# Properties

 Label 16T14 Order $$16$$ n $$16$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $Q_{16}$

# Related objects

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $D_4$

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

## Group invariants

 Order: $16=2^{4}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [16, 9]
 Character table:  2 4 4 3 3 3 2 2 1a 2a 8a 8b 4a 4b 4c 2P 1a 1a 4a 4a 2a 2a 2a 3P 1a 2a 8b 8a 4a 4b 4c 5P 1a 2a 8b 8a 4a 4b 4c 7P 1a 2a 8a 8b 4a 4b 4c X.1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 -1 1 X.3 1 1 -1 -1 1 1 -1 X.4 1 1 1 1 1 -1 -1 X.5 2 2 . . -2 . . X.6 2 -2 A -A . . . X.7 2 -2 -A A . . . A = -E(8)+E(8)^3 = -Sqrt(2) = -r2