Properties

Label 27T44
Order \(162\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $He_3:C_3:C_2$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $44$
Group :  $He_3:C_3:C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,24,2,23,3,22)(4,20,5,19,6,21)(7,25,8,27,9,26)(10,18)(11,17)(12,16)(14,15), (1,10,3,11,2,12)(4,16,6,17,5,18)(7,15,9,13,8,14)(19,23)(20,22)(21,24)(25,26), (1,14)(2,13)(3,15)(4,10)(5,12)(6,11)(7,18)(8,17)(9,16)(19,27)(20,26)(21,25)(23,24)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$ x 4
18:  $C_3^2:C_2$
54:  $(C_3^2:C_3):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 4

Degree 9: $C_3^2:C_2$

Low degree siblings

27T66 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 $ $3$ $3$ $(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $3$ $(10,12,11)(13,15,14)(16,18,17)(19,20,21)(22,23,24)(25,26,27)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $27$ $2$ $( 2, 3)( 4, 8)( 5, 7)( 6, 9)(10,25)(11,27)(12,26)(13,23)(14,22)(15,24)(16,19) (17,21)(18,20)$
$ 6, 6, 6, 2, 2, 2, 2, 1 $ $27$ $6$ $( 2, 3)( 4, 8)( 5, 7)( 6, 9)(10,26,11,25,12,27)(13,24,14,23,15,22) (16,20,17,19,18,21)$
$ 6, 6, 6, 2, 2, 2, 2, 1 $ $27$ $6$ $( 2, 3)( 4, 8)( 5, 7)( 6, 9)(10,27,12,25,11,26)(13,22,15,23,14,24) (16,21,18,19,17,20)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$
$ 9, 9, 9 $ $6$ $9$ $( 1, 4, 9, 2, 5, 7, 3, 6, 8)(10,13,18,11,14,16,12,15,17)(19,23,26,20,24,27,21, 22,25)$
$ 9, 9, 9 $ $6$ $9$ $( 1, 4, 9, 2, 5, 7, 3, 6, 8)(10,15,16,11,13,17,12,14,18)(19,24,25,20,22,26,21, 23,27)$
$ 9, 9, 9 $ $6$ $9$ $( 1, 5, 8, 2, 6, 9, 3, 4, 7)(10,13,18,11,14,16,12,15,17)(19,22,27,20,23,25,21, 24,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $18$ $3$ $( 1,10,25)( 2,11,26)( 3,12,27)( 4,14,19)( 5,15,20)( 6,13,21)( 7,18,24) ( 8,16,22)( 9,17,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $18$ $3$ $( 1,13,22)( 2,14,23)( 3,15,24)( 4,16,25)( 5,17,26)( 6,18,27)( 7,11,20) ( 8,12,21)( 9,10,19)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $18$ $3$ $( 1,16,21)( 2,17,19)( 3,18,20)( 4,10,22)( 5,11,23)( 6,12,24)( 7,15,26) ( 8,13,27)( 9,14,25)$

Group invariants

Order:  $162=2 \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [162, 21]
Character table:   
      2  1  1  1  1   1   1  .  .  .  .  .  .  .
      3  4  3  3  1   1   1  4  3  3  3  2  2  2

        1a 3a 3b 2a  6a  6b 3c 9a 9b 9c 3d 3e 3f
     2P 1a 3b 3a 1a  3a  3b 3c 9c 9a 9b 3d 3e 3f
     3P 1a 1a 1a 2a  2a  2a 1a 3c 3c 3c 1a 1a 1a
     5P 1a 3b 3a 2a  6b  6a 3c 9b 9c 9a 3d 3e 3f
     7P 1a 3a 3b 2a  6a  6b 3c 9c 9a 9b 3d 3e 3f

X.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  1  1
X.3      2  2  2  .   .   .  2  2  2  2 -1 -1 -1
X.4      2  2  2  .   .   .  2 -1 -1 -1  2 -1 -1
X.5      2  2  2  .   .   .  2 -1 -1 -1 -1 -1  2
X.6      2  2  2  .   .   .  2 -1 -1 -1 -1  2 -1
X.7      3  A /A -1   B  /B  3  .  .  .  .  .  .
X.8      3 /A  A -1  /B   B  3  .  .  .  .  .  .
X.9      3  A /A  1  -B -/B  3  .  .  .  .  .  .
X.10     3 /A  A  1 -/B  -B  3  .  .  .  .  .  .
X.11     6  .  .  .   .   . -3  C  E  D  .  .  .
X.12     6  .  .  .   .   . -3  D  C  E  .  .  .
X.13     6  .  .  .   .   . -3  E  D  C  .  .  .

A = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3
B = -E(3)
  = (1-Sqrt(-3))/2 = -b3
C = 2*E(9)^2+E(9)^4+E(9)^5+2*E(9)^7
D = -E(9)^2+E(9)^4+E(9)^5-E(9)^7
E = -E(9)^2-2*E(9)^4-2*E(9)^5-E(9)^7