Properties

 Label 16T27 Degree $16$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_4^2:C_2$

Related objects

Group action invariants

 Degree $n$: $16$ Transitive number $t$: $27$ Group: $C_4^2:C_2$ Parity: $1$ Primitive: no Nilpotency class: $2$ $|\Aut(F/K)|$: $4$ Generators: (1,5,9,14)(2,6,10,13)(3,16,12,7)(4,15,11,8), (1,2)(3,12)(4,11)(7,15)(8,16)(9,10), (1,4,2,3)(5,8,6,7)(9,11,10,12)(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
$8$:  $C_2^3$
$16$:  $Q_8:C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 8: $Q_8:C_2$ x 3

Low degree siblings

32T13

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

Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 33]
 Character table: 2 5 3 5 3 4 4 4 3 3 4 4 5 5 4 1a 2a 2b 4a 4b 4c 4d 4e 4f 4g 4h 2c 2d 4i 2P 1a 1a 1a 2b 2c 2c 2d 2c 2d 2b 2b 1a 1a 2d 3P 1a 2a 2b 4a 4c 4b 4i 4e 4f 4h 4g 2c 2d 4d 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 . A -A . . . . . -2 2 . X.10 2 . -2 . -A A . . . . . -2 2 . X.11 2 . -2 . . . . . . A -A 2 -2 . X.12 2 . -2 . . . . . . -A A 2 -2 . X.13 2 . 2 . . . A . . . . -2 -2 -A X.14 2 . 2 . . . -A . . . . -2 -2 A A = -2*E(4) = -2*Sqrt(-1) = -2i