# Properties

 Label 19T6 Degree $19$ Order $342$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $F_{19}$

# Related objects

## Group action invariants

 Degree $n$: $19$ Transitive number $t$: $6$ Group: $F_{19}$ 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), (2,3,5,9,17,14,8,15,10,19,18,16,12,4,7,13,6,11)

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

## 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