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Properties

Label 6T5
Order \(18\)
n \(6\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times C_3$

Related objects

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Group action invariants

Degree $n$ :  $6$
Transitive number $t$ :  $5$
Group :  $S_3\times C_3$
CHM label :  $F_{18}(6) = [3^{2}]2 = 3 wr 2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4)(2,5)(3,6), (2,4,6)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Low degree siblings

9T4, 18T3

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 $ $1$ $1$ $()$
$ 3, 1, 1, 1 $ $2$ $3$ $(2,4,6)$
$ 3, 1, 1, 1 $ $2$ $3$ $(2,6,4)$
$ 2, 2, 2 $ $3$ $2$ $(1,2)(3,4)(5,6)$
$ 6 $ $3$ $6$ $(1,2,3,4,5,6)$
$ 6 $ $3$ $6$ $(1,2,5,6,3,4)$
$ 3, 3 $ $1$ $3$ $(1,3,5)(2,4,6)$
$ 3, 3 $ $2$ $3$ $(1,3,5)(2,6,4)$
$ 3, 3 $ $1$ $3$ $(1,5,3)(2,6,4)$

Group invariants

Order:  $18=2 \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [18, 3]
Character table:    
     2  1   .   .  1   1   1  1  .  1
     3  2   2   2  1   1   1  2  2  2

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

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3