# Properties

 Label 16T38 Degree $16$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_8:C_2^2$

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## Group action invariants

 Degree $n$: $16$ Transitive number $t$: $38$ Group: $C_8:C_2^2$ Parity: $1$ Primitive: no Nilpotency class: $3$ $|\Aut(F/K)|$: $4$ Generators: (1,9)(2,10)(3,4)(5,13)(6,14)(7,8)(11,12)(15,16), (1,4,6,8,10,11,13,15)(2,3,5,7,9,12,14,16), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,16)(14,15)

## 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: $D_4\times C_2$, $Z_8 : Z_8^\times$ x 2

## Low degree siblings

8T15 x 2, 16T35, 16T38, 16T45, 32T21

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 43]
 Character table:  2 5 3 4 3 3 3 3 3 4 4 5 1a 2a 2b 2c 2d 8a 8b 4a 4b 4c 2e 2P 1a 1a 1a 1a 1a 4c 4c 2e 2e 2e 1a 3P 1a 2a 2b 2c 2d 8a 8b 4a 4b 4c 2e 5P 1a 2a 2b 2c 2d 8a 8b 4a 4b 4c 2e 7P 1a 2a 2b 2c 2d 8a 8b 4a 4b 4c 2e X.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 X.3 1 -1 -1 1 1 -1 1 -1 -1 1 1 X.4 1 -1 1 -1 -1 1 1 -1 1 1 1 X.5 1 -1 1 -1 1 -1 -1 1 1 1 1 X.6 1 1 -1 -1 -1 -1 1 1 -1 1 1 X.7 1 1 -1 -1 1 1 -1 -1 -1 1 1 X.8 1 1 1 1 -1 -1 -1 -1 1 1 1 X.9 2 . 2 . . . . . -2 -2 2 X.10 2 . -2 . . . . . 2 -2 2 X.11 4 . . . . . . . . . -4