# Properties

 Label 16T3 Degree $16$ Order $16$ Cyclic no Abelian yes Solvable yes Primitive no $p$-group yes Group: $C_2^4$

# Related objects

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_2^2$ x 35
$8$:  $C_2^3$ x 15

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 15

Degree 4: $C_2^2$ x 35

Degree 8: $C_2^3$ x 15

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

## Group invariants

 Order: $16=2^{4}$ Cyclic: no Abelian: yes Solvable: yes GAP id: [16, 14]
 Character table:  2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 1a 2a 2b 2c 2d 2e 2f 2g 2h 2i 2j 2k 2l 2m 2n 2o 2P 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 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 -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 X.7 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 X.9 1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 X.10 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 X.11 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 X.12 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 X.13 1 1 -1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 X.14 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 X.15 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 X.16 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1