# Properties

 Label 16T1 Degree $16$ Order $16$ Cyclic yes Abelian yes Solvable yes Primitive no $p$-group yes Group: $C_{16}$

# Related objects

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$8$:  $C_8$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $C_8$

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

## Group invariants

 Order: $16=2^{4}$ Cyclic: yes Abelian: yes Solvable: yes GAP id: [16, 1]
 Character table:  2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 1a 2a 16a 16b 8a 8b 16c 16d 4a 4b 16e 16f 8c 8d 16g 16h 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 A A -1 -1 -A -A 1 1 A A -1 -1 -A -A X.4 1 1 -A -A -1 -1 A A 1 1 -A -A -1 -1 A A X.5 1 1 B B A A -/B -/B -1 -1 -B -B -A -A /B /B X.6 1 1 -B -B A A /B /B -1 -1 B B -A -A -/B -/B 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 C -C B -B -/D /D -A A D -D /B -/B /C -/C X.10 1 -1 -C C B -B /D -/D -A A -D D /B -/B -/C /C X.11 1 -1 D -D -B B /C -/C -A A -C C -/B /B /D -/D X.12 1 -1 -D D -B B -/C /C -A A C -C -/B /B -/D /D X.13 1 -1 -/D /D -/B /B -C C A -A /C -/C -B B -D D X.14 1 -1 /D -/D -/B /B C -C A -A -/C /C -B B D -D X.15 1 -1 -/C /C /B -/B D -D A -A -/D /D B -B -C C X.16 1 -1 /C -/C /B -/B -D D A -A /D -/D B -B C -C A = -E(4) = -Sqrt(-1) = -i B = E(8)^3 C = -E(16)^3 D = E(16)^7