Properties

Label 27T10
Order \(54\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3:D_9$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $10$
Group :  $C_3:D_9$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (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), (1,23,18,11,4,25,20,13,7)(2,24,16,12,5,26,21,14,8)(3,22,17,10,6,27,19,15,9), (2,3)(4,25)(5,27)(6,26)(7,23)(8,22)(9,24)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$ x 4
18:  $D_{9}$ x 3, $C_3^2:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 4

Degree 9: $D_{9}$ x 3, $C_3^2:C_2$

Low degree siblings

There are no siblings 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, 2, 1 $ $27$ $2$ $( 2, 3)( 4,25)( 5,27)( 6,26)( 7,23)( 8,22)( 9,24)(10,21)(11,20)(12,19)(13,18) (14,17)(15,16)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$
$ 9, 9, 9 $ $2$ $9$ $( 1, 4, 7,11,13,18,20,23,25)( 2, 5, 8,12,14,16,21,24,26)( 3, 6, 9,10,15,17,19, 22,27)$
$ 9, 9, 9 $ $2$ $9$ $( 1, 5, 9,11,14,17,20,24,27)( 2, 6, 7,12,15,18,21,22,25)( 3, 4, 8,10,13,16,19, 23,26)$
$ 9, 9, 9 $ $2$ $9$ $( 1, 6, 8,11,15,16,20,22,26)( 2, 4, 9,12,13,17,21,23,27)( 3, 5, 7,10,14,18,19, 24,25)$
$ 9, 9, 9 $ $2$ $9$ $( 1, 7,13,20,25, 4,11,18,23)( 2, 8,14,21,26, 5,12,16,24)( 3, 9,15,19,27, 6,10, 17,22)$
$ 9, 9, 9 $ $2$ $9$ $( 1, 8,15,20,26, 6,11,16,22)( 2, 9,13,21,27, 4,12,17,23)( 3, 7,14,19,25, 5,10, 18,24)$
$ 9, 9, 9 $ $2$ $9$ $( 1, 9,14,20,27, 5,11,17,24)( 2, 7,15,21,25, 6,12,18,22)( 3, 8,13,19,26, 4,10, 16,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,10,21)( 2,11,19)( 3,12,20)( 4,15,24)( 5,13,22)( 6,14,23)( 7,17,26) ( 8,18,27)( 9,16,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,11,20)( 2,12,21)( 3,10,19)( 4,13,23)( 5,14,24)( 6,15,22)( 7,18,25) ( 8,16,26)( 9,17,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,12,19)( 2,10,20)( 3,11,21)( 4,14,22)( 5,15,23)( 6,13,24)( 7,16,27) ( 8,17,25)( 9,18,26)$
$ 9, 9, 9 $ $2$ $9$ $( 1,13,25,11,23, 7,20, 4,18)( 2,14,26,12,24, 8,21, 5,16)( 3,15,27,10,22, 9,19, 6,17)$
$ 9, 9, 9 $ $2$ $9$ $( 1,14,27,11,24, 9,20, 5,17)( 2,15,25,12,22, 7,21, 6,18)( 3,13,26,10,23, 8,19, 4,16)$
$ 9, 9, 9 $ $2$ $9$ $( 1,15,26,11,22, 8,20, 6,16)( 2,13,27,12,23, 9,21, 4,17)( 3,14,25,10,24, 7,19, 5,18)$

Group invariants

Order:  $54=2 \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [54, 7]
Character table:   
      2  1  1  .  .  .  .  .  .  .  .  .  .  .  .  .
      3  3  .  3  3  3  3  3  3  3  3  3  3  3  3  3

        1a 2a 3a 9a 9b 9c 9d 9e 9f 3b 3c 3d 9g 9h 9i
     2P 1a 1a 3a 9d 9f 9e 9g 9i 9h 3b 3c 3d 9a 9b 9c
     3P 1a 2a 1a 3c 3c 3c 3c 3c 3c 1a 1a 1a 3c 3c 3c
     5P 1a 2a 3a 9g 9h 9i 9a 9c 9b 3b 3c 3d 9d 9f 9e
     7P 1a 2a 3a 9d 9f 9e 9g 9i 9h 3b 3c 3d 9a 9b 9c

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

A = E(9)^2+E(9)^7
B = E(9)^4+E(9)^5
C = -E(9)^2-E(9)^4-E(9)^5-E(9)^7