ17−6 ζ176 ζ17−8 ζ178 ζ177 ζ17−7 ζ175 ζ17−5 ζ173 ζ17−3 ζ17 ζ17−1
17.1.1b14 C 1 ζ172 ζ17−2 ζ174 ζ17−4 ζ176 ζ17−6 ζ178 ζ17−8 ζ17−7 ζ177 ζ17−5 ζ175 ζ17−3 ζ173 ζ17−1 ζ17
17.1.1b15 C 1 ζ17−1 ζ17 ζ17−2 ζ172 ζ17−3 ζ173 ζ17−4 ζ174 ζ17−5 ζ175 ζ17−6 ζ176 ζ17−7 ζ177 ζ17−8 ζ178
17.1.1b16 C 1 ζ17 ζ17−1 ζ172 ζ17−2 ζ173 ζ17−3 ζ174 ζ17−4 ζ175 ζ17−5 ζ176 ζ17−6 ζ177 ζ17−7 ζ178 ζ17−8

magma: CharacterTable(G);