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Properties

Label 27T26
Order \(81\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $He_3.C_3$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $26$
Group :  $He_3.C_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,4,7,2,5,8,3,6,9)(10,13,18,11,14,16,12,15,17)(19,22,26,20,23,27,21,24,25), (1,12,25)(2,10,26)(3,11,27)(4,13,20)(5,14,21)(6,15,19)(7,16,24)(8,17,22)(9,18,23)
$|\Aut(F/K)|$:  $9$

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: $C_3^2$

Low degree siblings

27T20

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

Group invariants

Order:  $81=3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [81, 8]
Character table:    
      3  4  3  3  4  4  3  3  3  3  3  3  2  2  2  2  2  2

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

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

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