# Properties

 Label 8T7 Order $$16$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_8:C_2$

# Related objects

## Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $7$
Group :  $C_8:C_2$
CHM label :  $1/2[2^{3}]4$
Parity:  $-1$
Primitive:  No
Generators:  (1,2,3,4,5,6,7,8), (1,5)(3,7)
$|\Aut(F/K)|$:  $4$
Low degree resolvents:
 2: $C_2$ x 3 4: $C_4$ x 2, $V_4$ 8: $C_4\times C_2$

## Subfields

Degree 2: $C_2$

Degree 4: $C_4$

## Low degree siblings

16T6
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

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

## Group invariants

 Order: $16=2^{4}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [16, 6]
 Character table:  2 4 3 3 3 4 3 3 3 4 4 1a 2a 8a 8b 4a 4b 8c 8d 2b 4c 2P 1a 1a 4a 4c 2b 2b 4a 4c 1a 2b 3P 1a 2a 8d 8c 4c 4b 8b 8a 2b 4a 5P 1a 2a 8a 8b 4a 4b 8c 8d 2b 4c 7P 1a 2a 8d 8c 4c 4b 8b 8a 2b 4a X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 -1 1 -1 1 1 X.3 1 -1 1 -1 1 -1 -1 1 1 1 X.4 1 1 -1 -1 1 1 -1 -1 1 1 X.5 1 -1 A -A -1 1 A -A 1 -1 X.6 1 -1 -A A -1 1 -A A 1 -1 X.7 1 1 A A -1 -1 -A -A 1 -1 X.8 1 1 -A -A -1 -1 A A 1 -1 X.9 2 . . . B . . . -2 -B X.10 2 . . . -B . . . -2 B A = -E(4) = -Sqrt(-1) = -i B = -2*E(4) = -2*Sqrt(-1) = -2i