Properties

Label 18T19
Order \(54\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times D_9$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $19$
Group :  $C_3\times D_9$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5,7,2,6,8,3,4,9)(10,14,18,12,13,17,11,15,16), (1,15,2,13,3,14)(4,12,5,10,6,11)(7,18,8,16,9,17)
$|\Aut(F/K)|$:  $9$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$, $D_{9}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

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

Group invariants

Order:  $54=2 \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [54, 3]
Character table:   
      2  1   .   .  1  .  1  .   .   .   .   .  .   .  .   .  1   1   1
      3  3   3   3  3  3  3  3   3   3   3   3  3   3  3   3  1   1   1

        1a  3a  3b 3c 3d 3e 9a  9b  9c  9d  9e 9f  9g 9h  9i 2a  6a  6b
     2P 1a  3b  3a 3e 3d 3c 9h  9e  9b  9g  9d 9a  9i 9f  9c 1a  3c  3e
     3P 1a  1a  1a 1a 1a 1a 3d  3d  3d  3d  3d 3d  3d 3d  3d 2a  2a  2a
     5P 1a  3b  3a 3e 3d 3c 9f  9c  9i  9e  9b 9h  9d 9a  9g 2a  6b  6a
     7P 1a  3a  3b 3c 3d 3e 9h  9i  9g  9b  9c 9a  9e 9f  9d 2a  6a  6b

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  1  A  1   A  /A   A  /A  1  /A  1   A -1  -A -/A
X.4      1  /A   A  A  1 /A  1  /A   A  /A   A  1   A  1  /A -1 -/A  -A
X.5      1   A  /A /A  1  A  1   A  /A   A  /A  1  /A  1   A  1   A  /A
X.6      1  /A   A  A  1 /A  1  /A   A  /A   A  1   A  1  /A  1  /A   A
X.7      2   2   2  2  2  2 -1  -1  -1  -1  -1 -1  -1 -1  -1  .   .   .
X.8      2   B  /B /B  2  B -1 -/A  -A -/A  -A -1  -A -1 -/A  .   .   .
X.9      2  /B   B  B  2 /B -1  -A -/A  -A -/A -1 -/A -1  -A  .   .   .
X.10     2  -1  -1  2 -1  2  C   E   D   D   C  E   E  D   C  .   .   .
X.11     2  -1  -1  2 -1  2  D   C   E   E   D  C   C  E   D  .   .   .
X.12     2  -1  -1  2 -1  2  E   D   C   C   E  D   D  C   E  .   .   .
X.13     2 -/A  -A /B -1  B  C   F  /H   H  /G  E  /F  D   G  .   .   .
X.14     2 -/A  -A /B -1  B  D   G  /F   F  /H  C  /G  E   H  .   .   .
X.15     2 -/A  -A /B -1  B  E   H  /G   G  /F  D  /H  C   F  .   .   .
X.16     2  -A -/A  B -1 /B  C  /F   H  /H   G  E   F  D  /G  .   .   .
X.17     2  -A -/A  B -1 /B  D  /G   F  /F   H  C   G  E  /H  .   .   .
X.18     2  -A -/A  B -1 /B  E  /H   G  /G   F  D   H  C  /F  .   .   .

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(3)
  = -1+Sqrt(-3) = 2b3
C = E(9)^2+E(9)^7
D = E(9)^4+E(9)^5
E = -E(9)^2-E(9)^4-E(9)^5-E(9)^7
F = E(9)^2+E(9)^4
G = -E(9)^4+E(9)^5-E(9)^7
H = -E(9)^2-E(9)^5+E(9)^7