↑

Properties

Label 32T12882
Degree $32$
Order $512$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_4^2:C_2^3$

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(32, 12882);
 
Copy content sage:G = TransitiveGroup(32, 12882)
 
Copy content oscar:G = transitive_group(32, 12882)
 

Group invariants

Abstract group:  $D_4^2:C_2^3$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Order:  $512=2^{9}$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar: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)
 
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)
 
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)
 
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
 

Group action invariants

Degree $n$:  $32$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Transitive number $t$:  $12882$
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]
 
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)
 
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)
 
$\card{\Aut(F/K)}$:  $8$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(32).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(32), G)[1])
 
Generators:  $(1,28,17,10)(2,27,18,9)(3,26,19,12)(4,25,20,11)(5,6)(7,8)(13,14)(15,16)(21,22)(23,24)(29,30)(31,32)$, $(1,21,27,29,17,7,9,15)(2,22,28,30,18,8,10,16)(3,23,25,31,19,5,11,13)(4,24,26,32,20,6,12,14)$, $(1,14,9,24,17,32,27,6)(2,13,10,23,18,31,28,5)(3,16,11,22,19,30,25,8)(4,15,12,21,20,29,26,7)$, $(1,32)(2,31)(3,30)(4,29)(5,10,23,28)(6,9,24,27)(7,12,21,26)(8,11,22,25)(13,18)(14,17)(15,20)(16,19)$, $(1,16,27,8,17,30,9,22)(2,15,28,7,18,29,10,21)(3,14,25,6,19,32,11,24)(4,13,26,5,20,31,12,23)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 31
$4$:  $C_2^2$ x 155
$8$:  $D_{4}$ x 24, $C_2^3$ x 155
$16$:  $D_4\times C_2$ x 84, $C_2^4$ x 31
$32$:  $C_2^2 \wr C_2$ x 16, $C_2^2 \times D_4$ x 42, 32T39
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 4, 16T105 x 12, 32T273 x 3
$128$:  $C_2 \wr C_2\wr C_2$ x 4, 16T245 x 6, 32T1369
$256$:  16T509 x 6, 32T4287

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_2^3$, $D_4\times C_2$ x 6, $C_2 \wr C_2\wr C_2$ x 4

Degree 16: $C_2^2 \times D_4$, 16T509 x 6

Low degree siblings

32T12882 x 127, 32T12885 x 384

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Conjugacy classes not computed

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

Character table

80 x 80 character table

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

Regular extensions

Data not computed