# Properties

 Label 16T16 Degree $16$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $(C_8:C_2):C_2$

# Related objects

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $C_4\times C_2$ x 6, $C_2^3$
$16$:  $C_4\times C_2^2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 8: $C_4\times C_2$

## Low degree siblings

16T16 x 2, 32T2

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 38]
 Character table:  2 5 4 5 4 5 5 4 5 5 5 4 5 4 4 4 4 4 4 4 4 1a 2a 2b 8a 8b 8c 4a 4b 4c 8d 8e 8f 8g 8h 4d 2c 8i 8j 2d 4e 2P 1a 1a 1a 4c 4c 4c 2b 2b 2b 4b 4b 4b 4b 4c 2b 1a 4c 4b 1a 2b 3P 1a 2a 2b 8e 8f 8d 4a 4c 4b 8c 8a 8b 8i 8j 4d 2c 8g 8h 2d 4e 5P 1a 2a 2b 8a 8c 8b 4a 4b 4c 8f 8e 8d 8g 8h 4d 2c 8i 8j 2d 4e 7P 1a 2a 2b 8e 8d 8f 4a 4c 4b 8b 8a 8c 8i 8j 4d 2c 8g 8h 2d 4e X.1 1 1 1 1 1 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 1 -1 -1 1 1 -1 X.3 1 -1 1 -1 1 1 -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 -1 1 -1 1 -1 1 X.5 1 -1 1 1 -1 -1 -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 1 1 -1 -1 1 1 X.7 1 1 1 -1 -1 -1 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 -1 -1 -1 -1 -1 -1 X.9 1 -1 1 A -A -A 1 -1 -1 A -A A A -A -1 1 -A A 1 -1 X.10 1 -1 1 -A A A 1 -1 -1 -A A -A -A A -1 1 A -A 1 -1 X.11 1 -1 1 A -A -A 1 -1 -1 A -A A -A A 1 -1 A -A -1 1 X.12 1 -1 1 -A A A 1 -1 -1 -A A -A A -A 1 -1 -A A -1 1 X.13 1 1 1 A A A -1 -1 -1 -A -A -A A A -1 -1 -A -A 1 1 X.14 1 1 1 -A -A -A -1 -1 -1 A A A -A -A -1 -1 A A 1 1 X.15 1 1 1 A A A -1 -1 -1 -A -A -A -A -A 1 1 A A -1 -1 X.16 1 1 1 -A -A -A -1 -1 -1 A A A A A 1 1 -A -A -1 -1 X.17 2 . -2 . B -B . C -C /B . -/B . . . . . . . . X.18 2 . -2 . -/B /B . -C C -B . B . . . . . . . . X.19 2 . -2 . /B -/B . -C C B . -B . . . . . . . . X.20 2 . -2 . -B B . C -C -/B . /B . . . . . . . . A = -E(4) = -Sqrt(-1) = -i B = -2*E(8) C = -2*E(4) = -2*Sqrt(-1) = -2i