Properties

Label 16T3
Order \(16\)
n \(16\)
Cyclic No
Abelian Yes
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^4$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $3$
Group :  $C_2^4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $1$
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)
$|\Aut(F/K)|$:  $16$

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 TypeSizeOrderRepresentative
$ 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