# Properties

 Label 16T8 Order $$16$$ n $$16$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_4:C_4$

# Related objects

## Group action invariants

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

## 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:  $D_{4}$, $C_4\times C_2$, $Q_8$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 2

Degree 8: $C_4\times C_2$, $D_4$, $Q_8$

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

## Group invariants

 Order: $16=2^{4}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [16, 4]
 Character table: 2 4 4 3 4 4 3 3 3 3 3 1a 2a 4a 2b 2c 4b 4c 4d 4e 4f 2P 1a 1a 2c 1a 1a 2c 2a 2c 2a 2c 3P 1a 2a 4f 2b 2c 4d 4c 4b 4e 4a X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 1 1 -1 1 -1 1 -1 X.3 1 1 -1 1 1 1 -1 1 -1 -1 X.4 1 1 1 1 1 -1 -1 -1 -1 1 X.5 1 1 A -1 -1 A -1 -A 1 -A X.6 1 1 -A -1 -1 -A -1 A 1 A X.7 1 1 A -1 -1 -A 1 A -1 -A X.8 1 1 -A -1 -1 A 1 -A -1 A X.9 2 -2 . 2 -2 . . . . . X.10 2 -2 . -2 2 . . . . . A = -E(4) = -Sqrt(-1) = -i