Properties

Label 17T5
Order \(272\)
n \(17\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $F_{17}$

Related objects

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Group action invariants

Degree $n$ :  $17$
Transitive number $t$ :  $5$
Group :  $F_{17}$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $1$

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

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

Group invariants

Order:  $272=2^{4} \cdot 17$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [272, 50]
Character table:   
      2  4   4   4  4   4   4   4   4   4   4   4   4  4   4   4  4   .
     17  1   .   .  .   .   .   .   .   .   .   .   .  .   .   .  .   1

        1a  8a 16a 4a 16b 16c 16d  8b  8c 16e 16f 16g 4b 16h  8d 2a 17a
     2P 1a  4a  8c 2a  8b  8a  8d  4b  4b  8d  8a  8b 2a  8c  4a 1a 17a
     3P 1a  8b 16e 4b 16c 16g 16a  8a  8d 16h 16b 16f 4a 16d  8c 2a 17a
     5P 1a  8d 16b 4a 16h 16d 16f  8c  8b 16c 16e 16a 4b 16g  8a 2a 17a
     7P 1a  8c 16f 4b 16e 16h 16g  8d  8a 16b 16a 16d 4a 16c  8b 2a 17a
    11P 1a  8b 16d 4b 16f 16b 16h  8a  8d 16a 16g 16c 4a 16e  8c 2a 17a
    13P 1a  8d 16g 4a 16a 16e 16c  8c  8b 16f 16d 16h 4b 16b  8a 2a 17a
    17P 1a  8a 16a 4a 16b 16c 16d  8b  8c 16e 16f 16g 4b 16h  8d 2a  1a

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

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(8)
C = -E(16)^3
D = -E(16)