Properties

Label 34T7
Order \(272\)
n \(34\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_{17}:C_8$

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Group action invariants

Degree $n$ :  $34$
Transitive number $t$ :  $7$
Group :  $C_2\times C_{17}:C_8$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,34,17,25,21,24,5,31)(2,33,18,26,22,23,6,32)(3,16,10,29,20,8,13,27)(4,15,9,30,19,7,14,28), (1,7,29,19,5,33,11,22)(2,8,30,20,6,34,12,21)(3,4)(9,25,28,24,32,16,14,17)(10,26,27,23,31,15,13,18)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $C_8$ x 2, $C_4\times C_2$
16:  $C_8\times C_2$
136:  $C_{17}:C_{8}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: $C_{17}:C_{8}$

Low degree siblings

34T7

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

Group invariants

Order:  $272=2^{4} \cdot 17$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [272, 51]
Character table:   
      2  4   4  4   4   4  4   4  4  4   4  4   4   4  4   4  4   1   1   1
     17  1   .  .   .   .  .   .  .  1   .  .   .   .  .   .  .   1   1   1

        1a  8a 4a  8b  8c 4b  8d 2a 2b  8e 4c  8f  8g 4d  8h 2c 17a 34a 34b
     2P 1a  4a 2a  4b  4b 2a  4a 1a 1a  4a 2a  4b  4b 2a  4a 1a 17a 17a 17b
     3P 1a  8b 4b  8a  8d 4a  8c 2a 2b  8f 4d  8e  8h 4c  8g 2c 17b 34b 34a
     5P 1a  8d 4a  8c  8b 4b  8a 2a 2b  8h 4c  8g  8f 4d  8e 2c 17b 34b 34a
     7P 1a  8c 4b  8d  8a 4a  8b 2a 2b  8g 4d  8h  8e 4c  8f 2c 17b 34b 34a
    11P 1a  8b 4b  8a  8d 4a  8c 2a 2b  8f 4d  8e  8h 4c  8g 2c 17b 34b 34a
    13P 1a  8d 4a  8c  8b 4b  8a 2a 2b  8h 4c  8g  8f 4d  8e 2c 17a 34a 34b
    17P 1a  8a 4a  8b  8c 4b  8d 2a 2b  8e 4c  8f  8g 4d  8h 2c  1a  2b  2b
    19P 1a  8b 4b  8a  8d 4a  8c 2a 2b  8f 4d  8e  8h 4c  8g 2c 17a 34a 34b
    23P 1a  8c 4b  8d  8a 4a  8b 2a 2b  8g 4d  8h  8e 4c  8f 2c 17b 34b 34a
    29P 1a  8d 4a  8c  8b 4b  8a 2a 2b  8h 4c  8g  8f 4d  8e 2c 17b 34b 34a
    31P 1a  8c 4b  8d  8a 4a  8b 2a 2b  8g 4d  8h  8e 4c  8f 2c 17b 34b 34a

X.1      1   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  -1  -1
X.3      1  -1  1  -1  -1  1  -1  1  1  -1  1  -1  -1  1  -1  1   1   1   1
X.4      1   1  1   1   1  1   1  1 -1  -1 -1  -1  -1 -1  -1 -1   1  -1  -1
X.5      1   A -1  -A  -A -1   A  1 -1  -A  1   A   A  1  -A -1   1  -1  -1
X.6      1  -A -1   A   A -1  -A  1 -1   A  1  -A  -A  1   A -1   1  -1  -1
X.7      1   A -1  -A  -A -1   A  1  1   A -1  -A  -A -1   A  1   1   1   1
X.8      1  -A -1   A   A -1  -A  1  1  -A -1   A   A -1  -A  1   1   1   1
X.9      1   B -A -/B  /B  A  -B -1 -1  -B  A  /B -/B -A   B  1   1  -1  -1
X.10     1 -/B  A   B  -B -A  /B -1 -1  /B -A  -B   B  A -/B  1   1  -1  -1
X.11     1  /B  A  -B   B -A -/B -1 -1 -/B -A   B  -B  A  /B  1   1  -1  -1
X.12     1  -B -A  /B -/B  A   B -1 -1   B  A -/B  /B -A  -B  1   1  -1  -1
X.13     1   B -A -/B  /B  A  -B -1  1   B -A -/B  /B  A  -B -1   1   1   1
X.14     1 -/B  A   B  -B -A  /B -1  1 -/B  A   B  -B -A  /B -1   1   1   1
X.15     1  /B  A  -B   B -A -/B -1  1  /B  A  -B   B -A -/B -1   1   1   1
X.16     1  -B -A  /B -/B  A   B -1  1  -B -A  /B -/B  A   B -1   1   1   1
X.17     8   .  .   .   .  .   .  . -8   .  .   .   .  .   .  .   C  -C -*C
X.18     8   .  .   .   .  .   .  . -8   .  .   .   .  .   .  .  *C -*C  -C
X.19     8   .  .   .   .  .   .  .  8   .  .   .   .  .   .  .   C   C  *C
X.20     8   .  .   .   .  .   .  .  8   .  .   .   .  .   .  .  *C  *C   C

      2   1
     17   1

        17b
     2P 17b
     3P 17a
     5P 17a
     7P 17a
    11P 17a
    13P 17b
    17P  1a
    19P 17b
    23P 17a
    29P 17a
    31P 17a

X.1       1
X.2       1
X.3       1
X.4       1
X.5       1
X.6       1
X.7       1
X.8       1
X.9       1
X.10      1
X.11      1
X.12      1
X.13      1
X.14      1
X.15      1
X.16      1
X.17     *C
X.18      C
X.19     *C
X.20      C

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(8)
C = E(17)^3+E(17)^5+E(17)^6+E(17)^7+E(17)^10+E(17)^11+E(17)^12+E(17)^14
  = (-1-Sqrt(17))/2 = -1-b17