# Properties

 Label 16T17 Degree $16$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_4^2:C_2$

# Related objects

## Group action invariants

 Degree $n$: $16$ Transitive number $t$: $17$ Group: $C_4^2:C_2$ Parity: $1$ Primitive: no Nilpotency class: $2$ $|\Aut(F/K)|$: $8$ Generators: (1,2)(3,12)(4,11)(5,6)(7,16)(8,15)(9,10)(13,14), (1,15,2,16)(3,5,4,6)(7,9,8,10)(11,13,12,14), (1,14,10,6)(2,13,9,5)(3,7,11,15)(4,8,12,16)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $C_4\times C_2$ x 6, $C_2^3$
$16$:  $Q_8:C_2$ x 2, $C_4\times C_2^2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 8: $C_4\times C_2$, $Q_8:C_2$ x 2

## Low degree siblings

16T17, 32T3

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 24]
 Character table:  2 5 4 5 4 4 4 4 4 5 4 5 4 4 4 4 4 5 5 5 5 1a 2a 2b 2c 4a 4b 4c 4d 4e 4f 4g 4h 4i 4j 4k 4l 2d 2e 4m 4n 2P 1a 1a 1a 1a 2b 2d 2b 2d 2e 2e 2e 2e 2d 2b 2d 2b 1a 1a 2e 2e 3P 1a 2a 2b 2c 4c 4d 4a 4b 4m 4f 4n 4h 4k 4l 4i 4j 2d 2e 4e 4g X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 1 -1 -1 X.3 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 1 1 1 X.4 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 1 -1 -1 X.5 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 1 X.6 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 X.7 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 X.8 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 X.9 1 -1 -1 1 A -A -A A -1 1 1 -1 A -A -A A -1 1 -1 1 X.10 1 -1 -1 1 -A A A -A -1 1 1 -1 -A A A -A -1 1 -1 1 X.11 1 -1 -1 1 A -A -A A 1 -1 -1 1 -A A A -A -1 1 1 -1 X.12 1 -1 -1 1 -A A A -A 1 -1 -1 1 A -A -A A -1 1 1 -1 X.13 1 1 -1 -1 A A -A -A -1 -1 1 1 A A -A -A -1 1 -1 1 X.14 1 1 -1 -1 -A -A A A -1 -1 1 1 -A -A A A -1 1 -1 1 X.15 1 1 -1 -1 A A -A -A 1 1 -1 -1 -A -A A A -1 1 1 -1 X.16 1 1 -1 -1 -A -A A A 1 1 -1 -1 A A -A -A -1 1 1 -1 X.17 2 . -2 . . . . . B . -B . . . . . 2 -2 -B B X.18 2 . -2 . . . . . -B . B . . . . . 2 -2 B -B X.19 2 . 2 . . . . . B . B . . . . . -2 -2 -B -B X.20 2 . 2 . . . . . -B . -B . . . . . -2 -2 B B A = -E(4) = -Sqrt(-1) = -i B = -2*E(4) = -2*Sqrt(-1) = -2i