Properties

Label 27T9
Order \(54\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times D_9$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $9$
Group :  $C_3\times D_9$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,22,17,11,6,25,20,13,7)(2,23,18,12,4,26,21,14,8)(3,24,16,10,5,27,19,15,9), (1,2,3)(4,27,6,26,5,25)(7,23,9,22,8,24)(10,20,12,19,11,21)(13,18,15,17,14,16)
$|\Aut(F/K)|$:  $3$

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 3: $C_3$, $S_3$

Degree 9: $D_{9}$, $S_3\times C_3$

Low degree siblings

18T19

Siblings are shown 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, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $9$ $2$ $( 4,26)( 5,27)( 6,25)( 7,22)( 8,23)( 9,24)(10,19)(11,20)(12,21)(13,17)(14,18) (15,16)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$
$ 6, 6, 6, 6, 3 $ $9$ $6$ $( 1, 2, 3)( 4,27, 6,26, 5,25)( 7,23, 9,22, 8,24)(10,20,12,19,11,21) (13,18,15,17,14,16)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)$
$ 6, 6, 6, 6, 3 $ $9$ $6$ $( 1, 3, 2)( 4,25, 5,26, 6,27)( 7,24, 8,22, 9,23)(10,21,11,19,12,20) (13,16,14,17,15,18)$
$ 9, 9, 9 $ $2$ $9$ $( 1, 4, 9,11,14,16,20,23,27)( 2, 5, 7,12,15,17,21,24,25)( 3, 6, 8,10,13,18,19, 22,26)$
$ 9, 9, 9 $ $2$ $9$ $( 1, 5, 8,11,15,18,20,24,26)( 2, 6, 9,12,13,16,21,22,27)( 3, 4, 7,10,14,17,19, 23,25)$
$ 9, 9, 9 $ $2$ $9$ $( 1, 6, 7,11,13,17,20,22,25)( 2, 4, 8,12,14,18,21,23,26)( 3, 5, 9,10,15,16,19, 24,27)$
$ 9, 9, 9 $ $2$ $9$ $( 1, 7,13,20,25, 6,11,17,22)( 2, 8,14,21,26, 4,12,18,23)( 3, 9,15,19,27, 5,10, 16,24)$
$ 9, 9, 9 $ $2$ $9$ $( 1, 8,15,20,26, 5,11,18,24)( 2, 9,13,21,27, 6,12,16,22)( 3, 7,14,19,25, 4,10, 17,23)$
$ 9, 9, 9 $ $2$ $9$ $( 1, 9,14,20,27, 4,11,16,23)( 2, 7,15,21,25, 5,12,17,24)( 3, 8,13,19,26, 6,10, 18,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,10,21)( 2,11,19)( 3,12,20)( 4,13,24)( 5,14,22)( 6,15,23)( 7,16,26) ( 8,17,27)( 9,18,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,11,20)( 2,12,21)( 3,10,19)( 4,14,23)( 5,15,24)( 6,13,22)( 7,17,25) ( 8,18,26)( 9,16,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,12,19)( 2,10,20)( 3,11,21)( 4,15,22)( 5,13,23)( 6,14,24)( 7,18,27) ( 8,16,25)( 9,17,26)$
$ 9, 9, 9 $ $2$ $9$ $( 1,13,25,11,22, 7,20, 6,17)( 2,14,26,12,23, 8,21, 4,18)( 3,15,27,10,24, 9,19, 5,16)$
$ 9, 9, 9 $ $2$ $9$ $( 1,14,27,11,23, 9,20, 4,16)( 2,15,25,12,24, 7,21, 5,17)( 3,13,26,10,22, 8,19, 6,18)$
$ 9, 9, 9 $ $2$ $9$ $( 1,15,26,11,24, 8,20, 5,18)( 2,13,27,12,22, 9,21, 6,16)( 3,14,25,10,23, 7,19, 4,17)$

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  1  3   1  3   1   3   3  3  3   3   3   3  3   3  3   3   3

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

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3
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)^5+E(9)^7
G = -E(9)^2+E(9)^4-E(9)^5
H = E(9)^2-E(9)^4-E(9)^7