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Properties

Label 27T41
Order \(162\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $He_3.S_3$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $41$
Group :  $He_3.S_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10,25,2,11,26,3,12,27)(4,15,19,5,13,20,6,14,21)(7,16,23,8,17,24,9,18,22), (1,20,14,8,24,10)(2,19,15,7,22,12)(3,21,13,9,23,11)(4,26,16)(5,25,17,6,27,18)
$|\Aut(F/K)|$:  $1$

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

27T72

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

Group invariants

Order:  $162=2 \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [162, 13]
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  9d  6a 3c  9e  6b 3d
     2P 1a 3a 1a 3b 9b 9c 9a  9e  3d 3d  9d  3c 3c
     3P 1a 1a 2a 1a 3b 3b 3b  3b  2a 1a  3b  2a 1a
     5P 1a 3a 2a 3b 9c 9a 9b  9e  6b 3d  9d  6a 3c
     7P 1a 3a 2a 3b 9b 9c 9a  9d  6a 3c  9e  6b 3d

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  B  C   .   .  .   .   .  .
X.12     6  .  . -3  B  C  A   .   .  .   .   .  .
X.13     6  .  . -3  C  A  B   .   .  .   .   .  .

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