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Properties

Label 9T18
Order \(108\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^2 : D_{6} $

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $18$
Group :  $C_3^2 : D_{6} $
CHM label :  $E(9):D_{12}=[3^{2}:2]S(3)=[1/2.S(3)^{2}]S(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(3,5)(6,7), (1,2,9)(3,4,5)(6,7,8), (1,2)(3,6)(4,8)(5,7), (3,4,5)(6,8,7), (1,4,7)(2,5,8)(3,6,9)
$|\Aut(F/K)|$:  $1$

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: $S_3$

Low degree siblings

9T18, 18T51 x 2, 18T55 x 2, 18T56, 18T57 x 2, 27T29, 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 $ $1$ $1$ $()$
$ 3, 3, 1, 1, 1 $ $6$ $3$ $(3,4,5)(6,8,7)$
$ 2, 2, 2, 1, 1, 1 $ $9$ $2$ $(3,6)(4,7)(5,8)$
$ 2, 2, 2, 1, 1, 1 $ $9$ $2$ $(2,9)(4,5)(6,7)$
$ 6, 2, 1 $ $18$ $6$ $(2,9)(3,6,4,8,5,7)$
$ 2, 2, 2, 2, 1 $ $9$ $2$ $(2,9)(3,8)(4,7)(5,6)$
$ 3, 3, 3 $ $2$ $3$ $(1,2,9)(3,4,5)(6,7,8)$
$ 6, 3 $ $18$ $6$ $(1,2,9)(3,6,5,8,4,7)$
$ 3, 3, 3 $ $12$ $3$ $(1,3,6)(2,4,7)(5,8,9)$
$ 3, 3, 3 $ $6$ $3$ $(1,3,8)(2,4,6)(5,7,9)$
$ 6, 3 $ $18$ $6$ $(1,3,6,2,5,7)(4,8,9)$

Group invariants

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

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

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