Properties

 Label 16T29 Degree $16$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_2\times D_8$

Related objects

Group action invariants

 Degree $n$: $16$ Transitive number $t$: $29$ Group: $C_2\times D_8$ Parity: $1$ Primitive: no Nilpotency class: $3$ $|\Aut(F/K)|$: $4$ Generators: (1,9)(2,10)(3,12)(4,11)(5,13)(6,14)(7,15)(8,16), (1,12,5,16,10,4,14,7)(2,11,6,15,9,3,13,8), (1,12)(2,11)(3,9)(4,10)(5,7)(6,8)(13,15)(14,16)

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, $C_2^3$
$16$:  $D_{8}$ x 2, $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_{8}$ x 2, $D_4\times C_2$

Low degree siblings

16T29 x 3, 32T15

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

Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 39]
 Character table:  2 5 3 5 3 3 4 3 4 4 4 4 4 5 5 1a 2a 2b 2c 2d 8a 2e 8b 4a 4b 8c 8d 2f 2g 2P 1a 1a 1a 1a 1a 4a 1a 4a 2g 2g 4a 4a 1a 1a 3P 1a 2a 2b 2c 2d 8d 2e 8c 4a 4b 8b 8a 2f 2g 5P 1a 2a 2b 2c 2d 8d 2e 8c 4a 4b 8b 8a 2f 2g 7P 1a 2a 2b 2c 2d 8a 2e 8b 4a 4b 8c 8d 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 . . A . -A . . A -A 2 -2 X.12 2 . -2 . . -A . A . . -A A 2 -2 X.13 2 . 2 . . A . A . . -A -A -2 -2 X.14 2 . 2 . . -A . -A . . A A -2 -2 A = -E(8)+E(8)^3 = -Sqrt(2) = -r2