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Properties

Label 27T21
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 $t$ :  $21$
Group :  $C_3\wr C_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (4,5,6)(7,15,10)(8,13,11)(9,14,12)(16,20,22)(17,21,23)(18,19,24)(25,27,26), (1,11,19)(2,12,20)(3,10,21)(4,15,22)(5,13,23)(6,14,24)(7,16,25)(8,17,26)(9,18,27)
$|\Aut(F/K)|$:  $3$

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: $C_3^2:C_3$

Low degree siblings

9T17 x 3, 27T19, 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, 3, 3, 1, 1, 1 $ $9$ $3$ $( 4, 5, 6)( 7,15,10)( 8,13,11)( 9,14,12)(16,20,22)(17,21,23)(18,19,24) (25,27,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $9$ $3$ $( 4, 6, 5)( 7,10,15)( 8,11,13)( 9,12,14)(16,22,20)(17,23,21)(18,24,19) (25,26,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1, 4,25)( 2, 5,26)( 3, 6,27)( 7,11,15)( 8,12,13)( 9,10,14)(16,19,22) (17,20,23)(18,21,24)$
$ 9, 9, 9 $ $9$ $9$ $( 1, 7,18, 3, 9,17, 2, 8,16)( 4,10,19, 6,12,21, 5,11,20)(13,23,27,15,22,26,14, 24,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1, 7,22)( 2, 8,23)( 3, 9,24)( 4,11,16)( 5,12,17)( 6,10,18)(13,20,26) (14,21,27)(15,19,25)$
$ 9, 9, 9 $ $9$ $9$ $( 1, 7,19, 3, 9,21, 2, 8,20)( 4,12,24, 6,11,23, 5,10,22)(13,16,26,15,18,25,14, 17,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1, 8,24)( 2, 9,22)( 3, 7,23)( 4,12,18)( 5,10,16)( 6,11,17)(13,21,25) (14,19,26)(15,20,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1, 9,23)( 2, 7,24)( 3, 8,22)( 4,10,17)( 5,11,18)( 6,12,16)(13,19,27) (14,20,25)(15,21,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1,16,15)( 2,17,13)( 3,18,14)( 4,19, 7)( 5,20, 8)( 6,21, 9)(10,27,24) (11,25,22)(12,26,23)$
$ 9, 9, 9 $ $9$ $9$ $( 1,16, 8, 2,17, 9, 3,18, 7)( 4,20,11, 5,21,12, 6,19,10)(13,25,24,14,26,22,15, 27,23)$
$ 9, 9, 9 $ $9$ $9$ $( 1,16,11, 2,17,12, 3,18,10)( 4,21,13, 5,19,14, 6,20,15)( 7,27,22, 8,25,23, 9, 26,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1,17,14)( 2,18,15)( 3,16,13)( 4,20, 9)( 5,21, 7)( 6,19, 8)(10,25,23) (11,26,24)(12,27,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1,18,13)( 2,16,14)( 3,17,15)( 4,21, 8)( 5,19, 9)( 6,20, 7)(10,26,22) (11,27,23)(12,25,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1,25, 4)( 2,26, 5)( 3,27, 6)( 7,15,11)( 8,13,12)( 9,14,10)(16,22,19) (17,23,20)(18,24,21)$

Group invariants

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

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

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