# Properties

 Label 16T289 Degree $16$ Order $128$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_4.D_4:C_4$

# Related objects

## Group action invariants

 Degree $n$: $16$ Transitive number $t$: $289$ Group: $C_4.D_4:C_4$ Parity: $-1$ Primitive: no Nilpotency class: $4$ $|\Aut(F/K)|$: $8$ Generators: (1,2)(3,4)(5,6)(7,8)(9,15,13,12,10,16,14,11), (1,15,5,11,2,16,6,12)(3,9,7,14,4,10,8,13)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_8$ x 2, $C_4\times C_2$
$16$:  $D_{8}$, $C_8:C_2$, $QD_{16}$, $C_2^2:C_4$, $C_8\times C_2$
$32$:  $C_4\wr C_2$, $C_2^2 : C_8$, 16T26
$64$:  32T272

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $C_4\wr C_2$

## Low degree siblings

16T289 x 3, 32T667 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $2$ $2$ $( 9,10)(11,12)(13,14)(15,16)$ $8, 1, 1, 1, 1, 1, 1, 1, 1$ $2$ $8$ $( 9,11,14,16,10,12,13,15)$ $8, 1, 1, 1, 1, 1, 1, 1, 1$ $2$ $8$ $( 9,12,14,15,10,11,13,16)$ $4, 4, 1, 1, 1, 1, 1, 1, 1, 1$ $2$ $4$ $( 9,13,10,14)(11,15,12,16)$ $4, 4, 1, 1, 1, 1, 1, 1, 1, 1$ $2$ $4$ $( 9,14,10,13)(11,16,12,15)$ $8, 1, 1, 1, 1, 1, 1, 1, 1$ $2$ $8$ $( 9,15,13,12,10,16,14,11)$ $8, 1, 1, 1, 1, 1, 1, 1, 1$ $2$ $8$ $( 9,16,13,11,10,15,14,12)$ $2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ $8, 2, 2, 2, 2$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,11,14,16,10,12,13,15)$ $8, 2, 2, 2, 2$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12,14,15,10,11,13,16)$ $4, 4, 2, 2, 2, 2$ $2$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,13,10,14)(11,15,12,16)$ $4, 4, 2, 2, 2, 2$ $2$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,14,10,13)(11,16,12,15)$ $8, 2, 2, 2, 2$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,15,13,12,10,16,14,11)$ $8, 2, 2, 2, 2$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,16,13,11,10,15,14,12)$ $8, 8$ $1$ $8$ $( 1, 3, 5, 7, 2, 4, 6, 8)( 9,11,14,16,10,12,13,15)$ $8, 8$ $2$ $8$ $( 1, 3, 5, 7, 2, 4, 6, 8)( 9,12,14,15,10,11,13,16)$ $8, 4, 4$ $2$ $8$ $( 1, 3, 5, 7, 2, 4, 6, 8)( 9,13,10,14)(11,15,12,16)$ $8, 4, 4$ $2$ $8$ $( 1, 3, 5, 7, 2, 4, 6, 8)( 9,14,10,13)(11,16,12,15)$ $8, 8$ $2$ $8$ $( 1, 3, 5, 7, 2, 4, 6, 8)( 9,15,13,12,10,16,14,11)$ $8, 8$ $2$ $8$ $( 1, 3, 5, 7, 2, 4, 6, 8)( 9,16,13,11,10,15,14,12)$ $8, 8$ $1$ $8$ $( 1, 4, 5, 8, 2, 3, 6, 7)( 9,12,14,15,10,11,13,16)$ $8, 4, 4$ $2$ $8$ $( 1, 4, 5, 8, 2, 3, 6, 7)( 9,13,10,14)(11,15,12,16)$ $8, 4, 4$ $2$ $8$ $( 1, 4, 5, 8, 2, 3, 6, 7)( 9,14,10,13)(11,16,12,15)$ $8, 8$ $2$ $8$ $( 1, 4, 5, 8, 2, 3, 6, 7)( 9,15,13,12,10,16,14,11)$ $8, 8$ $2$ $8$ $( 1, 4, 5, 8, 2, 3, 6, 7)( 9,16,13,11,10,15,14,12)$ $4, 4, 4, 4$ $2$ $4$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,13,10,14)(11,15,12,16)$ $4, 4, 4, 4$ $1$ $4$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,14,10,13)(11,16,12,15)$ $8, 4, 4$ $2$ $8$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,15,13,12,10,16,14,11)$ $8, 4, 4$ $2$ $8$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,16,13,11,10,15,14,12)$ $4, 4, 4, 4$ $1$ $4$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,13,10,14)(11,15,12,16)$ $8, 4, 4$ $2$ $8$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,15,13,12,10,16,14,11)$ $8, 4, 4$ $2$ $8$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,16,13,11,10,15,14,12)$ $8, 8$ $2$ $8$ $( 1, 7, 6, 3, 2, 8, 5, 4)( 9,15,13,12,10,16,14,11)$ $8, 8$ $1$ $8$ $( 1, 7, 6, 3, 2, 8, 5, 4)( 9,16,13,11,10,15,14,12)$ $8, 8$ $1$ $8$ $( 1, 8, 6, 4, 2, 7, 5, 3)( 9,15,13,12,10,16,14,11)$ $2, 2, 2, 2, 2, 2, 2, 2$ $8$ $2$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,14)( 6,13)( 7,16)( 8,15)$ $4, 4, 4, 4$ $8$ $4$ $( 1, 9, 2,10)( 3,11, 4,12)( 5,14, 6,13)( 7,16, 8,15)$ $16$ $8$ $16$ $( 1, 9, 3,11, 5,14, 7,16, 2,10, 4,12, 6,13, 8,15)$ $16$ $8$ $16$ $( 1, 9, 4,12, 5,14, 8,15, 2,10, 3,11, 6,13, 7,16)$ $8, 8$ $8$ $8$ $( 1, 9, 5,14, 2,10, 6,13)( 3,11, 7,16, 4,12, 8,15)$ $8, 8$ $8$ $8$ $( 1, 9, 6,13, 2,10, 5,14)( 3,11, 8,15, 4,12, 7,16)$ $16$ $8$ $16$ $( 1, 9, 7,16, 6,13, 3,11, 2,10, 8,15, 5,14, 4,12)$ $16$ $8$ $16$ $( 1, 9, 8,15, 6,13, 4,12, 2,10, 7,16, 5,14, 3,11)$

## Group invariants

 Order: $128=2^{7}$ Cyclic: no Abelian: no Solvable: yes GAP id: [128, 67]
 Character table: not available.