# Properties

 Label 16T44 Degree $16$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $D_8:C_2$

# Related objects

## Group action invariants

 Degree $n$: $16$ Transitive number $t$: $44$ Group: $D_8:C_2$ Parity: $1$ Primitive: no Nilpotency class: $3$ $|\Aut(F/K)|$: $4$ Generators: (1,7,16,9,2,8,15,10)(3,13,6,12,4,14,5,11), (1,8)(2,7)(3,14)(4,13)(5,11)(6,12)(9,16)(10,15), (1,14,2,13)(3,7,4,8)(5,9,6,10)(11,16,12,15)

## 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, $C_2^3$
$16$:  $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4\times C_2$

## Low degree siblings

16T44, 16T47, 32T26

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 42]
 Character table:  2 5 3 5 3 5 5 4 4 3 4 4 3 4 4 1a 2a 2b 4a 4b 4c 2c 8a 2d 8b 8c 4d 8d 4e 2P 1a 1a 1a 2b 2b 2b 1a 4e 1a 4e 4e 2b 4e 2b 3P 1a 2a 2b 4a 4c 4b 2c 8b 2d 8a 8c 4d 8d 4e 5P 1a 2a 2b 4a 4b 4c 2c 8b 2d 8a 8d 4d 8c 4e 7P 1a 2a 2b 4a 4c 4b 2c 8a 2d 8b 8d 4d 8c 4e X.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 X.3 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 X.5 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 X.7 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 X.9 2 . 2 . -2 -2 2 . . . . . . -2 X.10 2 . 2 . 2 2 -2 . . . . . . -2 X.11 2 . -2 . A -A . B . -B C . -C . X.12 2 . -2 . A -A . -B . B -C . C . X.13 2 . -2 . -A A . B . -B -C . C . X.14 2 . -2 . -A A . -B . B C . -C . A = -2*E(4) = -2*Sqrt(-1) = -2i B = -E(8)+E(8)^3 = -Sqrt(2) = -r2 C = E(8)+E(8)^3 = Sqrt(-2) = i2