# Properties

 Label 16T19 Degree $16$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_4 \times D_4$

# Related objects

## Group action invariants

 Degree $n$: $16$ Transitive number $t$: $19$ Group: $C_4 \times D_4$ Parity: $1$ Primitive: no Nilpotency class: $2$ $|\Aut(F/K)|$: $8$ Generators: (1,11,2,12)(3,9,4,10)(5,7,6,8)(13,15,14,16), (1,6,2,5)(3,16,4,15)(7,11,8,12)(9,13,10,14), (1,13,16,11)(2,14,15,12)(3,7,5,9)(4,8,6,10)

## 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
$8$:  $D_{4}$ x 2, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$, $Q_8:C_2$, $C_4\times C_2^2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 2

Degree 8: $C_4\times C_2$, $D_4\times C_2$, $Q_8:C_2$

## Low degree siblings

16T19 x 3, 32T5

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 25]
 Character table:  2 5 4 5 4 4 5 4 5 5 5 4 4 4 4 4 4 4 4 5 5 1a 2a 2b 2c 4a 4b 4c 4d 4e 4f 2d 4g 2e 4h 4i 4j 4k 4l 2f 2g 2P 1a 1a 1a 1a 2b 2b 2b 2b 2b 2b 1a 2f 1a 2f 2g 2b 2g 2b 1a 1a 3P 1a 2a 2b 2c 4c 4d 4a 4b 4f 4e 2d 4g 2e 4h 4k 4l 4i 4j 2f 2g 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 A -A -1 1 1 -1 -A A A -A 1 -1 X.10 1 -1 -1 1 -A A A -A -A A -1 1 1 -1 A -A -A A 1 -1 X.11 1 -1 -1 1 A -A -A A A -A 1 -1 -1 1 A -A -A A 1 -1 X.12 1 -1 -1 1 -A A A -A -A A 1 -1 -1 1 -A A A -A 1 -1 X.13 1 1 -1 -1 A A -A -A -A A -1 -1 1 1 -A -A A A 1 -1 X.14 1 1 -1 -1 -A -A A A A -A -1 -1 1 1 A A -A -A 1 -1 X.15 1 1 -1 -1 A A -A -A -A A 1 1 -1 -1 A A -A -A 1 -1 X.16 1 1 -1 -1 -A -A A A A -A 1 1 -1 -1 -A -A A A 1 -1 X.17 2 . 2 . . -2 . -2 2 2 . . . . . . . . -2 -2 X.18 2 . 2 . . 2 . 2 -2 -2 . . . . . . . . -2 -2 X.19 2 . -2 . . B . -B B -B . . . . . . . . -2 2 X.20 2 . -2 . . -B . B -B B . . . . . . . . -2 2 A = -E(4) = -Sqrt(-1) = -i B = -2*E(4) = -2*Sqrt(-1) = -2i