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Properties

Label 16T127
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 $t$ :  $127$
Group :  $C_2\wr C_2^2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,5)(2,6)(3,16)(4,15)(7,12)(8,11)(9,13)(10,14), (1,2)(3,4)(5,6)(7,15)(8,16)(9,10)(11,12)(13,14), (1,16)(2,15)(3,5)(4,6)(7,10)(8,9)(11,13)(12,14)
$|\Aut(F/K)|$:  $4$

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 7

Degree 4: $C_2^2$ x 7

Degree 8: $C_2^3$, $(((C_4 \times C_2): C_2):C_2):C_2$ x 2

Low degree siblings

8T29 x 6, 8T31 x 2, 16T128 x 3, 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, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 5,14)( 6,13)( 7,16)( 8,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 3,12)( 4,11)( 7,16)( 8,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 3,12)( 4,11)( 5,14)( 6,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7,15)( 8,16)( 9,10)(11,12)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)( 9,10)$
$ 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 $ $4$ $4$ $( 1, 3,10,12)( 2, 4, 9,11)( 5, 7,14,16)( 6, 8,13,15)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 4)( 2, 3)( 5, 8,14,15)( 6, 7,13,16)( 9,12)(10,11)$
$ 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, 4, 4 $ $4$ $4$ $( 1, 5,10,14)( 2, 6, 9,13)( 3, 7,12,16)( 4, 8,11,15)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 6)( 2, 5)( 3, 8,12,15)( 4, 7,11,16)( 9,14)(10,13)$
$ 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 $ $4$ $4$ $( 1, 7,10,16)( 2, 8, 9,15)( 3, 5,12,14)( 4, 6,11,13)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 8,10,15)( 2, 7, 9,16)( 3, 6)( 4, 5)(11,14)(12,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$

Group invariants

Order:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [64, 138]
Character table:    
      2  6  5  5  5  4  4  4  4  3  4  4  3  4  4  3  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 2i 2a 1a 2i 2b 1a 2i 2c 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