# Properties

 Label 8T3 Degree $8$ Order $8$ Cyclic no Abelian yes Solvable yes Primitive no $p$-group yes Group: $C_2^3$

# Related objects

## Group action invariants

 Degree $n$: $8$ Transitive number $t$: $3$ Group: $C_2^3$ CHM label: $E(8)=2[x]2[x]2$ Parity: $1$ Primitive: no Nilpotency class: $1$ $|\Aut(F/K)|$: $8$ Generators: (1,3)(2,8)(4,6)(5,7), (1,8)(2,3)(4,5)(6,7), (1,5)(2,6)(3,7)(4,8)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

## Group invariants

 Order: $8=2^{3}$ Cyclic: no Abelian: yes Solvable: yes GAP id: [8, 5]
 Character table:  2 3 3 3 3 3 3 3 3 1a 2a 2b 2c 2d 2e 2f 2g 2P 1a 1a 1a 1a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 -1 -1 1 1 1 X.3 1 -1 -1 1 1 -1 -1 1 X.4 1 -1 1 -1 1 -1 1 -1 X.5 1 -1 1 1 -1 1 -1 -1 X.6 1 1 -1 -1 1 1 -1 -1 X.7 1 1 -1 1 -1 -1 1 -1 X.8 1 1 1 -1 -1 -1 -1 1