Properties

Label 19T2
Order \(38\)
n \(19\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $D_{19}$

Related objects

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Group action invariants

Degree $n$ :  $19$
Transitive number $t$ :  $2$
Group :  $D_{19}$
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), (2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$

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

Group invariants

Order:  $38=2 \cdot 19$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [38, 1]
Character table:   
      2  1  1   .   .   .   .   .   .   .   .   .
     19  1  .   1   1   1   1   1   1   1   1   1

        1a 2a 19a 19b 19c 19d 19e 19f 19g 19h 19i
     2P 1a 1a 19b 19d 19f 19h 19i 19g 19e 19c 19a
     3P 1a 2a 19c 19f 19i 19g 19d 19a 19b 19e 19h
     5P 1a 2a 19e 19i 19d 19a 19f 19h 19c 19b 19g
     7P 1a 2a 19g 19e 19b 19i 19c 19d 19h 19a 19f
    11P 1a 2a 19h 19c 19e 19f 19b 19i 19a 19g 19d
    13P 1a 2a 19f 19g 19a 19e 19h 19b 19d 19i 19c
    17P 1a 2a 19b 19d 19f 19h 19i 19g 19e 19c 19a
    19P 1a 2a  1a  1a  1a  1a  1a  1a  1a  1a  1a

X.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
X.3      2  .   A   C   G   D   F   B   E   H   I
X.4      2  .   B   E   A   F   H   C   D   I   G
X.5      2  .   C   D   B   H   I   E   F   G   A
X.6      2  .   D   H   E   G   A   F   I   B   C
X.7      2  .   E   F   C   I   G   D   H   A   B
X.8      2  .   F   I   D   A   B   H   G   C   E
X.9      2  .   G   B   I   E   D   A   C   F   H
X.10     2  .   H   G   F   B   C   I   A   E   D
X.11     2  .   I   A   H   C   E   G   B   D   F

A = E(19)^6+E(19)^13
B = E(19)^2+E(19)^17
C = E(19)^7+E(19)^12
D = E(19)^5+E(19)^14
E = E(19)^4+E(19)^15
F = E(19)^8+E(19)^11
G = E(19)+E(19)^18
H = E(19)^9+E(19)^10
I = E(19)^3+E(19)^16