# Properties

 Label 6T9 Degree $6$ Order $36$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $S_3^2$

# Related objects

## Group action invariants

 Degree $n$: $6$ Transitive number $t$: $9$ Group: $S_3^2$ CHM label: $F_{18}(6):2 = [1/2.S(3)^{2}]2$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,4)(2,5)(3,6), (2,4,6), (1,5)(2,4)

## 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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: None

## Low degree siblings

9T8, 12T16, 18T9, 18T11 x 2, 36T13

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

## Group invariants

 Order: $36=2^{2} \cdot 3^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [36, 10]
 Character table:  2 2 2 . 2 2 1 1 1 1 3 2 . 2 1 1 1 1 2 2 1a 2a 3a 2b 2c 6a 6b 3b 3c 2P 1a 1a 3a 1a 1a 3b 3c 3b 3c 3P 1a 2a 1a 2b 2c 2b 2c 1a 1a 5P 1a 2a 3a 2b 2c 6a 6b 3b 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 -1 1 1 -1 1 -1 1 1 X.4 1 1 1 -1 -1 -1 -1 1 1 X.5 2 . -1 . -2 . 1 2 -1 X.6 2 . -1 . 2 . -1 2 -1 X.7 2 . -1 -2 . 1 . -1 2 X.8 2 . -1 2 . -1 . -1 2 X.9 4 . 1 . . . . -2 -2