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Properties

Label 16T150
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 $t$ :  $150$
Group :  $C_2\wr C_2^2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,9,16,8)(2,10,15,7)(3,11,6,14)(4,12,5,13), (1,11,5,7)(2,12,6,8)(3,13,15,9)(4,14,16,10), (1,7)(2,8)(3,13)(4,14)(5,11)(6,12)(9,15)(10,16)
$|\Aut(F/K)|$:  $4$

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

Low degree siblings

8T29 x 6, 8T31 x 2, 16T127, 16T128 x 3, 16T129 x 3, 16T147, 16T149 x 6, 16T150 x 2, 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 $ $2$ $2$ $( 7,11)( 8,12)( 9,13)(10,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 3,15)( 4,16)( 9,13)(10,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 9)( 8,10)(11,13)(12,14)(15,16)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 2)( 3,16)( 4,15)( 5, 6)( 7, 9,11,13)( 8,10,12,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7,12)( 8,11)( 9,14)(10,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 4)( 2, 3)( 5,16)( 6,15)( 7,10)( 8, 9)(11,14)(12,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 4)( 2, 3)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 4, 5,16)( 2, 3, 6,15)( 7,10,11,14)( 8, 9,12,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 5)( 2, 6)( 3,15)( 4,16)( 7,11)( 8,12)( 9,13)(10,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)(13,15)(14,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 7, 5,11)( 2, 8, 6,12)( 3, 9,15,13)( 4,10,16,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 7)( 2, 8)( 3,13)( 4,14)( 5,11)( 6,12)( 9,15)(10,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 7, 5,11)( 2, 8, 6,12)( 3,13,15, 9)( 4,14,16,10)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 8, 4, 9)( 2, 7, 3,10)( 5,12,16,13)( 6,11,15,14)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 8, 4,13)( 2, 7, 3,14)( 5,12,16, 9)( 6,11,15,10)$

Group invariants

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

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

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  .  .  .  .  .  .