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Properties

Label 16T39
Order \(32\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^2\wr C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $39$
Group :  $C_2^2\wr C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,9,16,8)(2,10,15,7)(3,11,6,14)(4,12,5,13), (1,7,5,11)(2,8,6,12)(3,13,15,9)(4,14,16,10), (1,2)(3,4)(5,6)(7,9)(8,10)(11,13)(12,14)(15,16)
$|\Aut(F/K)|$:  $8$

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 6

Degree 8: $D_4$, $D_4\times C_2$ x 2, $C_2^2 \wr C_2$ x 4

Low degree siblings

8T18 x 8, 16T39 x 5, 16T46, 32T24

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 27]
Character table:    
      2  5  4  4  4  4  4  5  5  4  3  3  3  3  5

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

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