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Properties

Label 16T126
Order \(64\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $(C_2\times D_8):C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $126$
Group :  $(C_2\times D_8):C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,15,6,3,10,8,14,12)(2,16,5,4,9,7,13,11), (1,13)(2,14)(3,4)(5,10)(6,9)(7,15)(8,16)(11,12), (1,10)(2,9)(3,7)(4,8)(11,15)(12,16)
$|\Aut(F/K)|$:  $4$

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$ x 3

Degree 8: $D_{8}$, $Z_8 : Z_8^\times$, $C_2^2 \wr C_2$

Low degree siblings

16T126 x 7, 32T134 x 4, 32T135 x 2, 32T296

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

Group invariants

Order:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [64, 128]
Character table:    
      2  6  4  5  4  6  4  5  4  3  4  3  4  4  4  5  4  5  6  6

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

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 -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
X.7      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
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
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     2  . -2  . -2  .  2  .  .  A  . -A -A  A  .  .  .  2 -2
X.16     2  . -2  . -2  .  2  .  . -A  .  A  A -A  .  .  .  2 -2
X.17     2  .  2  . -2  . -2  .  .  A  .  A -A -A  .  .  .  2 -2
X.18     2  .  2  . -2  . -2  .  . -A  . -A  A  A  .  .  .  2 -2
X.19     4  .  .  .  4  .  .  .  .  .  .  .  .  .  .  .  . -4 -4

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