# Properties

 Label 8T23 Order $$48$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $\textrm{GL(2,3)}$

# Related objects

## Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $23$
Group :  $\textrm{GL(2,3)}$
CHM label :  $2S_{4}(8)=GL(2,3)$
Parity:  $-1$
Primitive:  No
Generators:   (1,2,3,4,5,6,7,8), (1,3,8)(4,5,7)
$|\Aut(F/K)|$:  $2$
Low degree resolvents:
 2: 2T1 6: 3T2 24: 4T5

## Subfields

Degree 2: None

Degree 4: $S_4$

## Low degree siblings

8T23b, 16T66
A number field with this Galois group has exactly one arithmetically equivalent field.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 1, 1$ $12$ $2$ $(2,3)(4,8)(6,7)$ $3, 3, 1, 1$ $8$ $3$ $(2,7,8)(3,4,6)$ $8$ $6$ $8$ $(1,2,3,4,5,6,7,8)$ $4, 4$ $6$ $4$ $(1,2,5,6)(3,8,7,4)$ $6, 2$ $8$ $6$ $(1,2,7,5,6,3)(4,8)$ $8$ $6$ $8$ $(1,2,8,3,5,6,4,7)$ $2, 2, 2, 2$ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$

## Group invariants

 Order: $48=2^{4} \cdot 3$ Cyclic: No Abelian: No Solvable: Yes GAP id: [48, 29]
 Character table:  2 4 2 1 3 3 1 3 4 3 1 . 1 . . 1 . 1 1a 2a 3a 8a 4a 6a 8b 2b 2P 1a 1a 3a 4a 2b 3a 4a 1a 3P 1a 2a 1a 8a 4a 2b 8b 2b 5P 1a 2a 3a 8b 4a 6a 8a 2b 7P 1a 2a 3a 8b 4a 6a 8a 2b X.1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 1 -1 1 X.3 2 . -1 . 2 -1 . 2 X.4 2 . -1 A . 1 -A -2 X.5 2 . -1 -A . 1 A -2 X.6 3 1 . -1 -1 . -1 3 X.7 3 -1 . 1 -1 . 1 3 X.8 4 . 1 . . -1 . -4 A = E(8)+E(8)^3 = Sqrt(-2) = i2