# Properties

 Label 16T5 Degree $16$ Order $16$ Cyclic no Abelian yes Solvable yes Primitive no $p$-group yes Group: $C_8\times C_2$

# Related objects

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_8$ x 2, $C_4\times C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $C_8$ x 2, $C_4\times C_2$

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

## Group invariants

 Order: $16=2^{4}$ Cyclic: no Abelian: yes Solvable: yes GAP id: [16, 5]
 Character table:  2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 1a 2a 8a 8b 4a 4b 8c 8d 2b 2c 8e 8f 4c 4d 8g 8h 2P 1a 1a 4a 4a 2b 2b 4c 4c 1a 1a 4a 4a 2b 2b 4c 4c 3P 1a 2a 8c 8d 4c 4d 8a 8b 2b 2c 8h 8g 4a 4b 8f 8e 5P 1a 2a 8f 8e 4a 4b 8g 8h 2b 2c 8b 8a 4c 4d 8c 8d 7P 1a 2a 8g 8h 4c 4d 8f 8e 2b 2c 8d 8c 4a 4b 8a 8b X.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 X.3 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 X.5 1 -1 A -A -1 1 -A A 1 -1 -A A -1 1 -A A X.6 1 -1 -A A -1 1 A -A 1 -1 A -A -1 1 A -A X.7 1 -1 B -B -A A -/B /B -1 1 B -B A -A /B -/B X.8 1 -1 -/B /B A -A B -B -1 1 -/B /B -A A -B B X.9 1 -1 /B -/B A -A -B B -1 1 /B -/B -A A B -B X.10 1 -1 -B B -A A /B -/B -1 1 -B B A -A -/B /B X.11 1 1 A A -1 -1 -A -A 1 1 A A -1 -1 -A -A X.12 1 1 -A -A -1 -1 A A 1 1 -A -A -1 -1 A A X.13 1 1 B B -A -A -/B -/B -1 -1 -B -B A A /B /B X.14 1 1 -/B -/B A A B B -1 -1 /B /B -A -A -B -B X.15 1 1 /B /B A A -B -B -1 -1 -/B -/B -A -A B B X.16 1 1 -B -B -A -A /B /B -1 -1 B B A A -/B -/B A = -E(4) = -Sqrt(-1) = -i B = -E(8)