Properties

Label 16T33
Order \(32\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^2.D_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $33$
Group :  $C_2^2.D_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,11,6,8)(2,12,5,7)(3,10,16,13)(4,9,15,14), (1,9,3,7)(2,10,4,8)(5,13,15,11)(6,14,16,12)
$|\Aut(F/K)|$:  $8$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4$, $C_2^3 : C_4 $, $C_2^3: C_4$

Low degree siblings

8T19 x 2, 8T20, 8T21, 16T33, 16T52, 16T53, 32T19

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 6]
Character table:   
      2  5  4  3  3  4  4  5  3  3  3  3

        1a 2a 4a 4b 2b 2c 2d 4c 4d 2e 4e
     2P 1a 1a 2a 2a 1a 1a 1a 2b 2b 1a 2d
     3P 1a 2a 4b 4a 2b 2c 2d 4d 4c 2e 4e

X.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
X.3      1  1 -1 -1  1  1  1  1  1 -1 -1
X.4      1  1  1  1  1  1  1 -1 -1 -1 -1
X.5      1 -1  A -A -1  1  1  A -A -1  1
X.6      1 -1 -A  A -1  1  1 -A  A -1  1
X.7      1 -1  A -A -1  1  1 -A  A  1 -1
X.8      1 -1 -A  A -1  1  1  A -A  1 -1
X.9      2  2  .  . -2 -2  2  .  .  .  .
X.10     2 -2  .  .  2 -2  2  .  .  .  .
X.11     4  .  .  .  .  . -4  .  .  .  .

A = -E(4)
  = -Sqrt(-1) = -i