Properties

Label 27T19
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 $t$ :  $19$
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)(16,17,18)(19,20,21)(22,23,24), (1,10,21)(2,11,19)(3,12,20)(4,13,24)(5,14,22)(6,15,23)(7,18,25)(8,16,26)(9,17,27)
$|\Aut(F/K)|$:  $9$

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: $C_3$ x 4

Degree 9: $C_3^2$, $C_3 \wr C_3 $ x 3

Low degree siblings

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

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

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   3   3  4  2  2  2  2  2  2  4

        1a 3a 3b  3c  3d  3e  3f  3g  3h 3i 9a 3j 9b 3k 9c 9d 3l
     2P 1a 3b 3a  3e  3f  3c  3d  3h  3g 3l 9c 3k 9d 3j 9a 9b 3i
     3P 1a 1a 1a  1a  1a  1a  1a  1a  1a 1a 3l 1a 3i 1a 3i 3l 1a
     5P 1a 3b 3a  3e  3f  3c  3d  3h  3g 3l 9c 3k 9d 3j 9a 9b 3i
     7P 1a 3a 3b  3c  3d  3e  3f  3g  3h 3i 9a 3j 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  B  B  B /B /B /B  1
X.3      1  1  1   1   1   1   1   1   1  1 /B /B /B  B  B  B  1
X.4      1  1  1   B   B  /B  /B   B  /B  1  1  B /B /B  1  B  1
X.5      1  1  1  /B  /B   B   B  /B   B  1  1 /B  B  B  1 /B  1
X.6      1  1  1   B   B  /B  /B   B  /B  1  B /B  1  B /B  1  1
X.7      1  1  1  /B  /B   B   B  /B   B  1 /B  B  1 /B  B  1  1
X.8      1  1  1   B   B  /B  /B   B  /B  1 /B  1  B  1  B /B  1
X.9      1  1  1  /B  /B   B   B  /B   B  1  B  1 /B  1 /B  B  1
X.10     3  A /A   .   .   .   .   .   .  3  .  .  .  .  .  .  3
X.11     3 /A  A   .   .   .   .   .   .  3  .  .  .  .  .  .  3
X.12     3  .  .   C   D  /C  -D -/C  -C /A  .  .  .  .  .  .  A
X.13     3  .  .  /C  -D   C   D  -C -/C  A  .  .  .  .  .  . /A
X.14     3  .  . -/C   C  -C  /C   D  -D /A  .  .  .  .  .  .  A
X.15     3  .  .  -C  /C -/C   C  -D   D  A  .  .  .  .  .  . /A
X.16     3  .  .   D -/C  -D  -C   C  /C /A  .  .  .  .  .  .  A
X.17     3  .  .  -D  -C   D -/C  /C   C  A  .  .  .  .  .  . /A

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