# Properties

 Label 16T9 Degree $16$ Order $16$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $D_4\times C_2$

# Related objects

## Group action invariants

 Degree $n$: $16$ Transitive number $t$: $9$ Group: $D_4\times C_2$ Parity: $1$ Primitive: no Nilpotency class: $2$ $|\Aut(F/K)|$: $16$ Generators: (1,3)(2,4)(5,15)(6,16)(7,11)(8,12)(9,13)(10,14), (1,11,16,14)(2,12,15,13)(3,10,6,7)(4,9,5,8), (1,13)(2,14)(3,8)(4,7)(5,10)(6,9)(11,15)(12,16)

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_2^3$, $D_4$ x 2, $D_4\times C_2$ x 4

## Low degree siblings

8T9 x 4

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

## Group invariants

 Order: $16=2^{4}$ Cyclic: no Abelian: no Solvable: yes GAP id: [16, 11]
 Character table:  2 4 3 3 4 4 3 3 3 3 4 1a 2a 2b 2c 2d 2e 4a 4b 2f 2g 2P 1a 1a 1a 1a 1a 1a 2g 2g 1a 1a 3P 1a 2a 2b 2c 2d 2e 4a 4b 2f 2g 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 . . 2 -2 . . . . -2 X.10 2 . . -2 2 . . . . -2