# Properties

 Label 19T5 Degree $19$ Order $171$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_{19}:C_{9}$

# Related objects

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$9$:  $C_9$

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

## Group invariants

 Order: $171=3^{2} \cdot 19$ Cyclic: no Abelian: no Solvable: yes GAP id: [171, 3]
 Character table:  3 2 2 2 2 2 2 2 2 2 . . 19 1 . . . . . . . . 1 1 1a 9a 9b 9c 3a 9d 3b 9e 9f 19a 19b 2P 1a 9e 9c 9f 3b 9b 3a 9d 9a 19b 19a 3P 1a 3a 3b 3a 1a 3a 1a 3b 3b 19b 19a 5P 1a 9f 9d 9b 3b 9e 3a 9a 9c 19a 19b 7P 1a 9c 9e 9d 3a 9a 3b 9f 9b 19a 19b 11P 1a 9e 9c 9f 3b 9b 3a 9d 9a 19a 19b 13P 1a 9d 9f 9a 3a 9c 3b 9b 9e 19b 19a 17P 1a 9b 9a 9e 3b 9f 3a 9c 9d 19a 19b 19P 1a 9a 9b 9c 3a 9d 3b 9e 9f 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 A /A A 1 A 1 /A /A 1 1 X.3 1 /A A /A 1 /A 1 A A 1 1 X.4 1 B /B /C A D /A C /D 1 1 X.5 1 /B B C /A /D A /C D 1 1 X.6 1 C /C /D /A /B A D B 1 1 X.7 1 D /D B A /C /A /B C 1 1 X.8 1 /D D /B /A C A B /C 1 1 X.9 1 /C C D A B /A /D /B 1 1 X.10 9 . . . . . . . . E /E X.11 9 . . . . . . . . /E E A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = -E(9)^2-E(9)^5 C = E(9)^7 D = E(9)^5 E = E(19)+E(19)^4+E(19)^5+E(19)^6+E(19)^7+E(19)^9+E(19)^11+E(19)^16+E(19)^17 = (-1+Sqrt(-19))/2 = b19