# Properties

 Label 16T11 Degree $16$ Order $16$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $Q_8 : C_2$

# Related objects

## Group action invariants

 Degree $n$: $16$ Transitive number $t$: $11$ Group: $Q_8 : C_2$ Parity: $1$ Primitive: no Nilpotency class: $2$ $|\Aut(F/K)|$: $16$ Generators: (1,9)(2,10)(3,4)(5,13)(6,14)(7,8)(11,12)(15,16), (1,5,10,14)(2,6,9,13)(3,7,12,16)(4,8,11,15), (1,7,10,16)(2,8,9,15)(3,5,12,14)(4,6,11,13)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $C_2^3$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

Degree 8: $C_2^3$, $Q_8:C_2$ x 3

## Low degree siblings

8T11 x 3

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

## 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