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Properties

Label 27T38
Order \(162\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_3\times C_9):C_6$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $38$
Group :  $(C_3\times C_9):C_6$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,19,13,8,23,10)(2,21,14,7,24,12)(3,20,15,9,22,11)(4,25,17)(5,27,18,6,26,16), (1,9,4,3,8,6,2,7,5)(10,17,13,12,16,15,11,18,14)(19,27,24,21,26,23,20,25,22)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$
54:  $C_3^2 : C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$, $S_3$

Degree 9: $S_3\times C_3$

Low degree siblings

27T64

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

Group invariants

Order:  $162=2 \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [162, 15]
Character table:    
      2  1  .  1  .  .  .  .   .   1  1   .   1  1
      3  4  3  1  4  3  3  3   2   1  2   2   1  2

        1a 3a 2a 3b 9a 9b 9c  3c  6a 3d  3e  6b 3f
     2P 1a 3a 1a 3b 9c 9a 9b  3e  3f 3f  3c  3d 3d
     3P 1a 1a 2a 1a 3b 3b 3b  1a  2a 1a  1a  2a 1a
     5P 1a 3a 2a 3b 9b 9c 9a  3e  6b 3f  3c  6a 3d
     7P 1a 3a 2a 3b 9c 9a 9b  3c  6a 3d  3e  6b 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      1  1 -1  1  1  1  1   D  -D  D  /D -/D /D
X.4      1  1 -1  1  1  1  1  /D -/D /D   D  -D  D
X.5      1  1  1  1  1  1  1   D   D  D  /D  /D /D
X.6      1  1  1  1  1  1  1  /D  /D /D   D   D  D
X.7      2  2  .  2 -1 -1 -1  -1   .  2  -1   .  2
X.8      2  2  .  2 -1 -1 -1  -D   .  E -/D   . /E
X.9      2  2  .  2 -1 -1 -1 -/D   . /E  -D   .  E
X.10     6 -3  .  6  .  .  .   .   .  .   .   .  .
X.11     6  .  . -3  A  C  B   .   .  .   .   .  .
X.12     6  .  . -3  B  A  C   .   .  .   .   .  .
X.13     6  .  . -3  C  B  A   .   .  .   .   .  .

A = 2*E(9)^2+E(9)^4+E(9)^5+2*E(9)^7
B = -E(9)^2+E(9)^4+E(9)^5-E(9)^7
C = -E(9)^2-2*E(9)^4-2*E(9)^5-E(9)^7
D = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
E = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3