Properties

Label 27T29
Order \(108\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3.S_3^2$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $29$
Group :  $C_3.S_3^2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,17,5,14,21)(2,8,18,4,15,20)(3,7,16,6,13,19)(10,22,27)(11,24,25,12,23,26), (1,26,5,3,27,4)(2,25,6)(7,19,11,23,15,18)(8,21,12,22,13,17)(9,20,10,24,14,16)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 2
12:  $D_{6}$ x 2
36:  $S_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 2

Degree 9: $S_3^2$, $C_3^2 : D_{6} $ x 2

Low degree siblings

9T18 x 2, 18T51 x 2, 18T55 x 2, 18T56, 18T57 x 2, 36T87 x 2, 36T90

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

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [108, 17]
Character table:   
      2  2  2  2  2  1  1  1  1  .  1  1
      3  3  1  1  1  2  1  1  3  2  1  2

        1a 2a 2b 2c 3a 6a 6b 3b 3c 6c 3d
     2P 1a 1a 1a 1a 3a 3a 3b 3b 3c 3d 3d
     3P 1a 2a 2b 2c 1a 2a 2b 1a 1a 2c 1a
     5P 1a 2a 2b 2c 3a 6a 6b 3b 3c 6c 3d

X.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
X.3      1 -1  1 -1  1 -1  1  1  1 -1  1
X.4      1  1 -1 -1  1  1 -1  1  1 -1  1
X.5      2  .  . -2  2  .  .  2 -1  1 -1
X.6      2  .  .  2  2  .  .  2 -1 -1 -1
X.7      2 -2  .  . -1  1  .  2 -1  .  2
X.8      2  2  .  . -1 -1  .  2 -1  .  2
X.9      4  .  .  . -2  .  .  4  1  . -2
X.10     6  . -2  .  .  .  1 -3  .  .  .
X.11     6  .  2  .  .  . -1 -3  .  .  .