↑

Properties

Label 16T29
Order \(32\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2\times D_8$

Related objects

Learn more about

Group action invariants

Degree $n$:  $16$
Transitive number $t$:  $29$
Group:  $C_2\times D_8$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,9)(2,10)(3,12)(4,11)(5,13)(6,14)(7,15)(8,16), (1,12,5,16,10,4,14,7)(2,11,6,15,9,3,13,8), (1,12)(2,11)(3,9)(4,10)(5,7)(6,8)(13,15)(14,16)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$
16:  $D_{8}$ x 2, $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_{8}$ x 2, $D_4\times C_2$

Low degree siblings

16T29 x 3, 32T15

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 39]
Character table:    
      2  5  3  5  3  3  4  3  4  4  4  4  4  5  5

        1a 2a 2b 2c 2d 8a 2e 8b 4a 4b 8c 8d 2f 2g
     2P 1a 1a 1a 1a 1a 4a 1a 4a 2g 2g 4a 4a 1a 1a
     3P 1a 2a 2b 2c 2d 8d 2e 8c 4a 4b 8b 8a 2f 2g
     5P 1a 2a 2b 2c 2d 8d 2e 8c 4a 4b 8b 8a 2f 2g
     7P 1a 2a 2b 2c 2d 8a 2e 8b 4a 4b 8c 8d 2f 2g

X.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
X.3      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
X.5      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
X.7      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
X.9      2  . -2  .  .  .  .  . -2  2  .  . -2  2
X.10     2  .  2  .  .  .  .  . -2 -2  .  .  2  2
X.11     2  . -2  .  .  A  . -A  .  .  A -A  2 -2
X.12     2  . -2  .  . -A  .  A  .  . -A  A  2 -2
X.13     2  .  2  .  .  A  .  A  .  . -A -A -2 -2
X.14     2  .  2  .  . -A  . -A  .  .  A  A -2 -2

A = -E(8)+E(8)^3
  = -Sqrt(2) = -r2