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Properties

Label 16T681
Order \(256\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_4:D_4.D_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $681$
Group :  $C_4:D_4.D_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $6$
Generators:  (1,13,6,9)(2,14,5,10)(3,15)(4,16)(7,11,8,12), (1,14,8,11,6,9,3,15,2,13,7,12,5,10,4,16)
$|\Aut(F/K)|$:  $2$

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 $
64:  $((C_8 : C_2):C_2):C_2$
128:  16T397

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $(((C_4 \times C_2): C_2):C_2):C_2$

Low degree siblings

16T680 x 2, 16T681, 32T3292, 32T3293 x 2, 32T3294, 32T3295, 32T3296, 32T7423, 32T7428

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 $ $2$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $16$ $4$ $( 5, 6)( 7, 8)( 9,11,10,12)(13,15,14,16)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $8$ $4$ $( 3, 4)( 7, 8)( 9,13,10,14)(11,16,12,15)$
$ 8, 4, 1, 1, 1, 1 $ $16$ $8$ $( 3, 7, 4, 8)( 9,11,14,15,10,12,13,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $16$ $2$ $( 3, 7)( 4, 8)( 5, 6)( 9,13)(10,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 8, 4, 2, 2 $ $16$ $8$ $( 1, 2)( 3, 7, 4, 8)( 5, 6)( 9,15,13,11,10,16,14,12)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,15,10,16)(11,14,12,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $16$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,13)(10,14)(11,15)(12,16)$
$ 8, 8 $ $8$ $8$ $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,11,14,16,10,12,13,15)$
$ 8, 8 $ $8$ $8$ $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,12,14,15,10,11,13,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,13,10,14)(11,15,12,16)$
$ 4, 4, 4, 2, 2 $ $32$ $4$ $( 1, 9, 5,13)( 2,10, 6,14)( 3,11, 4,12)( 7,16)( 8,15)$
$ 16 $ $16$ $16$ $( 1, 9, 3,11, 5,14, 8,16, 2,10, 4,12, 6,13, 7,15)$
$ 16 $ $16$ $16$ $( 1, 9, 4,12, 5,14, 7,15, 2,10, 3,11, 6,13, 8,16)$
$ 16 $ $16$ $16$ $( 1, 9, 7,11, 5,13, 3,15, 2,10, 8,12, 6,14, 4,16)$
$ 16 $ $16$ $16$ $( 1, 9, 8,12, 5,13, 4,16, 2,10, 7,11, 6,14, 3,15)$
$ 4, 4, 4, 2, 2 $ $32$ $4$ $( 1, 9, 5,14)( 2,10, 6,13)( 3,15, 4,16)( 7,12)( 8,11)$

Group invariants

Order:  $256=2^{8}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [256, 511]
Character table:    
      2  8  7  4  5  4  4  8  4  4  4  5  5  6  3   4   4   4   4  3

        1a 2a 4a 4b 8a 2b 2c 8b 4c 2d 8c 8d 4d 4e 16a 16b 16c 16d 4f
     2P 1a 1a 2a 2a 4b 1a 1a 4b 2a 1a 4d 4d 2c 2b  8c  8c  8d  8d 2b
     3P 1a 2a 4a 4b 8a 2b 2c 8b 4c 2d 8d 8c 4d 4f 16c 16d 16a 16b 4e
     5P 1a 2a 4a 4b 8a 2b 2c 8b 4c 2d 8c 8d 4d 4e 16a 16b 16c 16d 4f
     7P 1a 2a 4a 4b 8a 2b 2c 8b 4c 2d 8d 8c 4d 4f 16c 16d 16a 16b 4e
    11P 1a 2a 4a 4b 8a 2b 2c 8b 4c 2d 8d 8c 4d 4f 16c 16d 16a 16b 4e
    13P 1a 2a 4a 4b 8a 2b 2c 8b 4c 2d 8c 8d 4d 4e 16a 16b 16c 16d 4f

X.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
X.3      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
X.5      1  1 -1  1  1 -1  1  1  1 -1 -1 -1  1  B  -B  -B   B   B -B
X.6      1  1 -1  1  1 -1  1  1  1 -1 -1 -1  1 -B   B   B  -B  -B  B
X.7      1  1  1  1 -1 -1  1 -1  1  1 -1 -1  1  B   B   B  -B  -B -B
X.8      1  1  1  1 -1 -1  1 -1  1  1 -1 -1  1 -B  -B  -B   B   B  B
X.9      2  2  .  2  .  2  2  . -2  . -2 -2  2  .   .   .   .   .  .
X.10     2  2  .  2  . -2  2  . -2  .  2  2  2  .   .   .   .   .  .
X.11     4  4 -2  .  .  .  4  .  .  2  .  . -4  .   .   .   .   .  .
X.12     4  4  2  .  .  .  4  .  . -2  .  . -4  .   .   .   .   .  .
X.13     4  4  . -4  .  .  4  .  .  .  .  .  4  .   .   .   .   .  .
X.14     4 -4  .  .  .  .  4  .  .  .  A -A  .  .   C  -C  /C -/C  .
X.15     4 -4  .  .  .  .  4  .  .  . -A  A  .  .  /C -/C   C  -C  .
X.16     4 -4  .  .  .  .  4  .  .  .  A -A  .  .  -C   C -/C  /C  .
X.17     4 -4  .  .  .  .  4  .  .  . -A  A  .  . -/C  /C  -C   C  .
X.18     8  .  .  . -2  . -8  2  .  .  .  .  .  .   .   .   .   .  .
X.19     8  .  .  .  2  . -8 -2  .  .  .  .  .  .   .   .   .   .  .

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