↑

Properties

Label 27T31
Order \(108\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^2:(C_3:C_4)$

Learn more about

Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $31$
Group :  $C_3^2:(C_3:C_4)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5,8,14)(2,4,9,13)(3,6,7,15)(10,16,25,21)(11,18,26,20)(12,17,27,19)(23,24), (1,21,27,7)(2,20,25,9)(3,19,26,8)(4,5)(10,15,18,23)(11,14,16,22)(12,13,17,24)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
6:  $S_3$
12:  $C_3 : C_4$
36:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 9: $C_3^2:C_4$

Low degree siblings

12T72 x 2, 18T54 x 2, 36T89 x 2, 36T94 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $9$ $2$ $( 4,26)( 5,27)( 6,25)( 7,24)( 8,22)( 9,23)(10,20)(11,21)(12,19)(13,16)(14,17) (15,18)$
$ 4, 4, 4, 4, 4, 4, 2, 1 $ $27$ $4$ $( 2, 3)( 4,15,26,18)( 5,14,27,17)( 6,13,25,16)( 7,10,24,20)( 8,12,22,19) ( 9,11,23,21)$
$ 4, 4, 4, 4, 4, 4, 2, 1 $ $27$ $4$ $( 2, 3)( 4,18,26,15)( 5,17,27,14)( 6,16,25,13)( 7,20,24,10)( 8,19,22,12) ( 9,21,23,11)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$
$ 6, 6, 6, 6, 3 $ $18$ $6$ $( 1, 2, 3)( 4,27, 6,26, 5,25)( 7,22, 9,24, 8,23)(10,21,12,20,11,19) (13,17,15,16,14,18)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 4,25)( 2, 5,26)( 3, 6,27)( 7,10,14)( 8,11,15)( 9,12,13)(16,20,22) (17,21,23)(18,19,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 5,27)( 2, 6,25)( 3, 4,26)( 7,11,13)( 8,12,14)( 9,10,15)(16,21,24) (17,19,22)(18,20,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 6,26)( 2, 4,27)( 3, 5,25)( 7,12,15)( 8,10,13)( 9,11,14)(16,19,23) (17,20,24)(18,21,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 7,23)( 2, 8,24)( 3, 9,22)( 4,10,17)( 5,11,18)( 6,12,16)(13,20,27) (14,21,25)(15,19,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 8,22)( 2, 9,23)( 3, 7,24)( 4,11,16)( 5,12,17)( 6,10,18)(13,21,26) (14,19,27)(15,20,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 9,24)( 2, 7,22)( 3, 8,23)( 4,12,18)( 5,10,16)( 6,11,17)(13,19,25) (14,20,26)(15,21,27)$

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [108, 37]
Character table:    
      2  2  2  2  2  1  1  .  .  .  .  .  .
      3  3  1  .  .  3  1  3  3  3  3  3  3

        1a 2a 4a 4b 3a 6a 3b 3c 3d 3e 3f 3g
     2P 1a 1a 2a 2a 3a 3a 3d 3c 3b 3g 3f 3e
     3P 1a 2a 4b 4a 1a 2a 1a 1a 1a 1a 1a 1a
     5P 1a 2a 4a 4b 3a 6a 3d 3c 3b 3g 3f 3e

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

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