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Properties

Label 16T400
Order \(128\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2.C_2\wr C_2^2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $400$
Group :  $C_2.C_2\wr C_2^2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,12)(2,11)(3,10)(4,9)(5,13)(6,14)(7,15)(8,16), (1,6,2,5)(3,7,4,8)(9,15)(10,16)(11,14)(12,13), (1,7,2,8)(3,5,4,6)(9,11,10,12)(13,16,14,15)
$|\Aut(F/K)|$:  $4$

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:  $C_2^2 \wr C_2$
64:  $(((C_4 \times C_2): C_2):C_2):C_2$

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

16T345 x 2, 16T399 x 2, 16T400, 16T410 x 2, 32T789, 32T790, 32T791 x 2, 32T888, 32T889, 32T890 x 2, 32T891, 32T892, 32T893, 32T912, 32T913, 32T1570, 32T1695

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

Group invariants

Order:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [128, 929]
Character table:    
      2  7  5  6  5  5  7  4  4  5  5  5  5  4  5  5  3  3  4  3  4

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

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  .  .  .  .  .
X.10     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     4  . -4  .  .  4  .  .  .  .  .  .  .  .  .  .  . -2  .  2
X.16     4  . -4  .  .  4  .  .  .  .  .  .  .  .  .  .  .  2  . -2
X.17     4 -2  .  .  2 -4  .  .  A -A -A  A  .  .  .  .  .  .  .  .
X.18     4 -2  .  .  2 -4  .  . -A  A  A -A  .  .  .  .  .  .  .  .
X.19     4  2  .  . -2 -4  .  .  A  A -A -A  .  .  .  .  .  .  .  .
X.20     4  2  .  . -2 -4  .  . -A -A  A  A  .  .  .  .  .  .  .  .

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