Properties

 Label 6T10 Degree $6$ Order $36$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_3^2:C_4$

Related objects

Group action invariants

 Degree $n$: $6$ Transitive number $t$: $10$ Group: $C_3^2:C_4$ CHM label: $F_{36}(6) = 1/2[S(3)^{2}]2$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,4,5,2)(3,6), (2,4,6), (1,5)(2,4)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Low degree siblings

6T10, 9T9, 12T17 x 2, 18T10, 36T14

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

Group invariants

 Order: $36=2^{2} \cdot 3^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [36, 9]
 Character table:  2 2 2 . 2 2 . 3 2 . 2 . . 2 1a 2a 3a 4a 4b 3b 2P 1a 1a 3a 2a 2a 3b 3P 1a 2a 1a 4b 4a 1a X.1 1 1 1 1 1 1 X.2 1 1 1 -1 -1 1 X.3 1 -1 1 A -A 1 X.4 1 -1 1 -A A 1 X.5 4 . 1 . . -2 X.6 4 . -2 . . 1 A = -E(4) = -Sqrt(-1) = -i