Properties

Label 38T6
Order \(228\)
n \(38\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times D_{19}:C_3$

Learn more about

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $C_6$ x 3
12:  $C_6\times C_2$
114:  $C_{19}:C_{6}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: $C_{19}:C_{6}$

Low degree siblings

38T6

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

Group invariants

Order:  $228=2^{2} \cdot 3 \cdot 19$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [228, 7]
Character table:   
      2  2  2   2  2   2  2  2   2   2   2   2  2   1   1   1   1   1   1
      3  1  1   1  1   1  1  1   1   1   1   1  1   .   .   .   .   .   .
     19  1  .   .  .   .  .  1   .   .   .   .  .   1   1   1   1   1   1

        1a 3a  6a 3b  6b 2a 2b  6c  6d  6e  6f 2c 38a 19a 38b 19b 19c 38c
     2P 1a 3b  3a 3a  3b 1a 1a  3b  3a  3a  3b 1a 19b 19b 19c 19c 19a 19a
     3P 1a 1a  2a 1a  2a 2a 2b  2b  2c  2b  2c 2c 38b 19b 38c 19c 19a 38a
     5P 1a 3b  6b 3a  6a 2a 2b  6e  6f  6c  6d 2c 38b 19b 38c 19c 19a 38a
     7P 1a 3a  6a 3b  6b 2a 2b  6c  6d  6e  6f 2c 38a 19a 38b 19b 19c 38c
    11P 1a 3b  6b 3a  6a 2a 2b  6e  6f  6c  6d 2c 38a 19a 38b 19b 19c 38c
    13P 1a 3a  6a 3b  6b 2a 2b  6c  6d  6e  6f 2c 38c 19c 38a 19a 19b 38b
    17P 1a 3b  6b 3a  6a 2a 2b  6e  6f  6c  6d 2c 38b 19b 38c 19c 19a 38a
    19P 1a 3a  6a 3b  6b 2a 2b  6c  6d  6e  6f 2c  2b  1a  2b  1a  1a  2b
    23P 1a 3b  6b 3a  6a 2a 2b  6e  6f  6c  6d 2c 38c 19c 38a 19a 19b 38b
    29P 1a 3b  6b 3a  6a 2a 2b  6e  6f  6c  6d 2c 38c 19c 38a 19a 19b 38b
    31P 1a 3a  6a 3b  6b 2a 2b  6c  6d  6e  6f 2c 38a 19a 38b 19b 19c 38c
    37P 1a 3a  6a 3b  6b 2a 2b  6c  6d  6e  6f 2c 38a 19a 38b 19b 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  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      1  A -/A /A  -A -1 -1  -A  /A -/A   A  1  -1   1  -1   1   1  -1
X.6      1 /A  -A  A -/A -1 -1 -/A   A  -A  /A  1  -1   1  -1   1   1  -1
X.7      1  A -/A /A  -A -1  1   A -/A  /A  -A -1   1   1   1   1   1   1
X.8      1 /A  -A  A -/A -1  1  /A  -A   A -/A -1   1   1   1   1   1   1
X.9      1  A  /A /A   A  1 -1  -A -/A -/A  -A -1  -1   1  -1   1   1  -1
X.10     1 /A   A  A  /A  1 -1 -/A  -A  -A -/A -1  -1   1  -1   1   1  -1
X.11     1  A  /A /A   A  1  1   A  /A  /A   A  1   1   1   1   1   1   1
X.12     1 /A   A  A  /A  1  1  /A   A   A  /A  1   1   1   1   1   1   1
X.13     6  .   .  .   .  .  6   .   .   .   .  .   B   B   C   C   D   D
X.14     6  .   .  .   .  .  6   .   .   .   .  .   C   C   D   D   B   B
X.15     6  .   .  .   .  .  6   .   .   .   .  .   D   D   B   B   C   C
X.16     6  .   .  .   .  . -6   .   .   .   .  .  -B   B  -C   C   D  -D
X.17     6  .   .  .   .  . -6   .   .   .   .  .  -C   C  -D   D   B  -B
X.18     6  .   .  .   .  . -6   .   .   .   .  .  -D   D  -B   B   C  -C

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(19)^2+E(19)^3+E(19)^5+E(19)^14+E(19)^16+E(19)^17
C = E(19)^4+E(19)^6+E(19)^9+E(19)^10+E(19)^13+E(19)^15
D = E(19)+E(19)^7+E(19)^8+E(19)^11+E(19)^12+E(19)^18