Properties

Label 16T48
Order \(32\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2\times SD_{16}$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $48$
Group :  $C_2\times SD_{16}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,10)(2,9)(3,15)(4,16)(5,6)(7,11)(8,12)(13,14), (1,16,14,12,9,7,6,3)(2,15,13,11,10,8,5,4), (1,3,9,12)(2,4,10,11)(5,15,13,8)(6,16,14,7)
$|\Aut(F/K)|$:  $4$

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:  $QD_{16}$ 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: $QD_{16}$ x 2, $D_4\times C_2$

Low degree siblings

16T48, 32T27

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 40]
Character table:   
      2  5  3  5  3  4  3  4  3  4  4  5  5  4  4

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

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  .  .  . -2  2  A -A
X.12     2  . -2  . -A  .  A  .  .  . -2  2 -A  A
X.13     2  .  2  .  A  .  A  .  .  . -2 -2 -A -A
X.14     2  .  2  . -A  . -A  .  .  . -2 -2  A  A

A = -E(8)-E(8)^3
  = -Sqrt(-2) = -i2