# Properties

 Label 6T6 Order $$24$$ n $$6$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $A_4\times C_2$

# Related objects

## Group action invariants

Degree $n$ :  $6$
Transitive number $t$ :  $6$
Group :  $A_4\times C_2$
CHM label :  $2A_{4}(6) = [2^{3}]3 = 2 wr 3$
Parity:  $-1$
Primitive:  No
Generators:   (1,3,5)(2,4,6), (3,6)
$|\Aut(F/K)|$:  $2$
Low degree resolvents:
 2: 2T1 3: 3T1 6: 6T1 12: 4T4

## Subfields

Degree 2: None

Degree 3: $C_3$

## Low degree siblings

8T13, 12T6, 12T7
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

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

## Group invariants

 Order: $24=2^{3} \cdot 3$ Cyclic: No Abelian: No Solvable: Yes GAP id: [24, 13]
 Character table:  2 3 3 3 1 1 1 1 3 3 1 . . 1 1 1 1 1 1a 2a 2b 3a 6a 3b 6b 2c 2P 1a 1a 1a 3b 3b 3a 3a 1a 3P 1a 2a 2b 1a 2c 1a 2c 2c 5P 1a 2a 2b 3b 6b 3a 6a 2c X.1 1 1 1 1 1 1 1 1 X.2 1 -1 1 1 -1 1 -1 -1 X.3 1 -1 1 A -A /A -/A -1 X.4 1 -1 1 /A -/A A -A -1 X.5 1 1 1 A A /A /A 1 X.6 1 1 1 /A /A A A 1 X.7 3 1 -1 . . . . -3 X.8 3 -1 -1 . . . . 3 A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3