# Properties

 Label 16T41 Degree $16$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_2^3.C_4$

# Related objects

## Group action invariants

 Degree $n$: $16$ Transitive number $t$: $41$ Group: $C_2^3.C_4$ Parity: $1$ Primitive: no Nilpotency class: $3$ $|\Aut(F/K)|$: $4$ Generators: (3,12)(4,11)(5,13)(6,14)(7,8)(15,16), (1,3,6,7,9,11,14,15)(2,4,5,8,10,12,13,16)

## 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: $C_2^2:C_4$, $(C_8:C_2):C_2$

## Low degree siblings

8T16 x 2, 16T36, 16T41, 32T22

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

## Group invariants

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