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Properties

Label 27T5
Order \(27\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_9:C_3$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $5$
Group :  $C_9:C_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,4,8,11,14,18,21,24,25)(2,5,9,12,15,16,19,22,26)(3,6,7,10,13,17,20,23,27), (1,12,20)(2,10,21)(3,11,19)(4,22,13)(5,23,14)(6,24,15)(7,8,9)(16,17,18)(25,26,27)
$|\Aut(F/K)|$:  $27$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$ x 4
9:  $C_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$ x 4

Degree 9: $C_3^2$, $C_9:C_3$

Low degree siblings

9T6

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

Group invariants

Order:  $27=3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [27, 4]
Character table:    
      3  3  2  2  2  2  2  2  2  2  3  3

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

X.1      1  1  1  1  1  1  1  1  1  1  1
X.2      1  1  1  A  A  A /A /A /A  1  1
X.3      1  1  1 /A /A /A  A  A  A  1  1
X.4      1  A /A  1  A /A /A  1  A  1  1
X.5      1 /A  A  1 /A  A  A  1 /A  1  1
X.6      1  A /A  A /A  1  A /A  1  1  1
X.7      1 /A  A /A  A  1 /A  A  1  1  1
X.8      1  A /A /A  1  A  1  A /A  1  1
X.9      1 /A  A  A  1 /A  1 /A  A  1  1
X.10     3  .  .  .  .  .  .  .  .  B /B
X.11     3  .  .  .  .  .  .  .  . /B  B

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