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Properties

Label 34T8
Order \(272\)
n \(34\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $F_{17}$

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Group action invariants

Degree $n$ :  $34$
Transitive number $t$ :  $8$
Group :  $F_{17}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $2$

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

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

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 8, 8, 8, 8, 1, 1 $ $17$ $8$ $( 3, 5, 9,17,34,31,27,20)( 4, 6,10,18,33,32,28,19)( 7,13,25,16,29,23,12,21) ( 8,14,26,15,30,24,11,22)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ $17$ $4$ $( 3, 9,34,27)( 4,10,33,28)( 5,17,31,20)( 6,18,32,19)( 7,25,29,12)( 8,26,30,11) (13,16,23,21)(14,15,24,22)$
$ 8, 8, 8, 8, 1, 1 $ $17$ $8$ $( 3,17,27, 5,34,20, 9,31)( 4,18,28, 6,33,19,10,32)( 7,16,12,13,29,21,25,23) ( 8,15,11,14,30,22,26,24)$
$ 8, 8, 8, 8, 1, 1 $ $17$ $8$ $( 3,20,27,31,34,17, 9, 5)( 4,19,28,32,33,18,10, 6)( 7,21,12,23,29,16,25,13) ( 8,22,11,24,30,15,26,14)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ $17$ $4$ $( 3,27,34, 9)( 4,28,33,10)( 5,20,31,17)( 6,19,32,18)( 7,12,29,25)( 8,11,30,26) (13,21,23,16)(14,22,24,15)$
$ 8, 8, 8, 8, 1, 1 $ $17$ $8$ $( 3,31, 9,20,34, 5,27,17)( 4,32,10,19,33, 6,28,18)( 7,23,25,21,29,13,12,16) ( 8,24,26,22,30,14,11,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $17$ $2$ $( 3,34)( 4,33)( 5,31)( 6,32)( 7,29)( 8,30)( 9,27)(10,28)(11,26)(12,25)(13,23) (14,24)(15,22)(16,21)(17,20)(18,19)$
$ 16, 16, 2 $ $17$ $16$ $( 1, 2)( 3, 7,20,21,27,12,31,23,34,29,17,16, 9,25, 5,13)( 4, 8,19,22,28,11,32, 24,33,30,18,15,10,26, 6,14)$
$ 16, 16, 2 $ $17$ $16$ $( 1, 2)( 3,12,17,13,27,29, 5,21,34,25,20,23, 9, 7,31,16)( 4,11,18,14,28,30, 6, 22,33,26,19,24,10, 8,32,15)$
$ 16, 16, 2 $ $17$ $16$ $( 1, 2)( 3,13, 5,25, 9,16,17,29,34,23,31,12,27,21,20, 7)( 4,14, 6,26,10,15,18, 30,33,24,32,11,28,22,19, 8)$
$ 16, 16, 2 $ $17$ $16$ $( 1, 2)( 3,16,31, 7, 9,23,20,25,34,21, 5,29,27,13,17,12)( 4,15,32, 8,10,24,19, 26,33,22, 6,30,28,14,18,11)$
$ 16, 16, 2 $ $17$ $16$ $( 1, 2)( 3,21,31,29, 9,13,20,12,34,16, 5, 7,27,23,17,25)( 4,22,32,30,10,14,19, 11,33,15, 6, 8,28,24,18,26)$
$ 16, 16, 2 $ $17$ $16$ $( 1, 2)( 3,23, 5,12, 9,21,17, 7,34,13,31,25,27,16,20,29)( 4,24, 6,11,10,22,18, 8,33,14,32,26,28,15,19,30)$
$ 16, 16, 2 $ $17$ $16$ $( 1, 2)( 3,25,17,23,27, 7, 5,16,34,12,20,13, 9,29,31,21)( 4,26,18,24,28, 8, 6, 15,33,11,19,14,10,30,32,22)$
$ 16, 16, 2 $ $17$ $16$ $( 1, 2)( 3,29,20,16,27,25,31,13,34, 7,17,21, 9,12, 5,23)( 4,30,19,15,28,26,32, 14,33, 8,18,22,10,11, 6,24)$
$ 17, 17 $ $16$ $17$ $( 1, 3, 5, 8, 9,11,14,15,17,20,22,24,26,27,30,31,34)( 2, 4, 6, 7,10,12,13,16, 18,19,21,23,25,28,29,32,33)$

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 4a  8b  8c 4b  8d 2a 16a 16b 16c 16d 16e 16f 16g 16h 17a
     2P 1a  4a 2a  4b  4b 2a  4a 1a  8c  8b  8a  8d  8d  8a  8b  8c 17a
     3P 1a  8b 4b  8a  8d 4a  8c 2a 16e 16c 16g 16a 16h 16b 16f 16d 17a
     5P 1a  8d 4a  8c  8b 4b  8a 2a 16b 16h 16d 16f 16c 16e 16a 16g 17a
     7P 1a  8c 4b  8d  8a 4a  8b 2a 16f 16e 16h 16g 16b 16a 16d 16c 17a
    11P 1a  8b 4b  8a  8d 4a  8c 2a 16d 16f 16b 16h 16a 16g 16c 16e 17a
    13P 1a  8d 4a  8c  8b 4b  8a 2a 16g 16a 16e 16c 16f 16d 16h 16b 17a
    17P 1a  8a 4a  8b  8c 4b  8d 2a 16a 16b 16c 16d 16e 16f 16g 16h  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  1  -1  -1  1  -1  1   A   A  -A  -A  -A  -A   A   A   1
X.4      1  -1  1  -1  -1  1  -1  1  -A  -A   A   A   A   A  -A  -A   1
X.5      1   A -1  -A  -A -1   A  1   B  -B  /B -/B -/B  /B  -B   B   1
X.6      1   A -1  -A  -A -1   A  1  -B   B -/B  /B  /B -/B   B  -B   1
X.7      1  -A -1   A   A -1  -A  1 -/B  /B  -B   B   B  -B  /B -/B   1
X.8      1  -A -1   A   A -1  -A  1  /B -/B   B  -B  -B   B -/B  /B   1
X.9      1   B -A -/B  /B  A  -B -1   C  /D  /C   D  -D -/C -/D  -C   1
X.10     1   B -A -/B  /B  A  -B -1  -C -/D -/C  -D   D  /C  /D   C   1
X.11     1 -/B  A   B  -B -A  /B -1   D -/C  /D  -C   C -/D  /C  -D   1
X.12     1 -/B  A   B  -B -A  /B -1  -D  /C -/D   C  -C  /D -/C   D   1
X.13     1  /B  A  -B   B -A -/B -1 -/C  -D  -C -/D  /D   C   D  /C   1
X.14     1  /B  A  -B   B -A -/B -1  /C   D   C  /D -/D  -C  -D -/C   1
X.15     1  -B -A  /B -/B  A   B -1 -/D   C  -D  /C -/C   D  -C  /D   1
X.16     1  -B -A  /B -/B  A   B -1  /D  -C   D -/C  /C  -D   C -/D   1
X.17    16   .  .   .   .  .   .  .   .   .   .   .   .   .   .   .  -1

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