# Properties

 Label 8T11 Degree $8$ Order $16$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $Q_8:C_2$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(8, 11);

## Group action invariants

 Degree $n$: $8$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $11$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $Q_8:C_2$ CHM label: $1/2[2^{3}]E(4)=Q_{8}:2$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $2$ magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $4$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,3,5,7)(2,4,6,8), (1,5)(3,7), (1,4,5,8)(2,3,6,7) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $C_2^3$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

## Low degree siblings

8T11 x 2, 16T11

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 1, 1, 1, 1$ $2$ $2$ $(2,6)(4,8)$ $2, 2, 2, 2$ $2$ $2$ $(1,2)(3,4)(5,6)(7,8)$ $4, 4$ $2$ $4$ $(1,2,5,6)(3,4,7,8)$ $4, 4$ $1$ $4$ $(1,3,5,7)(2,4,6,8)$ $4, 4$ $2$ $4$ $(1,3,5,7)(2,8,6,4)$ $4, 4$ $2$ $4$ $(1,4,5,8)(2,3,6,7)$ $2, 2, 2, 2$ $2$ $2$ $(1,4)(2,7)(3,6)(5,8)$ $2, 2, 2, 2$ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$ $4, 4$ $1$ $4$ $(1,7,5,3)(2,8,6,4)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $16=2^{4}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 16.13 magma: IdentifyGroup(G);
 Character table:  2 4 3 3 3 4 3 3 3 4 4 1a 2a 2b 4a 4b 4c 4d 2c 2d 4e 2P 1a 1a 1a 2d 2d 2d 2d 1a 1a 2d 3P 1a 2a 2b 4a 4e 4c 4d 2c 2d 4b 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 . . . A . . . -2 -A X.10 2 . . . -A . . . -2 A A = -2*E(4) = -2*Sqrt(-1) = -2i 

magma: CharacterTable(G);