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Properties

Label 27T35
Degree $27$
Order $108$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^3:C_2^2$

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Group action invariants

Degree $n$:  $27$
Transitive number $t$:  $35$
Group:  $C_3^3:C_2^2$
Parity:  $1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $1$
Generators:  (1,19,3,21,2,20)(4,16,6,18,5,17)(7,14,9,13,8,15)(10,11,12)(22,25,24,27,23,26), (1,26,4)(2,25,5,3,27,6)(7,23,10,17,13,20)(8,22,11,16,14,19)(9,24,12,18,15,21)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 3

Degree 9: $S_3^2$ x 3

Low degree siblings

12T71, 18T53 x 3, 36T88 x 3, 36T93

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

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [108, 40]
Character table:    
      2  2  2  2  2  1  1  1  1  .  .  1  .  .  .  1
      3  3  1  1  1  3  1  3  1  3  3  1  3  3  3  3

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

X.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
X.3      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
X.5      2  .  . -2  2  .  2  .  2 -1  1 -1 -1 -1 -1
X.6      2  .  .  2  2  .  2  .  2 -1 -1 -1 -1 -1 -1
X.7      2 -2  .  . -1  1  2  . -1 -1  . -1  2 -1  2
X.8      2  . -2  .  2  . -1  1 -1 -1  . -1 -1  2  2
X.9      2  .  2  .  2  . -1 -1 -1 -1  . -1 -1  2  2
X.10     2  2  .  . -1 -1  2  . -1 -1  . -1  2 -1  2
X.11     4  .  .  .  4  . -2  . -2  1  .  1  1 -2 -2
X.12     4  .  .  . -2  .  4  . -2  1  .  1 -2  1 -2
X.13     4  .  .  . -2  . -2  .  1  1  .  1 -2 -2  4
X.14     4  .  .  . -2  . -2  .  1  A  . /A  1  1 -2
X.15     4  .  .  . -2  . -2  .  1 /A  .  A  1  1 -2

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