Properties

Label 27T32
Order \(108\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^2:S_3.C_2$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $32$
Group :  $C_3^2:S_3.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,15,9,4)(2,13,7,5)(3,14,8,6)(10,21,27,18)(11,19,25,16)(12,20,26,17), (1,25,22,18,3,27,24,17,2,26,23,16)(4,10,19,14,6,12,21,13,5,11,20,15)(7,9,8)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
36:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 9: $C_3^2:C_4$

Low degree siblings

18T49 x 2, 36T85 x 2

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

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [108, 15]
Character table:   
      2  2  2  2  2  2   2   2   2  2   2   2   2  .  .
      3  3  1  1  1  3   1   1   1  3   1   1   1  2  2

        1a 4a 4b 2a 3a 12a 12b  6a 3b 12c 12d  6b 3c 3d
     2P 1a 2a 2a 1a 3b  6b  6b  3b 3a  6a  6a  3a 3c 3d
     3P 1a 4b 4a 2a 1a  4b  4a  2a 1a  4b  4a  2a 1a 1a
     5P 1a 4a 4b 2a 3b 12c 12d  6b 3a 12a 12b  6a 3c 3d
     7P 1a 4b 4a 2a 3a 12b 12a  6a 3b 12d 12c  6b 3c 3d
    11P 1a 4b 4a 2a 3b 12d 12c  6b 3a 12b 12a  6a 3c 3d

X.1      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  1
X.3      1  A -A -1  1   A  -A  -1  1   A  -A  -1  1  1
X.4      1 -A  A -1  1  -A   A  -1  1  -A   A  -1  1  1
X.5      3 -1 -1 -1  B   C   C   C /B  /C  /C  /C  .  .
X.6      3 -1 -1 -1 /B  /C  /C  /C  B   C   C   C  .  .
X.7      3  1  1 -1  B  -C  -C   C /B -/C -/C  /C  .  .
X.8      3  1  1 -1 /B -/C -/C  /C  B  -C  -C   C  .  .
X.9      3  A -A  1  B   D  -D  -C /B -/D  /D -/C  .  .
X.10     3  A -A  1 /B -/D  /D -/C  B   D  -D  -C  .  .
X.11     3 -A  A  1  B  -D   D  -C /B  /D -/D -/C  .  .
X.12     3 -A  A  1 /B  /D -/D -/C  B  -D   D  -C  .  .
X.13     4  .  .  .  4   .   .   .  4   .   .   . -2  1
X.14     4  .  .  .  4   .   .   .  4   .   .   .  1 -2

A = -E(4)
  = -Sqrt(-1) = -i
B = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3
C = -E(3)^2
  = (1+Sqrt(-3))/2 = 1+b3
D = -E(12)^11