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Properties

Label 16T172
Order \(64\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2\wr C_4$

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $172$
Group :  $C_2\wr C_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,3,5,16,10,12,14,7)(2,4,6,15,9,11,13,8), (1,10)(2,9)(3,15,12,8)(4,16,11,7)
$|\Aut(F/K)|$:  $4$

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:  $C_2^2:C_4$
32:  $C_2^3 : C_4 $

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

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

Degree 8: $C_2^2:C_4$, $(((C_4 \times C_2): C_2):C_2):C_2$ x 2

Low degree siblings

8T27 x 2, 8T28 x 2, 16T130, 16T157 x 2, 16T158 x 2, 16T159 x 2, 16T166, 16T170, 16T171, 32T138 x 2, 32T139, 32T170, 32T176

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

Group invariants

Order:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [64, 32]
Character table:    
      2  6  3  5  4  4  3  3  3  3  4  4  4  6

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

X.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
X.3      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
X.5      1 -1  1 -1  1  A -A -A  A -1  1  1  1
X.6      1 -1  1 -1  1 -A  A  A -A -1  1  1  1
X.7      1  1  1 -1 -1  A  A -A -A -1 -1  1  1
X.8      1  1  1 -1 -1 -A -A  A  A -1 -1  1  1
X.9      2  .  2 -2  .  .  .  .  .  2  . -2  2
X.10     2  .  2  2  .  .  .  .  . -2  . -2  2
X.11     4  . -4  .  .  .  .  .  .  .  .  .  4
X.12     4  .  .  . -2  .  .  .  .  .  2  . -4
X.13     4  .  .  .  2  .  .  .  .  . -2  . -4

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