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Properties

Label 17T1
Order \(17\)
n \(17\)
Cyclic Yes
Abelian Yes
Solvable Yes
Primitive Yes
$p$-group Yes
Group: $C_{17}$

Related objects

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Group action invariants

Degree $n$ :  $17$
Transitive number $t$ :  $1$
Group :  $C_{17}$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $1$
Generators:  (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)
$|\Aut(F/K)|$:  $17$

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

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

Group invariants

Order:  $17$ (is prime)
Cyclic:  Yes
Abelian:  Yes
Solvable:  Yes
GAP id:  [17, 1]
Character table:    
     17  1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1

        1a 17a 17b 17c 17d 17e 17f 17g 17h 17i 17j 17k 17l 17m 17n 17o 17p

X.1      1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
X.2      1   A   B   C   D   E   F   G   H  /H  /G  /F  /E  /D  /C  /B  /A
X.3      1   B   D   F   H  /G  /E  /C  /A   A   C   E   G  /H  /F  /D  /B
X.4      1   C   F  /H  /E  /B   A   D   G  /G  /D  /A   B   E   H  /F  /C
X.5      1   D   H  /E  /A   C   G  /F  /B   B   F  /G  /C   A   E  /H  /D
X.6      1   E  /G  /B   C   H  /D   A   F  /F  /A   D  /H  /C   B   G  /E
X.7      1   F  /E   A   G  /D   B   H  /C   C  /H  /B   D  /G  /A   E  /F
X.8      1   G  /C   D  /F   A   H  /B   E  /E   B  /H  /A   F  /D   C  /G
X.9      1   H  /A   G  /B   F  /C   E  /D   D  /E   C  /F   B  /G   A  /H
X.10     1  /H   A  /G   B  /F   C  /E   D  /D   E  /C   F  /B   G  /A   H
X.11     1  /G   C  /D   F  /A  /H   B  /E   E  /B   H   A  /F   D  /C   G
X.12     1  /F   E  /A  /G   D  /B  /H   C  /C   H   B  /D   G   A  /E   F
X.13     1  /E   G   B  /C  /H   D  /A  /F   F   A  /D   H   C  /B  /G   E
X.14     1  /D  /H   E   A  /C  /G   F   B  /B  /F   G   C  /A  /E   H   D
X.15     1  /C  /F   H   E   B  /A  /D  /G   G   D   A  /B  /E  /H   F   C
X.16     1  /B  /D  /F  /H   G   E   C   A  /A  /C  /E  /G   H   F   D   B
X.17     1  /A  /B  /C  /D  /E  /F  /G  /H   H   G   F   E   D   C   B   A

A = E(17)
B = E(17)^2
C = E(17)^3
D = E(17)^4
E = E(17)^5
F = E(17)^6
G = E(17)^7
H = E(17)^8