↑

Properties

Label 34T5
Order \(136\)
n \(34\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times D_{17}.C_2$

Learn more about

Group action invariants

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

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:  $C_4\times C_2$
68:  $C_{17}:C_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: $C_{17}:C_{4}$

Low degree siblings

34T5

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

Group invariants

Order:  $136=2^{3} \cdot 17$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [136, 13]
Character table:    
      2  3  3  3  3  3  3  3  3   1   1   1   1   1   1   1   1
     17  1  .  .  .  1  .  .  .   1   1   1   1   1   1   1   1

        1a 4a 4b 2a 2b 4c 4d 2c 34a 17a 17b 34b 34c 17c 34d 17d
     2P 1a 2a 2a 1a 1a 2a 2a 1a 17b 17b 17a 17a 17d 17d 17c 17c
     3P 1a 4b 4a 2a 2b 4d 4c 2c 34c 17c 17d 34d 34b 17b 34a 17a
     5P 1a 4a 4b 2a 2b 4c 4d 2c 34c 17c 17d 34d 34b 17b 34a 17a
     7P 1a 4b 4a 2a 2b 4d 4c 2c 34d 17d 17c 34c 34a 17a 34b 17b
    11P 1a 4b 4a 2a 2b 4d 4c 2c 34d 17d 17c 34c 34a 17a 34b 17b
    13P 1a 4a 4b 2a 2b 4c 4d 2c 34a 17a 17b 34b 34c 17c 34d 17d
    17P 1a 4a 4b 2a 2b 4c 4d 2c  2b  1a  1a  2b  2b  1a  2b  1a
    19P 1a 4b 4a 2a 2b 4d 4c 2c 34b 17b 17a 34a 34d 17d 34c 17c
    23P 1a 4b 4a 2a 2b 4d 4c 2c 34d 17d 17c 34c 34a 17a 34b 17b
    29P 1a 4a 4b 2a 2b 4c 4d 2c 34c 17c 17d 34d 34b 17b 34a 17a
    31P 1a 4b 4a 2a 2b 4d 4c 2c 34c 17c 17d 34d 34b 17b 34a 17a

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

A = -E(4)
  = -Sqrt(-1) = -i
B = E(17)^3+E(17)^5+E(17)^12+E(17)^14
C = E(17)^2+E(17)^8+E(17)^9+E(17)^15
D = E(17)+E(17)^4+E(17)^13+E(17)^16
E = E(17)^6+E(17)^7+E(17)^10+E(17)^11