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Properties

Label 16T394
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 $t$ :  $394$
Group :  $C_2.C_2\wr C_2^2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,11,5,13)(2,12,6,14)(3,9,8,15)(4,10,7,16), (1,8,2,7)(3,5,4,6)(9,12)(10,11)(13,15)(14,16), (1,11,6,14,2,12,5,13)(3,10,7,15,4,9,8,16)
$|\Aut(F/K)|$:  $2$

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:  $D_4\times C_2$ 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

16T342 x 2, 16T365 x 4, 16T381, 16T386, 16T394, 32T782 x 2, 32T783 x 2, 32T784 x 2, 32T833 x 2, 32T834 x 4, 32T835 x 2, 32T862, 32T871 x 2, 32T881 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 $ $2$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 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, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 7, 8)(11,12)(15,16)$
$ 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 $ $8$ $4$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,13,10,14)(11,15,12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,13)(10,14)(11,15)(12,16)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,13,10,14)(11,16,12,15)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,13)(10,14)(11,16)(12,15)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,15,10,16)(11,14,12,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,15)(10,16)(11,13)(12,14)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,15,10,16)(11,13,12,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,15)( 6,16)( 7,14)( 8,13)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 9, 2,10)( 3,11, 4,12)( 5,15, 6,16)( 7,14, 8,13)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,15, 6,16)( 7,13, 8,14)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1,11, 6,14)( 2,12, 5,13)( 3, 9, 7,16)( 4,10, 8,15)$
$ 8, 8 $ $16$ $8$ $( 1,11, 5,13, 2,12, 6,14)( 3,10, 8,16, 4, 9, 7,15)$

Group invariants

Order:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [128, 931]
Character table:    
      2  7  6  5  4  7  4  4  4  4  4  5  5  4  4  3  3  3

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

X.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
X.3      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
X.5      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
X.7      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
X.9      2  2  2 -2  2  .  .  .  .  2 -2 -2  .  .  .  .  .
X.10     2  2  2  2  2  .  .  .  . -2 -2 -2  .  .  .  .  .
X.11     2  2 -2  .  2 -2  2  .  .  .  2 -2  .  .  .  .  .
X.12     2  2 -2  .  2  .  . -2  2  . -2  2  .  .  .  .  .
X.13     2  2 -2  .  2  .  .  2 -2  . -2  2  .  .  .  .  .
X.14     2  2 -2  .  2  2 -2  .  .  .  2 -2  .  .  .  .  .
X.15     4 -4  .  .  4  .  .  .  .  .  .  . -2  2  .  .  .
X.16     4 -4  .  .  4  .  .  .  .  .  .  .  2 -2  .  .  .
X.17     8  .  .  . -8  .  .  .  .  .  .  .  .  .  .  .  .