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Properties

Label 16T128
Order \(64\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2\wr C_2^2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $128$
Group :  $C_2\wr C_2^2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,16)(2,15)(3,6)(4,5)(7,10)(8,9)(11,14)(12,13), (1,5)(2,6)(3,15)(4,16)(7,11)(8,12)(9,13)(10,14), (1,2)(5,6)(7,15)(8,16)(9,10)(13,14)
$|\Aut(F/K)|$:  $4$

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:  $C_2^2 \wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $D_4\times C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$ x 2

Low degree siblings

8T29 x 6, 8T31 x 2, 16T127, 16T128 x 2, 16T129 x 3, 16T147, 16T149 x 6, 16T150 x 3, 32T136 x 3, 32T137 x 2, 32T163 x 3

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

Group invariants

Order:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [64, 138]
Character table:    
      2  6  4  5  4  5  5  4  3  4  4  3  4  4  3  4  6

        1a 2a 2b 2c 2d 2e 2f 4a 4b 2g 4c 4d 2h 4e 4f 2i
     2P 1a 1a 1a 1a 1a 1a 1a 2d 2i 1a 2b 2i 1a 2e 2i 1a
     3P 1a 2a 2b 2c 2d 2e 2f 4a 4b 2g 4c 4d 2h 4e 4f 2i

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