Properties

Label 27T42
Order \(162\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_9:C_3:S_3$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $42$
Group :  $C_9:C_3:S_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,3,10,2,11)(4,18,6,16,5,17)(7,14,9,15,8,13)(19,23)(20,22)(21,24)(25,26), (1,26,3,27,2,25)(4,23,6,24,5,22)(7,21,9,19,8,20)(11,12)(13,17)(14,16)(15,18), (1,16,19,2,17,20,3,18,21)(4,10,23,5,11,24,6,12,22)(7,15,27,8,13,25,9,14,26)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$ x 4
18:  $C_3^2:C_2$
54:  $(C_3^2:C_3):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 4

Degree 9: $C_3^2:C_2$

Low degree siblings

27T68

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

Group invariants

Order:  $162=2 \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [162, 20]
Character table:   
      2  1  1  1   1   1  1  .  .  .  .  .  .  .
      3  4  3  3   1   1  1  4  3  3  3  2  2  2

        1a 3a 3b  6a  6b 2a 3c 9a 9b 9c 3d 9d 9e
     2P 1a 3b 3a  3a  3b 1a 3c 9c 9a 9b 3d 9d 9e
     3P 1a 1a 1a  2a  2a 2a 1a 3c 3c 3c 1a 3c 3c
     5P 1a 3b 3a  6b  6a 2a 3c 9b 9c 9a 3d 9d 9e
     7P 1a 3a 3b  6a  6b 2a 3c 9c 9a 9b 3d 9d 9e

X.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
X.3      2  2  2   .   .  .  2  2  2  2 -1 -1 -1
X.4      2  2  2   .   .  .  2 -1 -1 -1  2 -1 -1
X.5      2  2  2   .   .  .  2 -1 -1 -1 -1 -1  2
X.6      2  2  2   .   .  .  2 -1 -1 -1 -1  2 -1
X.7      3  A /A   B  /B -1  3  .  .  .  .  .  .
X.8      3 /A  A  /B   B -1  3  .  .  .  .  .  .
X.9      3  A /A  -B -/B  1  3  .  .  .  .  .  .
X.10     3 /A  A -/B  -B  1  3  .  .  .  .  .  .
X.11     6  .  .   .   .  . -3  C  E  D  .  .  .
X.12     6  .  .   .   .  . -3  D  C  E  .  .  .
X.13     6  .  .   .   .  . -3  E  D  C  .  .  .

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