Properties

Label 16T17
Order \(32\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_4^2:C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $17$
Group :  $C_4^2:C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,2)(3,12)(4,11)(5,6)(7,16)(8,15)(9,10)(13,14), (1,15,2,16)(3,5,4,6)(7,9,8,10)(11,13,12,14), (1,14,10,6)(2,13,9,5)(3,7,11,15)(4,8,12,16)
$|\Aut(F/K)|$:  $8$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_4$ x 4, $C_2^2$ x 7
8:  $C_4\times C_2$ x 6, $C_2^3$
16:  $Q_8:C_2$ x 2, $C_4\times C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 8: $C_4\times C_2$, $Q_8:C_2$ x 2

Low degree siblings

16T17, 32T3

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 24]
Character table:   
      2  5  4  5  4  4  4  4  4  5  4  5  4  4  4  4  4  5  5  5  5

        1a 2a 2b 2c 4a 4b 4c 4d 4e 4f 4g 4h 4i 4j 4k 4l 2d 2e 4m 4n
     2P 1a 1a 1a 1a 2b 2d 2b 2d 2e 2e 2e 2e 2d 2b 2d 2b 1a 1a 2e 2e
     3P 1a 2a 2b 2c 4c 4d 4a 4b 4m 4f 4n 4h 4k 4l 4i 4j 2d 2e 4e 4g

X.1      1  1  1  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  1  1 -1 -1
X.3      1 -1  1 -1 -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  1  1 -1 -1
X.5      1 -1  1 -1  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  1  1 -1 -1
X.7      1  1  1  1 -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  1  1 -1 -1
X.9      1 -1 -1  1  A -A -A  A -1  1  1 -1  A -A -A  A -1  1 -1  1
X.10     1 -1 -1  1 -A  A  A -A -1  1  1 -1 -A  A  A -A -1  1 -1  1
X.11     1 -1 -1  1  A -A -A  A  1 -1 -1  1 -A  A  A -A -1  1  1 -1
X.12     1 -1 -1  1 -A  A  A -A  1 -1 -1  1  A -A -A  A -1  1  1 -1
X.13     1  1 -1 -1  A  A -A -A -1 -1  1  1  A  A -A -A -1  1 -1  1
X.14     1  1 -1 -1 -A -A  A  A -1 -1  1  1 -A -A  A  A -1  1 -1  1
X.15     1  1 -1 -1  A  A -A -A  1  1 -1 -1 -A -A  A  A -1  1  1 -1
X.16     1  1 -1 -1 -A -A  A  A  1  1 -1 -1  A  A -A -A -1  1  1 -1
X.17     2  . -2  .  .  .  .  .  B  . -B  .  .  .  .  .  2 -2 -B  B
X.18     2  . -2  .  .  .  .  . -B  .  B  .  .  .  .  .  2 -2  B -B
X.19     2  .  2  .  .  .  .  .  B  .  B  .  .  .  .  . -2 -2 -B -B
X.20     2  .  2  .  .  .  .  . -B  . -B  .  .  .  .  . -2 -2  B  B

A = -E(4)
  = -Sqrt(-1) = -i
B = -2*E(4)
  = -2*Sqrt(-1) = -2i