# Properties

 Label 16T43 Degree $16$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_2^2:D_4$

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## Group action invariants

 Degree $n$: $16$ Transitive number $t$: $43$ Group: $C_2^2:D_4$ Parity: $1$ Primitive: no Nilpotency class: $2$ $|\Aut(F/K)|$: $8$ Generators: (1,16)(2,15)(3,5)(4,6)(7,8)(9,10)(11,12)(13,14), (1,10,15,7)(2,9,16,8)(3,11,6,14)(4,12,5,13), (1,3,2,4)(5,15,6,16)(7,12,8,11)(9,14,10,13)

## 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 4, $C_2^3$
$16$:  $D_4\times C_2$ x 2, $Q_8:C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

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

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

## Low degree siblings

16T34 x 2, 16T43, 32T20

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 28]
 Character table:  2 5 4 5 4 4 4 4 4 3 3 3 3 5 5 1a 2a 2b 2c 4a 4b 4c 4d 2d 4e 4f 2e 2f 2g 2P 1a 1a 1a 1a 2b 2b 2b 2b 1a 2f 2f 1a 1a 1a 3P 1a 2a 2b 2c 4c 4b 4a 4d 2d 4e 4f 2e 2f 2g 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 . . -2 . 2 . . . . -2 -2 X.12 2 . 2 . . 2 . -2 . . . . -2 -2 X.13 2 . -2 . A . -A . . . . . -2 2 X.14 2 . -2 . -A . A . . . . . -2 2 A = -2*E(4) = -2*Sqrt(-1) = -2i