# Properties

 Label 8T48 Order $1344$ n $8$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $C_2^3:\GL(3,2)$

# Related objects

## Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $48$
Group :  $C_2^3:\GL(3,2)$
CHM label :  $E(8):L_{7}=AL(8)$
Parity:  $1$
Primitive:  Yes
Generators:   (2,4,8,3,6,7,5), (1,4)(2,5)(3,6)(7,8)
$|\textrm{Aut}(F/K)|$:  $1$
Low degree resolvents:
 168: 7T5

Degree 2: None

Degree 4: None

## Low degree siblings

8T48b, 14T34a, 14T34b
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2$ $7$ $2$ $(1,8)(2,3)(4,5)(6,7)$ $2, 2, 1, 1, 1, 1$ $42$ $2$ $(4,5)(6,7)$ $2, 2, 2, 2$ $42$ $2$ $(1,3)(2,8)(4,7)(5,6)$ $4, 4$ $84$ $4$ $(1,7,8,6)(2,4,3,5)$ $3, 3, 1, 1$ $224$ $3$ $(3,4,5)(6,8,7)$ $6, 2$ $224$ $6$ $(1,3,6,2,8,5)(4,7)$ $4, 2, 1, 1$ $168$ $4$ $(3,4,8,7)(5,6)$ $4, 4$ $168$ $4$ $(1,8,6,4)(2,3,5,7)$ $7, 1$ $192$ $7$ $(2,3,4,7,5,8,6)$ $7, 1$ $192$ $7$ $(2,3,4,8,6,5,7)$

## Group invariants

 Order: $1344=2^{6} \cdot 3 \cdot 7$ Cyclic: No Abelian: No Solvable: No
 Character table: ``` 2 6 6 5 5 4 1 1 3 3 . . 3 1 1 . . . 1 1 . . . . 7 1 . . . . . . . . 1 1 1a 2a 2b 2c 4a 3a 6a 4b 4c 7a 7b 2P 1a 1a 1a 1a 2a 3a 3a 2b 2c 7a 7b 3P 1a 2a 2b 2c 4a 1a 2a 4b 4c 7b 7a 5P 1a 2a 2b 2c 4a 3a 6a 4b 4c 7b 7a 7P 1a 2a 2b 2c 4a 3a 6a 4b 4c 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 3 3 -1 -1 -1 . . 1 1 A /A X.3 3 3 -1 -1 -1 . . 1 1 /A A X.4 6 6 2 2 2 . . . . -1 -1 X.5 7 -1 3 -1 -1 1 -1 1 -1 . . X.6 7 7 -1 -1 -1 1 1 -1 -1 . . X.7 7 -1 -1 3 -1 1 -1 -1 1 . . X.8 8 8 . . . -1 -1 . . 1 1 X.9 14 -2 2 2 -2 -1 1 . . . . X.10 21 -3 1 -3 1 . . -1 1 . . X.11 21 -3 -3 1 1 . . 1 -1 . . A = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7 ```