Properties

Label 27T3
Order \(27\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $He_3$

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Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$ x 4

Degree 9: $C_3^2$, $C_3^2:C_3$ x 4

Low degree siblings

9T7 x 4

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, 3 $ $3$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7,12,14)( 8,10,15)( 9,11,13)(16,22,19)(17,23,20) (18,24,21)(25,26,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7,14,12)( 8,15,10)( 9,13,11)(16,19,22)(17,20,23) (18,21,24)(25,27,26)$
$ 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 $ $3$ $3$ $( 1, 7,16)( 2, 8,17)( 3, 9,18)( 4,10,19)( 5,11,20)( 6,12,21)(13,22,25) (14,23,26)(15,24,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1, 8,19)( 2, 9,20)( 3, 7,21)( 4,11,22)( 5,12,23)( 6,10,24)(13,18,27) (14,16,25)(15,17,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1, 9,22)( 2, 7,23)( 3, 8,24)( 4,12,16)( 5,10,17)( 6,11,18)(13,20,26) (14,21,27)(15,19,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1,16, 7)( 2,17, 8)( 3,18, 9)( 4,19,10)( 5,20,11)( 6,21,12)(13,25,22) (14,26,23)(15,27,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1,17,14)( 2,18,15)( 3,16,13)( 4,20, 8)( 5,21, 9)( 6,19, 7)(10,27,22) (11,25,23)(12,26,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1,18,12)( 2,16,10)( 3,17,11)( 4,21,15)( 5,19,13)( 6,20,14)( 7,26,22) ( 8,27,23)( 9,25,24)$
$ 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:  $27=3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [27, 3]
Character table:   
      3  3  2  2  3  2  2  2  2  2  2  3

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

X.1      1  1  1  1  1  1  1  1  1  1  1
X.2      1  1  1  1  A  A  A /A /A /A  1
X.3      1  1  1  1 /A /A /A  A  A  A  1
X.4      1  A /A  1  1  A /A  1  A /A  1
X.5      1 /A  A  1  1 /A  A  1 /A  A  1
X.6      1  A /A  1  A /A  1 /A  1  A  1
X.7      1 /A  A  1 /A  A  1  A  1 /A  1
X.8      1  A /A  1 /A  1  A  A /A  1  1
X.9      1 /A  A  1  A  1 /A /A  A  1  1
X.10     3  .  .  B  .  .  .  .  .  . /B
X.11     3  .  . /B  .  .  .  .  .  .  B

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3