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Properties

Label 8T22
8T22 1 3 1->3 5 1->5 8 1->8 2 2->3 2->3 2->3 6 2->6 2->8 7 3->7 4 4->5 4->5 4->6 4->8 5->7 6->7 6->7
Degree $8$
Order $32$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $Q_8:C_2^2$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(8, 22);
 
Copy content sage:G = TransitiveGroup(8, 22)
 
Copy content oscar:G = transitive_group(8, 22)
 
Copy content gap:G := TransitiveGroup(8, 22);
 

Group invariants

Abstract group:  $Q_8:C_2^2$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Copy content oscar:small_group_identification(G)
 
Copy content gap:IdGroup(G);
 
Order:  $32=2^{5}$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Copy content gap:Order(G);
 
Cyclic:  no
Copy content comment:Determine if group is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content oscar:is_cyclic(G)
 
Copy content gap:IsCyclic(G);
 
Abelian:  no
Copy content comment:Determine if group is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content oscar:is_abelian(G)
 
Copy content gap:IsAbelian(G);
 
Solvable:  yes
Copy content comment:Determine if group is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content sage:G.is_solvable()
 
Copy content oscar:is_solvable(G)
 
Copy content gap:IsSolvable(G);
 
Nilpotency class:  $2$
Copy content comment:Nilpotency class
 
Copy content magma:NilpotencyClass(G);
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 
Copy content gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;
 

Group action invariants

Degree $n$:  $8$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Copy content gap:NrMovedPoints(G);
 
Transitive number $t$:  $22$
Copy content comment:Transitive number
 
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Copy content sage:G.transitive_number()
 
Copy content oscar:transitive_group_identification(G)[2]
 
Copy content gap:TransitiveIdentification(G);
 
CHM label:   $E(8):D_{4}=[2^{3}]2^{2}$
Parity:  $1$
Copy content comment:Parity
 
Copy content magma:IsEven(G);
 
Copy content sage:all(g.SignPerm() == 1 for g in libgap(G).GeneratorsOfGroup())
 
Copy content oscar:is_even(G)
 
Copy content gap:ForAll(GeneratorsOfGroup(G), g -> SignPerm(g) = 1);
 
Transitivity:  1
Primitive:  no
Copy content comment:Determine if group is primitive
 
Copy content magma:IsPrimitive(G);
 
Copy content sage:G.is_primitive()
 
Copy content oscar:is_primitive(G)
 
Copy content gap:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(8).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(8), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(8), G));
 
Generators:  $(1,3)(2,8)(4,6)(5,7)$, $(2,3)(6,7)$, $(2,3)(4,5)$, $(1,8)(2,3)(4,5)(6,7)$, $(1,5)(2,6)(3,7)(4,8)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 
Copy content gap:GeneratorsOfGroup(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_2^2$ x 35
$8$:  $C_2^3$ x 15
$16$:  $C_2^4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Low degree siblings

8T22 x 5, 16T23 x 9, 32T9

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 $2^{4}$ $1$ $2$ $4$ $(1,8)(2,3)(4,5)(6,7)$
2B $2^{4}$ $2$ $2$ $4$ $(1,6)(2,4)(3,5)(7,8)$
2C $2^{4}$ $2$ $2$ $4$ $(1,5)(2,7)(3,6)(4,8)$
2D $2^{4}$ $2$ $2$ $4$ $(1,3)(2,8)(4,7)(5,6)$
2E $2^{2},1^{4}$ $2$ $2$ $2$ $(2,3)(4,5)$
2F $2^{4}$ $2$ $2$ $4$ $(1,6)(2,5)(3,4)(7,8)$
2G $2^{2},1^{4}$ $2$ $2$ $2$ $(1,8)(4,5)$
2H $2^{4}$ $2$ $2$ $4$ $(1,4)(2,7)(3,6)(5,8)$
2I $2^{2},1^{4}$ $2$ $2$ $2$ $(1,8)(2,3)$
2J $2^{4}$ $2$ $2$ $4$ $(1,2)(3,8)(4,7)(5,6)$
4A $4^{2}$ $2$ $4$ $6$ $(1,4,8,5)(2,7,3,6)$
4B $4^{2}$ $2$ $4$ $6$ $(1,2,8,3)(4,7,5,6)$
4C $4^{2}$ $2$ $4$ $6$ $(1,6,8,7)(2,5,3,4)$
4D $4^{2}$ $2$ $4$ $6$ $(1,3,8,2)(4,7,5,6)$
4E $4^{2}$ $2$ $4$ $6$ $(1,6,8,7)(2,4,3,5)$
4F $4^{2}$ $2$ $4$ $6$ $(1,5,8,4)(2,7,3,6)$

Malle's constant $a(G)$:     $1/2$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 
Copy content gap:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 4A 4B 4C 4D 4E 4F
Size 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A
Type
32.49.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.49.1b R 1 1 −1 −1 1 −1 1 −1 1 1 1 1 −1 1 −1 −1 −1
32.49.1c R 1 1 −1 −1 1 −1 1 1 −1 −1 −1 1 −1 −1 1 1 1
32.49.1d R 1 1 −1 −1 1 1 −1 −1 1 −1 −1 −1 1 1 −1 1 1
32.49.1e R 1 1 −1 −1 1 1 −1 1 −1 1 1 −1 1 −1 1 −1 −1
32.49.1f R 1 1 −1 1 −1 −1 1 −1 −1 1 −1 −1 1 1 1 −1 1
32.49.1g R 1 1 −1 1 −1 −1 1 1 1 −1 1 −1 1 −1 −1 1 −1
32.49.1h R 1 1 −1 1 −1 1 −1 −1 −1 −1 1 1 −1 1 1 1 −1
32.49.1i R 1 1 −1 1 −1 1 −1 1 1 1 −1 1 −1 −1 −1 −1 1
32.49.1j R 1 1 1 −1 −1 −1 −1 −1 1 1 −1 1 1 −1 1 1 −1
32.49.1k R 1 1 1 −1 −1 −1 −1 1 −1 −1 1 1 1 1 −1 −1 1
32.49.1l R 1 1 1 −1 −1 1 1 −1 1 −1 1 −1 −1 −1 1 −1 1
32.49.1m R 1 1 1 −1 −1 1 1 1 −1 1 −1 −1 −1 1 −1 1 −1
32.49.1n R 1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 −1 −1 1 1
32.49.1o R 1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1 −1 −1
32.49.1p R 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 −1 −1
32.49.4a R 4 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 
Copy content gap:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{8} + \left(t^{2} - 5\right) x^{6} + \left(-t^{2} + 9\right) x^{4} - 8 x^{2} + 4$ Copy content Toggle raw display