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Properties

Label 19T4
Degree $19$
Order $114$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_{19}:C_{6}$

Related objects

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Group action invariants

Degree $n$:  $19$
Transitive number $t$:  $4$
Group:  $C_{19}:C_{6}$
Parity:  $-1$
Primitive:  yes
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $1$
Generators:  (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19), (2,9,8,19,12,13)(3,17,15,18,4,6)(5,14,10,16,7,11)

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

Prime degree - none

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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 1 $ $19$ $3$ $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)$
$ 6, 6, 6, 1 $ $19$ $6$ $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)$
$ 3, 3, 3, 3, 3, 3, 1 $ $19$ $3$ $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)$
$ 6, 6, 6, 1 $ $19$ $6$ $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $19$ $2$ $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)$
$ 19 $ $6$ $19$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)$
$ 19 $ $6$ $19$ $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)$
$ 19 $ $6$ $19$ $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)$

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