# Properties

 Label 8T2 Degree $8$ Order $8$ Cyclic no Abelian yes Solvable yes Primitive no $p$-group yes Group: $C_4\times C_2$

# Related objects

## Group action invariants

 Degree $n$: $8$ Transitive number $t$: $2$ Group: $C_4\times C_2$ CHM label: $4[x]2$ Parity: $1$ Primitive: no Nilpotency class: $1$ $|\Aut(F/K)|$: $8$ Generators: (1,2,3,8)(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 3
$4$:  $C_4$ x 2, $C_2^2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

## 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 $1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 4$ $1$ $4$ $(1,2,3,8)(4,5,6,7)$ $2, 2, 2, 2$ $1$ $2$ $(1,3)(2,8)(4,6)(5,7)$ $4, 4$ $1$ $4$ $(1,4,3,6)(2,5,8,7)$ $2, 2, 2, 2$ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$ $4, 4$ $1$ $4$ $(1,6,3,4)(2,7,8,5)$ $2, 2, 2, 2$ $1$ $2$ $(1,7)(2,4)(3,5)(6,8)$ $4, 4$ $1$ $4$ $(1,8,3,2)(4,7,6,5)$

## Group invariants

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