Properties

Label 38T8
Order \(342\)
n \(38\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $F_{19}$

Learn more about

Group action invariants

Degree $n$ :  $38$
Transitive number $t$ :  $8$
Group :  $F_{19}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,22,31,18,29,36,19,12,8,5,24,14,27,16,9,26,34,37)(2,21,32,17,30,35,20,11,7,6,23,13,28,15,10,25,33,38)(3,4), (1,35,9,11,6,24,8,17,25)(2,36,10,12,5,23,7,18,26)(3,29,27,34,15,31,21,13,38)(4,30,28,33,16,32,22,14,37)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
9:  $C_9$
18:  $C_{18}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: $F_{19}$

Low degree siblings

19T6

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, 1, 1 $ $1$ $1$ $()$
$ 9, 9, 9, 9, 1, 1 $ $19$ $9$ $( 3, 9,34,15,19,35,24,13,11)( 4,10,33,16,20,36,23,14,12)( 5,18,28,30,37,32, 7, 26,22)( 6,17,27,29,38,31, 8,25,21)$
$ 9, 9, 9, 9, 1, 1 $ $19$ $9$ $( 3,11,13,24,35,19,15,34, 9)( 4,12,14,23,36,20,16,33,10)( 5,22,26, 7,32,37,30, 28,18)( 6,21,25, 8,31,38,29,27,17)$
$ 9, 9, 9, 9, 1, 1 $ $19$ $9$ $( 3,13,35,15, 9,11,24,19,34)( 4,14,36,16,10,12,23,20,33)( 5,26,32,30,18,22, 7, 37,28)( 6,25,31,29,17,21, 8,38,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $19$ $3$ $( 3,15,24)( 4,16,23)( 5,30, 7)( 6,29, 8)( 9,19,13)(10,20,14)(11,34,35) (12,33,36)(17,38,25)(18,37,26)(21,27,31)(22,28,32)$
$ 9, 9, 9, 9, 1, 1 $ $19$ $9$ $( 3,19,11,15,13,34,24, 9,35)( 4,20,12,16,14,33,23,10,36)( 5,37,22,30,26,28, 7, 18,32)( 6,38,21,29,25,27, 8,17,31)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $19$ $3$ $( 3,24,15)( 4,23,16)( 5, 7,30)( 6, 8,29)( 9,13,19)(10,14,20)(11,35,34) (12,36,33)(17,25,38)(18,26,37)(21,31,27)(22,32,28)$
$ 9, 9, 9, 9, 1, 1 $ $19$ $9$ $( 3,34,19,24,11, 9,15,35,13)( 4,33,20,23,12,10,16,36,14)( 5,28,37, 7,22,18,30, 32,26)( 6,27,38, 8,21,17,29,31,25)$
$ 9, 9, 9, 9, 1, 1 $ $19$ $9$ $( 3,35, 9,24,34,13,15,11,19)( 4,36,10,23,33,14,16,12,20)( 5,32,18, 7,28,26,30, 22,37)( 6,31,17, 8,27,25,29,21,38)$
$ 18, 18, 2 $ $19$ $18$ $( 1, 2)( 3, 5, 9,18,34,28,15,30,19,37,35,32,24, 7,13,26,11,22)( 4, 6,10,17,33, 27,16,29,20,38,36,31,23, 8,14,25,12,21)$
$ 18, 18, 2 $ $19$ $18$ $( 1, 2)( 3, 7,19,18,11,32,15, 5,13,37,34,22,24,30, 9,26,35,28)( 4, 8,20,17,12, 31,16, 6,14,38,33,21,23,29,10,25,36,27)$
$ 6, 6, 6, 6, 6, 6, 2 $ $19$ $6$ $( 1, 2)( 3,18,15,37,24,26)( 4,17,16,38,23,25)( 5,34,30,35, 7,11) ( 6,33,29,36, 8,12)( 9,28,19,32,13,22)(10,27,20,31,14,21)$
$ 18, 18, 2 $ $19$ $18$ $( 1, 2)( 3,22,11,26,13, 7,24,32,35,37,19,30,15,28,34,18, 9, 5)( 4,21,12,25,14, 8,23,31,36,38,20,29,16,27,33,17,10, 6)$
$ 6, 6, 6, 6, 6, 6, 2 $ $19$ $6$ $( 1, 2)( 3,26,24,37,15,18)( 4,25,23,38,16,17)( 5,11, 7,35,30,34) ( 6,12, 8,36,29,33)( 9,22,13,32,19,28)(10,21,14,31,20,27)$
$ 18, 18, 2 $ $19$ $18$ $( 1, 2)( 3,28,35,26, 9,30,24,22,34,37,13, 5,15,32,11,18,19, 7)( 4,27,36,25,10, 29,23,21,33,38,14, 6,16,31,12,17,20, 8)$
$ 18, 18, 2 $ $19$ $18$ $( 1, 2)( 3,30,13,18,35,22,15, 7, 9,37,11,28,24, 5,19,26,34,32)( 4,29,14,17,36, 21,16, 8,10,38,12,27,23, 6,20,25,33,31)$
$ 18, 18, 2 $ $19$ $18$ $( 1, 2)( 3,32,34,26,19, 5,24,28,11,37, 9, 7,15,22,35,18,13,30)( 4,31,33,25,20, 6,23,27,12,38,10, 8,16,21,36,17,14,29)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $19$ $2$ $( 1, 2)( 3,37)( 4,38)( 5,35)( 6,36)( 7,34)( 8,33)( 9,32)(10,31)(11,30)(12,29) (13,28)(14,27)(15,26)(16,25)(17,23)(18,24)(19,22)(20,21)$
$ 19, 19 $ $18$ $19$ $( 1, 3, 6, 8, 9,11,13,15,17,19,21,24,25,27,29,31,34,35,38)( 2, 4, 5, 7,10,12, 14,16,18,20,22,23,26,28,30,32,33,36,37)$

Group invariants

Order:  $342=2 \cdot 3^{2} \cdot 19$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [342, 7]
Character table:   
      2  1  1  1  1  1  1  1  1  1   1   1   1   1   1   1   1   1  1   .
      3  2  2  2  2  2  2  2  2  2   2   2   2   2   2   2   2   2  2   .
     19  1  .  .  .  .  .  .  .  .   .   .   .   .   .   .   .   .  .   1

        1a 9a 9b 9c 3a 9d 3b 9e 9f 18a 18b  6a 18c  6b 18d 18e 18f 2a 19a
     2P 1a 9e 9c 9f 3b 9b 3a 9d 9a  9a  9d  3a  9b  3b  9f  9c  9e 1a 19a
     3P 1a 3a 3b 3a 1a 3a 1a 3b 3b  6a  6a  2a  6b  2a  6b  6a  6b 2a 19a
     5P 1a 9f 9d 9b 3b 9e 3a 9a 9c 18d 18f  6b 18b  6a 18e 18c 18a 2a 19a
     7P 1a 9c 9e 9d 3a 9a 3b 9f 9b 18e 18a  6a 18f  6b 18c 18b 18d 2a 19a
    11P 1a 9e 9c 9f 3b 9b 3a 9d 9a 18f 18c  6b 18e  6a 18a 18d 18b 2a 19a
    13P 1a 9d 9f 9a 3a 9c 3b 9b 9e 18b 18e  6a 18d  6b 18f 18a 18c 2a 19a
    17P 1a 9b 9a 9e 3b 9f 3a 9c 9d 18c 18d  6b 18a  6a 18b 18f 18e 2a 19a
    19P 1a 9a 9b 9c 3a 9d 3b 9e 9f 18a 18b  6a 18c  6b 18d 18e 18f 2a  1a

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = -E(9)^2-E(9)^5
C = E(9)^7
D = E(9)^5