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Properties

Label 8T35
8T35 1 2 1->2 3 1->3 2->3 8 3->8 4 5 4->5 4->8 6 5->6 7 5->7 6->7 7->4 8->1
Degree $8$
Order $128$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_2 \wr C_2\wr C_2$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(8, 35);
 
Copy content sage:G = TransitiveGroup(8, 35)
 
Copy content oscar:G = transitive_group(8, 35)
 
Copy content gap:G := TransitiveGroup(8, 35);
 

Group invariants

Abstract group:  $C_2 \wr C_2\wr C_2$
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:  $128=2^{7}$
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:  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:  $4$
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 $t$:  $35$
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:   $[2^{4}]D(4)$
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:  $(1,2,3,8)(4,5,6,7)$, $(4,8)$, $(1,3)(5,7)$
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 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 3
$32$:  $C_2^2 \wr C_2$
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Low degree siblings

8T35 x 7, 16T376 x 4, 16T388 x 4, 16T390 x 4, 16T391 x 4, 16T393 x 4, 16T395 x 4, 16T396 x 4, 16T401 x 4, 32T852 x 4, 32T853 x 2, 32T854 x 2, 32T872 x 2, 32T876 x 4, 32T877 x 2, 32T880 x 2, 32T882 x 2, 32T883 x 4, 32T884 x 2, 32T885 x 2

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 $2^{2},1^{4}$ $2$ $2$ $2$ $(2,6)(4,8)$
2C $2^{2},1^{4}$ $4$ $2$ $2$ $(2,8)(4,6)$
2D $2,1^{6}$ $4$ $2$ $1$ $(4,8)$
2E $2^{4}$ $4$ $2$ $4$ $(1,3)(2,6)(4,8)(5,7)$
2F $2^{3},1^{2}$ $4$ $2$ $3$ $(1,5)(3,7)(4,8)$
2G $2^{4}$ $4$ $2$ $4$ $(1,3)(2,4)(5,7)(6,8)$
2H $2^{2},1^{4}$ $4$ $2$ $2$ $(1,5)(2,6)$
2I $2^{3},1^{2}$ $8$ $2$ $3$ $(1,3)(4,8)(5,7)$
2J $2^{4}$ $8$ $2$ $4$ $(1,2)(3,8)(4,7)(5,6)$
4A $4,1^{4}$ $4$ $4$ $3$ $(2,4,6,8)$
4B $4,2^{2}$ $4$ $4$ $5$ $(1,5)(2,4,6,8)(3,7)$
4C $4^{2}$ $4$ $4$ $6$ $(1,7,5,3)(2,4,6,8)$
4D $4,2^{2}$ $8$ $4$ $5$ $(1,3)(2,4,6,8)(5,7)$
4E $4,2,1^{2}$ $8$ $4$ $4$ $(2,4,6,8)(3,7)$
4F $4^{2}$ $8$ $4$ $6$ $(1,2,5,6)(3,8,7,4)$
4G $4^{2}$ $16$ $4$ $6$ $(1,2,3,4)(5,6,7,8)$
4H $4,2^{2}$ $16$ $4$ $5$ $(1,6,5,2)(3,4)(7,8)$
8A $8$ $16$ $8$ $7$ $(1,4,7,6,5,8,3,2)$

Malle's constant $a(G)$:     $1$

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 4G 4H 8A
Size 1 1 2 4 4 4 4 4 4 8 8 4 4 4 8 8 8 16 16 16
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2B 2B 2A 2B 2B 2A 2G 2H 4C
Type
128.928.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
128.928.1b R 1 1 1 −1 1 −1 −1 1 −1 1 −1 1 1 1 −1 −1 −1 1 1 −1
128.928.1c R 1 1 1 −1 1 −1 −1 1 −1 1 1 1 1 1 −1 −1 1 −1 −1 1
128.928.1d R 1 1 1 −1 1 1 1 1 −1 −1 −1 −1 1 −1 −1 1 −1 −1 1 1
128.928.1e R 1 1 1 −1 1 1 1 1 −1 −1 1 −1 1 −1 −1 1 1 1 −1 −1
128.928.1f R 1 1 1 1 1 −1 −1 1 1 −1 −1 −1 1 −1 1 −1 −1 1 −1 1
128.928.1g R 1 1 1 1 1 −1 −1 1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1
128.928.1h R 1 1 1 1 1 1 1 1 1 1 −1 1 1 1 1 1 −1 −1 −1 −1
128.928.2a R 2 2 2 −2 2 0 0 −2 −2 0 0 0 −2 0 2 0 0 0 0 0
128.928.2b R 2 2 2 0 −2 −2 −2 2 0 0 0 0 −2 0 0 2 0 0 0 0
128.928.2c R 2 2 2 0 −2 0 0 −2 0 −2 0 2 2 2 0 0 0 0 0 0
128.928.2d R 2 2 2 0 −2 0 0 −2 0 2 0 −2 2 −2 0 0 0 0 0 0
128.928.2e R 2 2 2 0 −2 2 2 2 0 0 0 0 −2 0 0 −2 0 0 0 0
128.928.2f R 2 2 2 2 2 0 0 −2 2 0 0 0 −2 0 −2 0 0 0 0 0
128.928.4a R 4 4 −4 0 0 0 0 0 0 0 −2 0 0 0 0 0 2 0 0 0
128.928.4b R 4 4 −4 0 0 0 0 0 0 0 2 0 0 0 0 0 −2 0 0 0
128.928.4c R 4 −4 0 −2 0 −2 2 0 2 0 0 −2 0 2 0 0 0 0 0 0
128.928.4d R 4 −4 0 −2 0 2 −2 0 2 0 0 2 0 −2 0 0 0 0 0 0
128.928.4e R 4 −4 0 2 0 −2 2 0 −2 0 0 2 0 −2 0 0 0 0 0 0
128.928.4f R 4 −4 0 2 0 2 −2 0 −2 0 0 −2 0 2 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

$f_{ 1 } =$ $x^{8} + 2 x^{6} - x^{2} + t$ Copy content Toggle raw display