Properties

Label 27T27
Order \(81\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_3\wr C_3$

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Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 9: $C_3^2:C_3$, $C_3 \wr C_3 $ x 2

Low degree siblings

9T17 x 3, 27T19, 27T21, 27T27 x 2

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

Group invariants

Order:  $81=3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [81, 7]
Character table:   
      3  4   3   3   3  3  3   3  4   3   3  2  2  2  2  2  2  4

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

X.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  A  A  A /A /A /A  1
X.3      1   1   1   1  1  1   1  1   1   1 /A /A /A  A  A  A  1
X.4      1   A  /A  /A  1  1   A  1   A  /A  1  A /A  1  A /A  1
X.5      1  /A   A   A  1  1  /A  1  /A   A  1 /A  A  1 /A  A  1
X.6      1   A  /A  /A  1  1   A  1   A  /A  A /A  1 /A  1  A  1
X.7      1  /A   A   A  1  1  /A  1  /A   A /A  A  1  A  1 /A  1
X.8      1   A  /A  /A  1  1   A  1   A  /A /A  1  A  A /A  1  1
X.9      1  /A   A   A  1  1  /A  1  /A   A  A  1 /A /A  A  1  1
X.10     3   B  /B  -C  .  .   C /D -/B  -B  .  .  .  .  .  .  D
X.11     3  /B   B   C  .  .  -C  D  -B -/B  .  .  .  .  .  . /D
X.12     3   C  -C  -B  .  . -/B /D   B  /B  .  .  .  .  .  .  D
X.13     3  -C   C -/B  .  .  -B  D  /B   B  .  .  .  .  .  . /D
X.14     3  -B -/B   B  .  .  /B  D  -C   C  .  .  .  .  .  . /D
X.15     3 -/B  -B  /B  .  .   B /D   C  -C  .  .  .  .  .  .  D
X.16     3   .   .   .  D /D   .  3   .   .  .  .  .  .  .  .  3
X.17     3   .   .   . /D  D   .  3   .   .  .  .  .  .  .  .  3

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