# Properties

 Label 16T12 Degree $16$ Order $16$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $QD_{16}$

# Related objects

## Group action invariants

 Degree $n$: $16$ Transitive number $t$: $12$ Group: $QD_{16}$ Parity: $1$ Primitive: no Nilpotency class: $3$ $|\Aut(F/K)|$: $16$ Generators: (1,6)(2,5)(3,11)(4,12)(7,8)(9,13)(10,14)(15,16), (1,3,5,7,9,12,14,15)(2,4,6,8,10,11,13,16)

## 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: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $D_4$, $QD_{16}$

## Low degree siblings

8T8

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2$ $4$ $2$ $( 1, 2)( 3, 8)( 4, 7)( 5,13)( 6,14)( 9,10)(11,15)(12,16)$ $8, 8$ $2$ $8$ $( 1, 3, 5, 7, 9,12,14,15)( 2, 4, 6, 8,10,11,13,16)$ $4, 4, 4, 4$ $4$ $4$ $( 1, 4, 9,11)( 2, 3,10,12)( 5,16,14, 8)( 6,15,13, 7)$ $4, 4, 4, 4$ $2$ $4$ $( 1, 5, 9,14)( 2, 6,10,13)( 3, 7,12,15)( 4, 8,11,16)$ $2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14)( 6,13)( 7,15)( 8,16)$ $8, 8$ $2$ $8$ $( 1,12, 5,15, 9, 3,14, 7)( 2,11, 6,16,10, 4,13, 8)$

## 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