Properties

Label 27T34
Order \(108\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times C_3:S_3$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $34$
Group :  $S_3\times C_3:S_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,19,11)(2,20,12)(3,21,10)(4,17,15,26,23,8)(5,18,13,27,24,9)(6,16,14,25,22,7), (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), (1,26,4)(2,25,5,3,27,6)(7,24,10,18,14,20)(8,23,11,17,15,19)(9,22,12,16,13,21)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 5
12:  $D_{6}$ x 5
18:  $C_3^2:C_2$
36:  $S_3^2$ x 4, 18T12

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 5

Degree 9: $C_3^2:C_2$, $S_3^2$ x 4

Low degree siblings

18T58 x 4, 36T91 x 4

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

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [108, 39]
Character table:   
      2  2  2  2  2  1  1  1  1  .  .  1  .  1  .  1  1  1  1
      3  3  2  1  .  3  2  3  1  3  3  2  3  2  3  2  3  3  3

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

X.1      1  1  1  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 -1  1  1  1
X.3      1 -1  1 -1  1 -1  1  1  1  1 -1  1 -1  1 -1  1  1  1
X.4      1  1 -1 -1  1  1  1 -1  1  1  1  1  1  1  1  1  1  1
X.5      2 -2  .  .  2 -2  2  .  2 -1  1 -1  1 -1  1 -1 -1 -1
X.6      2  2  .  .  2  2  2  .  2 -1 -1 -1 -1 -1 -1 -1 -1 -1
X.7      2 -2  .  . -1  1  2  . -1  2 -2 -1  1 -1  1  2 -1 -1
X.8      2  . -2  .  2  . -1  1 -1 -1  . -1  . -1  .  2  2  2
X.9      2  .  2  .  2  . -1 -1 -1 -1  . -1  . -1  .  2  2  2
X.10     2  2  .  . -1 -1  2  . -1  2  2 -1 -1 -1 -1  2 -1 -1
X.11     2 -2  .  . -1  1  2  . -1 -1  1 -1  1  2 -2 -1 -1  2
X.12     2 -2  .  . -1  1  2  . -1 -1  1  2 -2 -1  1 -1  2 -1
X.13     2  2  .  . -1 -1  2  . -1 -1 -1 -1 -1  2  2 -1 -1  2
X.14     2  2  .  . -1 -1  2  . -1 -1 -1  2  2 -1 -1 -1  2 -1
X.15     4  .  .  .  4  . -2  . -2  1  .  1  .  1  . -2 -2 -2
X.16     4  .  .  . -2  . -2  .  1 -2  .  1  .  1  .  4 -2 -2
X.17     4  .  .  . -2  . -2  .  1  1  . -2  .  1  . -2  4 -2
X.18     4  .  .  . -2  . -2  .  1  1  .  1  . -2  . -2 -2  4