Properties

Label 27T13
Order \(54\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times C_3:S_3$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $13$
Group :  $C_3\times C_3:S_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (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), (1,27,6)(2,25,4)(3,26,5)(7,14,11)(8,15,12)(9,13,10)(16,22,20)(17,23,21)(18,24,19), (1,10,19)(2,12,20,3,11,21)(4,8,22,26,14,17)(5,7,23,25,15,16)(6,9,24,27,13,18)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$ x 4, $C_6$
18:  $S_3\times C_3$ x 4, $C_3^2:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$, $S_3$ x 4

Degree 9: $S_3\times C_3$ x 4, $C_3^2:C_2$

Low degree siblings

18T23 x 4

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

Group invariants

Order:  $54=2 \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [54, 13]
Character table:   
      2  1  1  .  .  .  .   .   1   .   .  1   .   .   1   .   .  1   .
      3  3  1  3  3  3  3   3   1   3   3  3   3   3   1   3   3  3   3

        1a 2a 3a 3b 3c 3d  3e  6a  3f  3g 3h  3i  3j  6b  3k  3l 3m  3n
     2P 1a 1a 3a 3b 3c 3d  3j  3m  3k  3l 3m  3n  3e  3h  3f  3g 3h  3i
     3P 1a 2a 1a 1a 1a 1a  1a  2a  1a  1a 1a  1a  1a  2a  1a  1a 1a  1a
     5P 1a 2a 3a 3b 3c 3d  3j  6b  3k  3l 3m  3n  3e  6a  3f  3g 3h  3i

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  1  1  1  1   A  -A   A   A  A   A  /A -/A  /A  /A /A  /A
X.4      1 -1  1  1  1  1  /A -/A  /A  /A /A  /A   A  -A   A   A  A   A
X.5      1  1  1  1  1  1   A   A   A   A  A   A  /A  /A  /A  /A /A  /A
X.6      1  1  1  1  1  1  /A  /A  /A  /A /A  /A   A   A   A   A  A   A
X.7      2  .  2 -1 -1 -1  -1   .  -1  -1  2   2  -1   .  -1  -1  2   2
X.8      2  . -1  2 -1 -1  -1   .   2  -1  2  -1  -1   .   2  -1  2  -1
X.9      2  . -1 -1 -1  2  -1   .  -1   2  2  -1  -1   .  -1   2  2  -1
X.10     2  . -1 -1  2 -1   2   .  -1  -1  2  -1   2   .  -1  -1  2  -1
X.11     2  .  2 -1 -1 -1  -A   .  -A  -A  B   B -/A   . -/A -/A /B  /B
X.12     2  .  2 -1 -1 -1 -/A   . -/A -/A /B  /B  -A   .  -A  -A  B   B
X.13     2  . -1  2 -1 -1  -A   .   B  -A  B  -A -/A   .  /B -/A /B -/A
X.14     2  . -1  2 -1 -1 -/A   .  /B -/A /B -/A  -A   .   B  -A  B  -A
X.15     2  . -1 -1 -1  2 -/A   . -/A  /B /B -/A  -A   .  -A   B  B  -A
X.16     2  . -1 -1 -1  2  -A   .  -A   B  B  -A -/A   . -/A  /B /B -/A
X.17     2  . -1 -1  2 -1   B   .  -A  -A  B  -A  /B   . -/A -/A /B -/A
X.18     2  . -1 -1  2 -1  /B   . -/A -/A /B -/A   B   .  -A  -A  B  -A

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3