# Properties

 Label 8T11 Order $$16$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $Q_8:C_2$

# Related objects

## Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $11$
Group :  $Q_8:C_2$
CHM label :  $1/2[2^{3}]E(4)=Q_{8}:2$
Parity:  $1$
Primitive:  No
Generators:   (1,3,5,7)(2,4,6,8), (1,4,5,8)(2,3,6,7), (1,5)(3,7)
$|\Aut(F/K)|$:  $4$
Low degree resolvents:
 2: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1 4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2 8: 8T3

## Subfields

Degree 2: $C_2$, $C_2$, $C_2$

Degree 4: $V_4$

## Low degree siblings

8T11b, 8T11c, 16T11
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

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

## Group invariants

 Order: $16=2^{4}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [16, 13]
 Character table:  2 4 3 3 3 4 3 3 3 4 4 1a 2a 2b 4a 4b 4c 4d 2c 2d 4e 2P 1a 1a 1a 2d 2d 2d 2d 1a 1a 2d 3P 1a 2a 2b 4a 4e 4c 4d 2c 2d 4b 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 1 -1 1 -1 1 -1 1 1 X.6 1 1 -1 -1 -1 -1 1 1 1 -1 X.7 1 1 -1 -1 1 1 -1 -1 1 1 X.8 1 1 1 1 -1 -1 -1 -1 1 -1 X.9 2 . . . A . . . -2 -A X.10 2 . . . -A . . . -2 A A = -2*E(4) = -2*Sqrt(-1) = -2i