Properties

 Label 16T28 Degree $16$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_4\wr C_2$

Related objects

Group action invariants

 Degree $n$: $16$ Transitive number $t$: $28$ Group: $C_4\wr C_2$ Parity: $1$ Primitive: no Nilpotency class: $3$ $|\Aut(F/K)|$: $4$ Generators: (1,13,2,14)(3,15,4,16)(5,10,6,9)(7,11,8,12), (1,7,6,12,2,8,5,11)(3,10,15,13,4,9,16,14)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

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

Degree 8: $C_2^2:C_4$

Low degree siblings

8T17 x 2, 16T42, 32T14

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

Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 11]
 Character table:  2 5 3 5 3 4 4 3 5 5 4 3 4 4 4 1a 2a 2b 8a 4a 4b 4c 4d 4e 4f 8b 4g 4h 2c 2P 1a 1a 1a 4d 2c 2c 2b 2b 2b 2c 4e 2c 2b 1a 3P 1a 2a 2b 8b 4g 4f 4c 4e 4d 4b 8a 4a 4h 2c 5P 1a 2a 2b 8a 4a 4b 4c 4d 4e 4f 8b 4g 4h 2c 7P 1a 2a 2b 8b 4g 4f 4c 4e 4d 4b 8a 4a 4h 2c 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 A -A -A 1 -1 -1 A -A A 1 -1 X.6 1 -1 1 -A A A 1 -1 -1 -A A -A 1 -1 X.7 1 1 1 A A A -1 -1 -1 -A -A -A 1 -1 X.8 1 1 1 -A -A -A -1 -1 -1 A A A 1 -1 X.9 2 . 2 . . . . -2 -2 . . . -2 2 X.10 2 . 2 . . . . 2 2 . . . -2 -2 X.11 2 . -2 . B -B . C -C -/B . /B . . X.12 2 . -2 . /B -/B . -C C -B . B . . X.13 2 . -2 . -/B /B . -C C B . -B . . X.14 2 . -2 . -B B . C -C /B . -/B . . A = -E(4) = -Sqrt(-1) = -i B = -1-E(4) = -1-Sqrt(-1) = -1-i C = 2*E(4) = 2*Sqrt(-1) = 2i