# Properties

 Label 8T12 Order $$24$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $\SL(2,3)$

# Related objects

## Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $12$
Group :  $\SL(2,3)$
CHM label :  $2A_{4}(8)=[2]A(4)=SL(2,3)$
Parity:  $1$
Primitive:  No
Generators:   (1,3,5,7)(2,4,6,8), (1,3,8)(4,5,7)
$|\Aut(F/K)|$:  $2$
Low degree resolvents:
 3: 3T1 12: 4T4

## Subfields

Degree 2: None

Degree 4: $A_4$

## Low degree siblings

There is no other low degree representation.
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 1, 1$ $4$ $3$ $(2,7,8)(3,4,6)$ $3, 3, 1, 1$ $4$ $3$ $(2,8,7)(3,6,4)$ $4, 4$ $6$ $4$ $(1,2,5,6)(3,8,7,4)$ $6, 2$ $4$ $6$ $(1,2,7,5,6,3)(4,8)$ $6, 2$ $4$ $6$ $(1,3,6,5,7,2)(4,8)$ $2, 2, 2, 2$ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$

## Group invariants

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