Properties

Label 38T4
Order \(114\)
n \(38\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_{19}:C_3$

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Group action invariants

Degree $n$ :  $38$
Transitive number $t$ :  $4$
Group :  $C_2\times C_{19}:C_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,29,36,2,30,35)(3,5,19,4,6,20)(7,33,26,8,34,25)(9,10)(11,23,31,12,24,32)(13,38,15,14,37,16)(17,28,22,18,27,21), (1,24,25,2,23,26)(3,37,10,4,38,9)(5,14,31,6,13,32)(7,28,15,8,27,16)(11,18,22,12,17,21)(19,35,33,20,36,34)(29,30)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
57:  $C_{19}:C_{3}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: $C_{19}:C_{3}$

Low degree siblings

There are no siblings 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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $19$ $3$ $( 3,16,23)( 4,15,24)( 5,29, 7)( 6,30, 8)( 9,20,13)(10,19,14)(11,34,35) (12,33,36)(17,37,26)(18,38,25)(21,28,32)(22,27,31)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $19$ $3$ $( 3,23,16)( 4,24,15)( 5, 7,29)( 6, 8,30)( 9,13,20)(10,14,19)(11,35,34) (12,36,33)(17,26,37)(18,25,38)(21,32,28)(22,31,27)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)$
$ 6, 6, 6, 6, 6, 6, 2 $ $19$ $6$ $( 1, 2)( 3,15,23, 4,16,24)( 5,30, 7, 6,29, 8)( 9,19,13,10,20,14) (11,33,35,12,34,36)(17,38,26,18,37,25)(21,27,32,22,28,31)$
$ 6, 6, 6, 6, 6, 6, 2 $ $19$ $6$ $( 1, 2)( 3,24,16, 4,23,15)( 5, 8,29, 6, 7,30)( 9,14,20,10,13,19) (11,36,34,12,35,33)(17,25,37,18,26,38)(21,31,28,22,32,27)$
$ 19, 19 $ $3$ $19$ $( 1, 3, 6, 8,10,12,14,16,18,19,21,23,25,28,30,32,33,36,38)( 2, 4, 5, 7, 9,11, 13,15,17,20,22,24,26,27,29,31,34,35,37)$
$ 38 $ $3$ $38$ $( 1, 4, 6, 7,10,11,14,15,18,20,21,24,25,27,30,31,33,35,38, 2, 3, 5, 8, 9,12, 13,16,17,19,22,23,26,28,29,32,34,36,37)$
$ 38 $ $3$ $38$ $( 1, 5,10,13,18,22,25,29,33,37, 3, 7,12,15,19,24,28,31,36, 2, 6, 9,14,17,21, 26,30,34,38, 4, 8,11,16,20,23,27,32,35)$
$ 19, 19 $ $3$ $19$ $( 1, 6,10,14,18,21,25,30,33,38, 3, 8,12,16,19,23,28,32,36)( 2, 5, 9,13,17,22, 26,29,34,37, 4, 7,11,15,20,24,27,31,35)$
$ 38 $ $3$ $38$ $( 1, 9,18,26,33, 4,12,20,28,35, 6,13,21,29,38, 7,16,24,32, 2,10,17,25,34, 3, 11,19,27,36, 5,14,22,30,37, 8,15,23,31)$
$ 19, 19 $ $3$ $19$ $( 1,10,18,25,33, 3,12,19,28,36, 6,14,21,30,38, 8,16,23,32)( 2, 9,17,26,34, 4, 11,20,27,35, 5,13,22,29,37, 7,15,24,31)$
$ 38 $ $3$ $38$ $( 1,11,21,31, 3,13,23,34, 6,15,25,35, 8,17,28,37,10,20,30, 2,12,22,32, 4,14, 24,33, 5,16,26,36, 7,18,27,38, 9,19,29)$
$ 19, 19 $ $3$ $19$ $( 1,12,21,32, 3,14,23,33, 6,16,25,36, 8,18,28,38,10,19,30)( 2,11,22,31, 4,13, 24,34, 5,15,26,35, 7,17,27,37, 9,20,29)$
$ 38 $ $3$ $38$ $( 1,17,33,11,28, 5,21,37,16,31,10,26, 3,20,36,13,30, 7,23, 2,18,34,12,27, 6, 22,38,15,32, 9,25, 4,19,35,14,29, 8,24)$
$ 19, 19 $ $3$ $19$ $( 1,18,33,12,28, 6,21,38,16,32,10,25, 3,19,36,14,30, 8,23)( 2,17,34,11,27, 5, 22,37,15,31, 9,26, 4,20,35,13,29, 7,24)$
$ 19, 19 $ $3$ $19$ $( 1,21, 3,23, 6,25, 8,28,10,30,12,32,14,33,16,36,18,38,19)( 2,22, 4,24, 5,26, 7,27, 9,29,11,31,13,34,15,35,17,37,20)$
$ 38 $ $3$ $38$ $( 1,22, 3,24, 6,26, 8,27,10,29,12,31,14,34,16,35,18,37,19, 2,21, 4,23, 5,25, 7,28, 9,30,11,32,13,33,15,36,17,38,20)$

Group invariants

Order:  $114=2 \cdot 3 \cdot 19$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [114, 2]
Character table:   
      2  1  1  1  1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
      3  1  1  1  1   1   1   .   .   .   .   .   .   .   .   .   .   .   .
     19  1  .  .  1   .   .   1   1   1   1   1   1   1   1   1   1   1   1

        1a 3a 3b 2a  6a  6b 19a 38a 38b 19b 38c 19c 38d 19d 38e 19e 19f 38f
     2P 1a 3b 3a 1a  3b  3a 19b 19b 19c 19c 19e 19e 19f 19f 19d 19d 19a 19a
     3P 1a 1a 1a 2a  2a  2a 19b 38b 38c 19c 38e 19e 38f 19f 38d 19d 19a 38a
     5P 1a 3b 3a 2a  6b  6a 19d 38d 38f 19f 38a 19a 38c 19c 38b 19b 19e 38e
     7P 1a 3a 3b 2a  6a  6b 19a 38a 38b 19b 38c 19c 38d 19d 38e 19e 19f 38f
    11P 1a 3b 3a 2a  6b  6a 19a 38a 38b 19b 38c 19c 38d 19d 38e 19e 19f 38f
    13P 1a 3a 3b 2a  6a  6b 19f 38f 38a 19a 38b 19b 38e 19e 38c 19c 19d 38d
    17P 1a 3b 3a 2a  6b  6a 19d 38d 38f 19f 38a 19a 38c 19c 38b 19b 19e 38e
    19P 1a 3a 3b 2a  6a  6b  1a  2a  2a  1a  2a  1a  2a  1a  2a  1a  1a  2a
    23P 1a 3b 3a 2a  6b  6a 19c 38c 38e 19e 38d 19d 38a 19a 38f 19f 19b 38b
    29P 1a 3b 3a 2a  6b  6a 19f 38f 38a 19a 38b 19b 38e 19e 38c 19c 19d 38d
    31P 1a 3a 3b 2a  6a  6b 19e 38e 38d 19d 38f 19f 38b 19b 38a 19a 19c 38c
    37P 1a 3a 3b 2a  6a  6b 19e 38e 38d 19d 38f 19f 38b 19b 38a 19a 19c 38c

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(19)^2+E(19)^3+E(19)^14
C = E(19)^4+E(19)^6+E(19)^9
D = E(19)+E(19)^7+E(19)^11