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Properties

Label 8T9
Order \(16\)
n \(8\)
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 $n$ :  $8$
Transitive number $t$ :  $9$
Group :  $D_4\times C_2$
CHM label :  $E(8):2=D(4)[x]2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,3)(2,8)(4,6)(5,7), (4,5)(6,7), (1,8)(2,3)(4,5)(6,7), (1,5)(2,6)(3,7)(4,8)
$|\Aut(F/K)|$:  $4$

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 2, $C_2^3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Low degree siblings

8T9 x 3, 16T9

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 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1 $ $2$ $2$ $(4,5)(6,7)$
$ 2, 2, 2, 2 $ $2$ $2$ $(1,2)(3,8)(4,6)(5,7)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,2)(3,8)(4,7)(5,6)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,3)(2,8)(4,6)(5,7)$
$ 2, 2, 2, 2 $ $2$ $2$ $(1,4)(2,7)(3,6)(5,8)$
$ 4, 4 $ $2$ $4$ $(1,4,8,5)(2,7,3,6)$
$ 4, 4 $ $2$ $4$ $(1,6,8,7)(2,5,3,4)$
$ 2, 2, 2, 2 $ $2$ $2$ $(1,6)(2,5)(3,4)(7,8)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,8)(2,3)(4,5)(6,7)$

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