# 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

## 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 Type Size Order Representative $1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 1, 1, 1$ $2$ $3$ $(2,4,6)$ $3, 1, 1, 1$ $2$ $3$ $(2,6,4)$ $2, 2, 2$ $3$ $2$ $(1,2)(3,4)(5,6)$ $6$ $3$ $6$ $(1,2,3,4,5,6)$ $6$ $3$ $6$ $(1,2,5,6,3,4)$ $3, 3$ $1$ $3$ $(1,3,5)(2,4,6)$ $3, 3$ $2$ $3$ $(1,3,5)(2,6,4)$ $3, 3$ $1$ $3$ $(1,5,3)(2,6,4)$

## 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