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Properties

Label 16T28
Order \(32\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_4\wr C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $28$
Group :  $C_4\wr C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,13,2,14)(3,15,4,16)(5,10,6,9)(7,11,8,12), (1,7,6,12,2,8,5,11)(3,10,15,13,4,9,16,14)
$|\Aut(F/K)|$:  $4$

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: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 8: $C_2^2:C_4$

Low degree siblings

8T17 x 2, 16T42, 32T14

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 11]
Character table:    
      2  5  3  5  3   4   4  3  5  5   4  3   4  4  4

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

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  A  -A  -A  1 -1 -1   A -A   A  1 -1
X.6      1 -1  1 -A   A   A  1 -1 -1  -A  A  -A  1 -1
X.7      1  1  1  A   A   A -1 -1 -1  -A -A  -A  1 -1
X.8      1  1  1 -A  -A  -A -1 -1 -1   A  A   A  1 -1
X.9      2  .  2  .   .   .  . -2 -2   .  .   . -2  2
X.10     2  .  2  .   .   .  .  2  2   .  .   . -2 -2
X.11     2  . -2  .   B  -B  .  C -C -/B  .  /B  .  .
X.12     2  . -2  .  /B -/B  . -C  C  -B  .   B  .  .
X.13     2  . -2  . -/B  /B  . -C  C   B  .  -B  .  .
X.14     2  . -2  .  -B   B  .  C -C  /B  . -/B  .  .

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