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Properties

Label 16T385
Order \(128\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_4.C_2^2\wr C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $385$
Group :  $C_4.C_2^2\wr C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,8,11,16,4,5,10,13,2,7,12,15,3,6,9,14), (1,3,2,4)(5,7,6,8)(9,11,10,12)(13,16,14,15), (1,10,2,9)(3,12,4,11)(5,8)(6,7)(15,16)
$|\Aut(F/K)|$:  $2$

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:  $D_{8}$ x 2, $D_4\times C_2$ x 3
32:  $Z_8 : Z_8^\times$, $C_2^2 \wr C_2$, 16T29
64:  16T126

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $D_{8}$

Low degree siblings

16T385, 32T868, 32T869, 32T870, 32T1523 x 2, 32T1937

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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

Group invariants

Order:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [128, 922]
Character table:    
      2  7  5  6  3  3  7  5  6  6  4  4  4  4   4   4   4   4  5  4  5

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

X.1      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  1
X.3      1 -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  1
X.5      1 -1  1  1 -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   1  1  1  1
X.7      1  1  1 -1 -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  -1  1  1  1
X.9      2 -2  2  .  .  2 -2  2  2  .  .  .  .   .   .   .   . -2  2 -2
X.10     2  2  2  .  .  2  2  2  2  .  .  .  .   .   .   .   . -2 -2 -2
X.11     2  . -2  .  .  2  . -2  2 -2  .  2  .   .   .   .   . -2  .  2
X.12     2  . -2  .  .  2  . -2  2  . -2  .  2   .   .   .   .  2  . -2
X.13     2  . -2  .  .  2  . -2  2  .  2  . -2   .   .   .   .  2  . -2
X.14     2  . -2  .  .  2  . -2  2  2  . -2  .   .   .   .   . -2  .  2
X.15     2 -2  2  .  .  2  2 -2 -2  .  .  .  .   A  -A   A  -A  .  .  .
X.16     2 -2  2  .  .  2  2 -2 -2  .  .  .  .  -A   A  -A   A  .  .  .
X.17     2  2  2  .  .  2 -2 -2 -2  .  .  .  .   A   A  -A  -A  .  .  .
X.18     2  2  2  .  .  2 -2 -2 -2  .  .  .  .  -A  -A   A   A  .  .  .
X.19     4  . -4  .  .  4  .  4 -4  .  .  .  .   .   .   .   .  .  .  .
X.20     8  .  .  .  . -8  .  .  .  .  .  .  .   .   .   .   .  .  .  .

A = -E(8)+E(8)^3
  = -Sqrt(2) = -r2