Properties

Label 27T20
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 $t$ :  $20$
Group :  $He_3.C_3$
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), (1,12,20,3,11,19,2,10,21)(4,13,23,6,15,22,5,14,24)(7,16,26,9,18,25,8,17,27)
$|\Aut(F/K)|$:  $3$

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

27T26

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 $ $9$ $9$ $( 1, 7,17, 2, 8,18, 3, 9,16)( 4,10,21, 5,11,19, 6,12,20)(13,22,26,14,23,27,15, 24,25)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 7,24, 3, 9,23, 2, 8,22)( 4,11,18, 6,10,17, 5,12,16)(13,19,25,15,21,27,14, 20,26)$
$ 9, 9, 9 $ $9$ $9$ $( 1, 7,21, 2, 8,19, 3, 9,20)( 4,12,23, 5,10,24, 6,11,22)(13,18,25,14,16,26,15, 17,27)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 8,23, 3, 7,22, 2, 9,24)( 4,12,17, 6,11,16, 5,10,18)(13,20,27,15,19,26,14, 21,25)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 9,22, 3, 8,24, 2, 7,23)( 4,10,16, 6,12,18, 5,11,17)(13,21,26,15,20,25,14, 19,27)$
$ 9, 9, 9 $ $3$ $9$ $( 1,16,13, 2,17,14, 3,18,15)( 4,19, 8, 5,20, 9, 6,21, 7)(10,27,24,11,25,22,12, 26,23)$
$ 9, 9, 9 $ $9$ $9$ $( 1,16, 9, 3,18, 8, 2,17, 7)( 4,20,12, 6,19,11, 5,21,10)(13,25,24,15,27,23,14, 26,22)$
$ 9, 9, 9 $ $9$ $9$ $( 1,16,12, 3,18,11, 2,17,10)( 4,21,14, 6,20,13, 5,19,15)( 7,27,22, 9,26,24, 8, 25,23)$
$ 9, 9, 9 $ $3$ $9$ $( 1,17,15, 2,18,13, 3,16,14)( 4,20, 7, 5,21, 8, 6,19, 9)(10,25,23,11,26,24,12, 27,22)$
$ 9, 9, 9 $ $3$ $9$ $( 1,18,14, 2,16,15, 3,17,13)( 4,21, 9, 5,19, 7, 6,20, 8)(10,26,22,11,27,23,12, 25,24)$
$ 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, 8]
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 9b 9c 9d 9e 9f 9g 9h 9i 9j 3f
     2P 1a 3b 3a 3d 3c 3f 9g 9j 9h 9i 9f 9d 9a 9c 9b 9e 3e
     3P 1a 1a 1a 1a 1a 1a 3c 3d 3c 3d 3d 3c 3d 3d 3c 3c 1a
     5P 1a 3b 3a 3d 3c 3f 9g 9i 9h 9f 9j 9e 9a 9c 9d 9b 3e
     7P 1a 3a 3b 3c 3d 3e 9a 9d 9c 9e 9b 9j 9g 9h 9f 9i 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  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  E /C  .  . /E /D  .
X.13     3  .  .  B /B  .  .  D  .  E  C /D  .  . /C /E  .
X.14     3  .  .  B /B  .  .  E  .  C  D /E  .  . /D /C  .
X.15     3  .  . /B  B  .  . /D  . /E /C  D  .  .  C  E  .
X.16     3  .  . /B  B  .  . /C  . /D /E  C  .  .  E  D  .
X.17     3  .  . /B  B  .  . /E  . /C /D  E  .  .  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(9)^4-2*E(9)^7
D = -E(9)^4+E(9)^7
E = 2*E(9)^4+E(9)^7