Properties

Label 38T5
Order \(114\)
n \(38\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{19}:C_3$

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Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

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

Low degree siblings

19T4

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

Group invariants

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

       1a 3a 3b  6a  6b 2a 19a 19b 19c
    2P 1a 3b 3a  3a  3b 1a 19b 19c 19a
    3P 1a 1a 1a  2a  2a 2a 19b 19c 19a
    5P 1a 3b 3a  6b  6a 2a 19b 19c 19a
    7P 1a 3a 3b  6a  6b 2a 19a 19b 19c
   11P 1a 3b 3a  6b  6a 2a 19a 19b 19c
   13P 1a 3a 3b  6a  6b 2a 19c 19a 19b
   17P 1a 3b 3a  6b  6a 2a 19b 19c 19a
   19P 1a 3a 3b  6a  6b 2a  1a  1a  1a

X.1     1  1  1   1   1  1   1   1   1
X.2     1  1  1  -1  -1 -1   1   1   1
X.3     1  A /A -/A  -A -1   1   1   1
X.4     1 /A  A  -A -/A -1   1   1   1
X.5     1  A /A  /A   A  1   1   1   1
X.6     1 /A  A   A  /A  1   1   1   1
X.7     6  .  .   .   .  .   B   C   D
X.8     6  .  .   .   .  .   C   D   B
X.9     6  .  .   .   .  .   D   B   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