# Properties

 Label 8T8 Order $$16$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $QD_{16}$

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## Group action invariants

 Degree $n$ : $8$ Transitive number $t$ : $8$ Group : $QD_{16}$ CHM label : $2D_{8}(8)=[D(4)]2$ Parity: $-1$ Primitive: No Nilpotency class: $3$ Generators: (1,2,3,4,5,6,7,8), (1,3)(2,6)(5,7) $|\Aut(F/K)|$: $2$

## 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

16T12

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

## Group invariants

 Order: $16=2^{4}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [16, 8]
 Character table:  2 4 2 3 2 3 4 3 1a 2a 8a 4a 4b 2b 8b 2P 1a 1a 4b 2b 2b 1a 4b 3P 1a 2a 8a 4a 4b 2b 8b 5P 1a 2a 8b 4a 4b 2b 8a 7P 1a 2a 8b 4a 4b 2b 8a 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 . . -2 -A X.7 2 . -A . . -2 A A = -E(8)-E(8)^3 = -Sqrt(-2) = -i2