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Properties

Label 16T9
Order \(16\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $D_4\times C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $9$
Group :  $D_4\times C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,3)(2,4)(5,15)(6,16)(7,11)(8,12)(9,13)(10,14), (1,11,16,14)(2,12,15,13)(3,10,6,7)(4,9,5,8), (1,13)(2,14)(3,8)(4,7)(5,10)(6,9)(11,15)(12,16)
$|\Aut(F/K)|$:  $16$

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 2, $C_2^3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_2^3$, $D_4$ x 2, $D_4\times C_2$ x 4

Low degree siblings

8T9 x 4

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, 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 $ $2$ $2$ $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7,11)( 8,12)( 9,13)(10,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 4)( 2, 3)( 5,16)( 6,15)( 7,13)( 8,14)( 9,11)(10,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 5)( 2, 6)( 3,15)( 4,16)( 7,12)( 8,11)( 9,14)(10,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 7)( 2, 8)( 3,14)( 4,13)( 5,12)( 6,11)( 9,15)(10,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 8,16, 9)( 2, 7,15,10)( 3,13, 6,12)( 4,14, 5,11)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1,11,16,14)( 2,12,15,13)( 3,10, 6, 7)( 4, 9, 5, 8)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1,12)( 2,11)( 3, 9)( 4,10)( 5, 7)( 6, 8)(13,16)(14,15)$
$ 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:  $16=2^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [16, 11]
Character table:    
      2  4  3  3  4  4  3  3  3  3  4

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

X.1      1  1  1  1  1  1  1  1  1  1
X.2      1 -1 -1  1  1 -1  1  1 -1  1
X.3      1 -1 -1  1  1  1 -1 -1  1  1
X.4      1 -1  1 -1 -1 -1  1 -1  1  1
X.5      1 -1  1 -1 -1  1 -1  1 -1  1
X.6      1  1 -1 -1 -1 -1 -1  1  1  1
X.7      1  1 -1 -1 -1  1  1 -1 -1  1
X.8      1  1  1  1  1 -1 -1 -1 -1  1
X.9      2  .  .  2 -2  .  .  .  . -2
X.10     2  .  . -2  2  .  .  .  . -2