# Properties

 Label 8T6 Degree $8$ Order $16$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $D_{8}$

# Related objects

## Group action invariants

 Degree $n$: $8$ Transitive number $t$: $6$ Group: $D_{8}$ CHM label: $D(8)$ Parity: $-1$ Primitive: no Nilpotency class: $3$ $|\Aut(F/K)|$: $2$ Generators: (1,2,3,4,5,6,7,8), (1,6)(2,5)(3,4)(7,8)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

## Low degree siblings

8T6, 16T13

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

## Group invariants

 Order: $16=2^{4}$ Cyclic: no Abelian: no Solvable: yes GAP id: [16, 7]
 Character table:  2 4 2 2 3 3 3 4 1a 2a 2b 8a 4a 8b 2c 2P 1a 1a 1a 4a 2c 4a 1a 3P 1a 2a 2b 8b 4a 8a 2c 5P 1a 2a 2b 8b 4a 8a 2c 7P 1a 2a 2b 8a 4a 8b 2c X.1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 1 1 X.3 1 -1 1 -1 1 -1 1 X.4 1 1 -1 -1 1 -1 1 X.5 2 . . . -2 . 2 X.6 2 . . A . -A -2 X.7 2 . . -A . A -2 A = -E(8)+E(8)^3 = -Sqrt(2) = -r2