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Properties

Label 8T11
8T11 1 3 1->3 4 1->4 5 1->5 2 2->3 2->4 3->5 6 3->6 7 3->7 4->5 4->6 5->7 8 5->8 6->7 6->8 7->1 7->2 8->1 8->2
Degree $8$
Order $16$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $Q_8:C_2$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(8, 11);
 

Group invariants

Abstract group:  $Q_8:C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $16=2^{4}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:  $2$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$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

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{8}$ $1$ $1$ $0$ $()$
2A $2^{4}$ $1$ $2$ $4$ $(1,5)(2,6)(3,7)(4,8)$
2B $2^{4}$ $2$ $2$ $4$ $(1,4)(2,7)(3,6)(5,8)$
2C $2^{2},1^{4}$ $2$ $2$ $2$ $(1,5)(3,7)$
2D $2^{4}$ $2$ $2$ $4$ $(1,6)(2,5)(3,8)(4,7)$
4A1 $4^{2}$ $1$ $4$ $6$ $(1,3,5,7)(2,4,6,8)$
4A-1 $4^{2}$ $1$ $4$ $6$ $(1,7,5,3)(2,8,6,4)$
4B $4^{2}$ $2$ $4$ $6$ $(1,3,5,7)(2,8,6,4)$
4C $4^{2}$ $2$ $4$ $6$ $(1,6,5,2)(3,8,7,4)$
4D $4^{2}$ $2$ $4$ $6$ $(1,8,5,4)(2,7,6,3)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 4A1 4A-1 4B 4C 4D
Size 1 1 2 2 2 1 1 2 2 2
2 P 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A
Type
16.13.1a R 1 1 1 1 1 1 1 1 1 1
16.13.1b R 1 1 −1 −1 −1 −1 −1 1 1 1
16.13.1c R 1 1 −1 −1 1 1 1 −1 1 −1
16.13.1d R 1 1 −1 1 −1 1 1 −1 −1 1
16.13.1e R 1 1 −1 1 1 −1 −1 1 −1 −1
16.13.1f R 1 1 1 −1 −1 1 1 1 −1 −1
16.13.1g R 1 1 1 −1 1 −1 −1 −1 −1 1
16.13.1h R 1 1 1 1 −1 −1 −1 −1 1 −1
16.13.2a1 C 2 −2 0 0 0 −2i 2i 0 0 0
16.13.2a2 C 2 −2 0 0 0 2i −2i 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $\left(t^{2} - 1\right) x^{8} + 2 t x^{7} - 14 t x^{5} + \left(14 t^{2} - 14\right) x^{4} + 14 t x^{3} - 2 t x + \left(t^{2} - 1\right)$ Copy content Toggle raw display