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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

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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 TypeSizeOrderRepresentative
$ 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