# Properties

 Label 16T13 Degree $16$ Order $16$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $D_{8}$

# Related objects

## Group action invariants

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

## 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: $D_4$, $D_{8}$ x 2

## Low degree siblings

8T6 x 2

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

## Group invariants

 Order: $16=2^{4}$ Cyclic: no Abelian: no Solvable: yes GAP id: [16, 7]
 Character table:  2 4 2 2 3 3 3 4 1a 2a 2b 8a 4a 8b 2c 2P 1a 1a 1a 4a 2c 4a 1a 3P 1a 2a 2b 8b 4a 8a 2c 5P 1a 2a 2b 8b 4a 8a 2c 7P 1a 2a 2b 8a 4a 8b 2c 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 . -A -2 X.7 2 . . -A . A -2 A = -E(8)+E(8)^3 = -Sqrt(2) = -r2