Properties

Label 34T3
Order \(68\)
n \(34\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{34}$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
34:  $D_{17}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: $D_{17}$

Low degree siblings

34T3

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

Group invariants

Order:  $68=2^{2} \cdot 17$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [68, 4]
Character table:   
      2  2  2  2  2   1   1   1   1   1   1   1   1   1   1   1   1   1   1
     17  1  .  1  .   1   1   1   1   1   1   1   1   1   1   1   1   1   1

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

X.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
X.3      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
X.5      2  .  2  .   A   A   E   E   D   D   B   B   F   F   G   G   H   H
X.6      2  .  2  .   B   B   C   C   F   F   A   A   D   D   H   H   G   G
X.7      2  .  2  .   C   C   A   A   H   H   E   E   G   G   D   D   F   F
X.8      2  .  2  .   D   D   G   G   C   C   F   F   E   E   A   A   B   B
X.9      2  .  2  .   E   E   B   B   G   G   C   C   H   H   F   F   D   D
X.10     2  .  2  .   F   F   H   H   E   E   D   D   C   C   B   B   A   A
X.11     2  .  2  .   G   G   F   F   A   A   H   H   B   B   E   E   C   C
X.12     2  .  2  .   H   H   D   D   B   B   G   G   A   A   C   C   E   E
X.13     2  . -2  .  -A   A  -E   E  -D   D  -B   B  -F   F   G  -G  -H   H
X.14     2  . -2  .  -B   B  -C   C  -F   F  -A   A  -D   D   H  -H  -G   G
X.15     2  . -2  .  -C   C  -A   A  -H   H  -E   E  -G   G   D  -D  -F   F
X.16     2  . -2  .  -D   D  -G   G  -C   C  -F   F  -E   E   A  -A  -B   B
X.17     2  . -2  .  -E   E  -B   B  -G   G  -C   C  -H   H   F  -F  -D   D
X.18     2  . -2  .  -F   F  -H   H  -E   E  -D   D  -C   C   B  -B  -A   A
X.19     2  . -2  .  -G   G  -F   F  -A   A  -H   H  -B   B   E  -E  -C   C
X.20     2  . -2  .  -H   H  -D   D  -B   B  -G   G  -A   A   C  -C  -E   E

      2   1   1
     17   1   1

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

X.1       1   1
X.2       1  -1
X.3       1   1
X.4       1  -1
X.5       C   C
X.6       E   E
X.7       B   B
X.8       H   H
X.9       A   A
X.10      G   G
X.11      D   D
X.12      F   F
X.13      C  -C
X.14      E  -E
X.15      B  -B
X.16      H  -H
X.17      A  -A
X.18      G  -G
X.19      D  -D
X.20      F  -F

A = E(17)^5+E(17)^12
B = E(17)^3+E(17)^14
C = E(17)^6+E(17)^11
D = E(17)^2+E(17)^15
E = E(17)^7+E(17)^10
F = E(17)^8+E(17)^9
G = E(17)^4+E(17)^13
H = E(17)+E(17)^16