Properties

Label 19T6
Order \(342\)
n \(19\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $F_{19}$

Related objects

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Group action invariants

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

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

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

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   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .  .

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

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

      2   .
      3   .
     19   1

        19a
     2P 19a
     3P 19a
     5P 19a
     7P 19a
    11P 19a
    13P 19a
    17P 19a
    19P  1a

X.1       1
X.2       1
X.3       1
X.4       1
X.5       1
X.6       1
X.7       1
X.8       1
X.9       1
X.10      1
X.11      1
X.12      1
X.13      1
X.14      1
X.15      1
X.16      1
X.17      1
X.18      1
X.19     -1

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