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Properties

Label 9T10
Order \(54\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_9:C_3):C_2$

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $10$
Group :  $(C_9:C_3):C_2$
CHM label :  $[3^{2}]S(3)_{6}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4,7)(2,8,5), (1,2,3,4,5,6,7,8,9), (1,8)(2,7)(3,6)(4,5)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Low degree siblings

18T18, 27T14

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 $ $1$ $1$ $()$
$ 6, 2, 1 $ $9$ $6$ $(2,3,5,9,8,6)(4,7)$
$ 3, 3, 1, 1, 1 $ $3$ $3$ $(2,5,8)(3,9,6)$
$ 6, 2, 1 $ $9$ $6$ $(2,6,8,9,5,3)(4,7)$
$ 3, 3, 1, 1, 1 $ $3$ $3$ $(2,8,5)(3,6,9)$
$ 2, 2, 2, 2, 1 $ $9$ $2$ $(2,9)(3,8)(4,7)(5,6)$
$ 9 $ $6$ $9$ $(1,2,3,4,5,6,7,8,9)$
$ 9 $ $6$ $9$ $(1,2,6,4,5,9,7,8,3)$
$ 9 $ $6$ $9$ $(1,2,9,4,5,3,7,8,6)$
$ 3, 3, 3 $ $2$ $3$ $(1,4,7)(2,5,8)(3,6,9)$

Group invariants

Order:  $54=2 \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [54, 6]
Character table:    
      2  1   1   1   1   1  1  .   .   .  .
      3  3   1   2   1   2  1  2   2   2  3

        1a  6a  3a  6b  3b 2a 9a  9b  9c 3c
     2P 1a  3a  3b  3b  3a 1a 9a  9c  9b 3c
     3P 1a  2a  1a  2a  1a 2a 3c  3c  3c 1a
     5P 1a  6b  3b  6a  3a 2a 9a  9c  9b 3c
     7P 1a  6a  3a  6b  3b 2a 9a  9b  9c 3c

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

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