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Properties

Label 16T175
Order \(64\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $D_4.D_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $175$
Group :  $D_4.D_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,8,10,13,2,7,9,14)(3,5,11,16,4,6,12,15), (1,15)(2,16)(3,13)(4,14)(5,9)(6,10)(7,12)(8,11), (1,4,2,3)(5,14)(6,13)(7,15)(8,16)(9,11,10,12)
$|\Aut(F/K)|$:  $4$

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 6, $C_2^3$
16:  $D_4\times C_2$ x 3
32:  $C_2^2 \wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4$

Low degree siblings

16T138, 16T145 x 2, 16T175, 32T151, 32T160 x 2, 32T161 x 2

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

Group invariants

Order:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [64, 136]
Character table:    
      2  6  4  5  6  3  4  4  4  4  4  4  3  3  4  5  5

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

X.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
X.3      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
X.5      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
X.7      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
X.9      2 -2  2  2  .  .  .  .  .  .  .  .  .  2 -2 -2
X.10     2  2  2  2  .  .  .  .  .  .  .  .  . -2 -2 -2
X.11     2  . -2  2  .  .  . -2  .  2  .  .  .  . -2  2
X.12     2  . -2  2  .  .  .  . -2  .  2  .  .  .  2 -2
X.13     2  . -2  2  .  .  .  .  2  . -2  .  .  .  2 -2
X.14     2  . -2  2  .  .  .  2  . -2  .  .  .  . -2  2
X.15     4  .  . -4  .  A -A  .  .  .  .  .  .  .  .  .
X.16     4  .  . -4  . -A  A  .  .  .  .  .  .  .  .  .

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