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Properties

Label 8T49
8T49 1 2 1->2 1->2 3 2->3 3->1 4 3->4 5 4->5 6 5->6 7 6->7 8 7->8 8->3
Degree $8$
Order $20160$
Cyclic no
Abelian no
Solvable no
Transitivity $6$
Primitive yes
$p$-group no
Group: $A_8$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(8, 49);
 
Copy content sage:G = TransitiveGroup(8, 49)
 
Copy content oscar:G = transitive_group(8, 49)
 
Copy content gap:G := TransitiveGroup(8, 49);
 

Group invariants

Abstract group:  $A_8$
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:  $20160=2^{6} \cdot 3^{2} \cdot 5 \cdot 7$
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:  no
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:   not nilpotent
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 $t$:  $49$
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:   $A8$
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:  6
Primitive:  yes
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);
 
$\card{\Aut(F/K)}$:  $1$
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:  $(1,2)(3,4,5,6,7,8)$, $(1,2,3)$
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

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Low degree siblings

15T72 x 2, 28T433, 35T36

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 $2^{4}$ $105$ $2$ $4$ $(1,8)(2,5)(3,4)(6,7)$
2B $2^{2},1^{4}$ $210$ $2$ $2$ $(2,4)(6,7)$
3A $3,1^{5}$ $112$ $3$ $2$ $(2,3,4)$
3B $3^{2},1^{2}$ $1120$ $3$ $4$ $(1,6,3)(4,8,7)$
4A $4^{2}$ $1260$ $4$ $6$ $(1,7,3,2)(4,8,6,5)$
4B $4,2,1^{2}$ $2520$ $4$ $4$ $(2,6,4,7)(3,5)$
5A $5,1^{3}$ $1344$ $5$ $4$ $(1,8,7,6,5)$
6A $3,2^{2},1$ $1680$ $6$ $4$ $(1,6)(3,4,7)(5,8)$
6B $6,2$ $3360$ $6$ $6$ $(1,4,6,8,3,7)(2,5)$
7A1 $7,1$ $2880$ $7$ $6$ $(1,8,6,7,5,2,4)$
7A-1 $7,1$ $2880$ $7$ $6$ $(1,4,2,5,7,6,8)$
15A1 $5,3$ $1344$ $15$ $6$ $(1,7,5,8,6)(2,4,3)$
15A-1 $5,3$ $1344$ $15$ $6$ $(1,6,8,5,7)(2,3,4)$

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 3A 3B 4A 4B 5A 6A 6B 7A1 7A-1 15A1 15A-1
Size 1 105 210 112 1120 1260 2520 1344 1680 3360 2880 2880 1344 1344
2 P 1A 1A 1A 3A 3B 2A 2B 5A 3A 3B 7A1 7A-1 15A1 15A-1
3 P 1A 2A 2B 1A 1A 4A 4B 5A 2B 2A 7A-1 7A1 5A 5A
5 P 1A 2A 2B 3A 3B 4A 4B 1A 6A 6B 7A-1 7A1 3A 3A
7 P 1A 2A 2B 3A 3B 4A 4B 5A 6A 6B 1A 1A 15A-1 15A1
Type
20160.a.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
20160.a.7a R 7 −1 3 4 1 −1 1 2 0 −1 0 0 −1 −1
20160.a.14a R 14 6 2 −1 2 2 0 −1 −1 0 0 0 −1 −1
20160.a.20a R 20 4 4 5 −1 0 0 0 1 1 −1 −1 0 0
20160.a.21a R 21 −3 1 6 0 1 −1 1 −2 0 0 0 1 1
20160.a.21b1 C 21 −3 1 −3 0 1 −1 1 1 0 0 0 1−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 −2+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157
20160.a.21b2 C 21 −3 1 −3 0 1 −1 1 1 0 0 0 −2+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 1−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157
20160.a.28a R 28 −4 4 1 1 0 0 −2 1 −1 0 0 1 1
20160.a.35a R 35 3 −5 5 2 −1 −1 0 1 0 0 0 0 0
20160.a.45a1 C 45 −3 −3 0 0 1 1 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 0 0
20160.a.45a2 C 45 −3 −3 0 0 1 1 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 0 0
20160.a.56a R 56 8 0 −4 −1 0 0 1 0 −1 0 0 1 1
20160.a.64a R 64 0 0 4 −2 0 0 −1 0 0 1 1 −1 −1
20160.a.70a R 70 −2 2 −5 1 −2 0 0 −1 1 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

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