# Properties

 Label 8T43 Order $336$ n $8$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $\PGL(2,7)$

# Related objects

## Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $43$
Group :  $\PGL(2,7)$
CHM label :  $L(8):2=PGL(2,7)$
Parity:  $-1$
Primitive:  Yes
Generators:   (2,4,6,3,7,5), (1,8,4,7,2,6,5)
$|\Aut(F/K)|$:  $1$
Low degree resolvents:
 2: 2T1

Degree 2: None

Degree 4: None

## Low degree siblings

14T16, 16T713, 21T20
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 1, 1$ $28$ $2$ $(3,4)(5,7)(6,8)$ $6, 1, 1$ $56$ $6$ $(3,5,8,4,7,6)$ $3, 3, 1, 1$ $56$ $3$ $(3,7,8)(4,5,6)$ $7, 1$ $48$ $7$ $(2,3,7,6,8,5,4)$ $2, 2, 2, 2$ $21$ $2$ $(1,2)(3,4)(5,8)(6,7)$ $8$ $42$ $8$ $(1,2,3,6,4,7,8,5)$ $4, 4$ $42$ $4$ $(1,2,3,7)(4,6,5,8)$ $8$ $42$ $8$ $(1,2,3,8,6,7,5,4)$

## Group invariants

 Order: $336=2^{4} \cdot 3 \cdot 7$ Cyclic: No Abelian: No Solvable: No GAP id: [336, 208]
 Character table: ``` 2 4 2 1 1 . 4 3 3 3 3 1 1 1 1 . . . . . 7 1 . . . 1 . . . . 1a 2a 6a 3a 7a 2b 8a 4a 8b 2P 1a 1a 3a 3a 7a 1a 4a 2b 4a 3P 1a 2a 2a 1a 7a 2b 8b 4a 8a 5P 1a 2a 6a 3a 7a 2b 8b 4a 8a 7P 1a 2a 6a 3a 1a 2b 8a 4a 8b X.1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 1 -1 1 -1 X.3 6 . . . -1 -2 . 2 . X.4 6 . . . -1 2 A . -A X.5 6 . . . -1 2 -A . A X.6 7 -1 -1 1 . -1 1 -1 1 X.7 7 1 1 1 . -1 -1 -1 -1 X.8 8 -2 1 -1 1 . . . . X.9 8 2 -1 -1 1 . . . . A = -E(8)+E(8)^3 = -Sqrt(2) = -r2 ```