Properties

Label 32T4
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $D_4:C_2^2$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(32, 4);
 
Copy content sage:G = TransitiveGroup(32, 4)
 
Copy content oscar:G = transitive_group(32, 4)
 
Copy content gap:G := TransitiveGroup(32, 4);
 

Group invariants

Abstract group:  $D_4:C_2^2$
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 $n$:  $32$
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 $t$:  $4$
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);
 
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);
 
$\card{\Aut(F/K)}$:  $32$
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])
 
Copy content gap:Order(Centralizer(SymmetricGroup(32), G));
 
Generators:  $(1,6)(2,5)(3,8)(4,7)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22)(25,30)(26,29)(27,31)(28,32)$, $(1,5,19,22)(2,6,20,21)(3,7,17,24)(4,8,18,23)(9,16,26,30)(10,15,25,29)(11,14,28,31)(12,13,27,32)$, $(1,14,19,31)(2,13,20,32)(3,15,17,29)(4,16,18,30)(5,11,22,28)(6,12,21,27)(7,10,24,25)(8,9,23,26)$, $(1,9,19,26)(2,10,20,25)(3,12,17,27)(4,11,18,28)(5,16,22,30)(6,15,21,29)(7,13,24,32)(8,14,23,31)$
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
$16$:  $Q_8:C_2$ x 2, $C_2^4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 15

Degree 4: $C_2^2$ x 35

Degree 8: $C_2^3$ x 15, $Q_8:C_2$ x 6

Degree 16: $C_2^4$, $Q_8 : C_2$ x 2, $C_2 \times (C_4\times C_2):C_2$ x 6

Low degree siblings

16T18 x 6

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{32}$ $1$ $1$ $0$ $()$
2A $2^{16}$ $1$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32)$
2B $2^{16}$ $1$ $2$ $16$ $( 1,17)( 2,18)( 3,19)( 4,20)( 5,24)( 6,23)( 7,22)( 8,21)( 9,27)(10,28)(11,25)(12,26)(13,30)(14,29)(15,31)(16,32)$
2C $2^{16}$ $1$ $2$ $16$ $( 1,19)( 2,20)( 3,17)( 4,18)( 5,22)( 6,21)( 7,24)( 8,23)( 9,26)(10,25)(11,28)(12,27)(13,32)(14,31)(15,29)(16,30)$
2D $2^{16}$ $2$ $2$ $16$ $( 1,13)( 2,14)( 3,16)( 4,15)( 5,27)( 6,28)( 7,26)( 8,25)( 9,24)(10,23)(11,21)(12,22)(17,30)(18,29)(19,32)(20,31)$
2E $2^{16}$ $2$ $2$ $16$ $( 1,16)( 2,15)( 3,13)( 4,14)( 5,26)( 6,25)( 7,27)( 8,28)( 9,22)(10,21)(11,23)(12,24)(17,32)(18,31)(19,30)(20,29)$
2F $2^{16}$ $2$ $2$ $16$ $( 1,18)( 2,17)( 3,20)( 4,19)( 5, 8)( 6, 7)( 9,28)(10,27)(11,26)(12,25)(13,15)(14,16)(21,24)(22,23)(29,32)(30,31)$
2G $2^{16}$ $2$ $2$ $16$ $( 1,20)( 2,19)( 3,18)( 4,17)( 5, 6)( 7, 8)( 9,25)(10,26)(11,27)(12,28)(13,14)(15,16)(21,22)(23,24)(29,30)(31,32)$
2H $2^{16}$ $2$ $2$ $16$ $( 1,23)( 2,24)( 3,21)( 4,22)( 5,18)( 6,17)( 7,20)( 8,19)( 9,31)(10,32)(11,30)(12,29)(13,25)(14,26)(15,27)(16,28)$
2I $2^{16}$ $2$ $2$ $16$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22)(25,30)(26,29)(27,31)(28,32)$
4A1 $4^{8}$ $1$ $4$ $24$ $( 1,27,19,12)( 2,28,20,11)( 3,26,17, 9)( 4,25,18,10)( 5,32,22,13)( 6,31,21,14)( 7,30,24,16)( 8,29,23,15)$
4A-1 $4^{8}$ $1$ $4$ $24$ $( 1,12,19,27)( 2,11,20,28)( 3, 9,17,26)( 4,10,18,25)( 5,13,22,32)( 6,14,21,31)( 7,16,24,30)( 8,15,23,29)$
4B1 $4^{8}$ $1$ $4$ $24$ $( 1, 9,19,26)( 2,10,20,25)( 3,12,17,27)( 4,11,18,28)( 5,16,22,30)( 6,15,21,29)( 7,13,24,32)( 8,14,23,31)$
4B-1 $4^{8}$ $1$ $4$ $24$ $( 1,26,19, 9)( 2,25,20,10)( 3,27,17,12)( 4,28,18,11)( 5,30,22,16)( 6,29,21,15)( 7,32,24,13)( 8,31,23,14)$
4C $4^{8}$ $2$ $4$ $24$ $( 1,29,19,15)( 2,30,20,16)( 3,31,17,14)( 4,32,18,13)( 5,25,22,10)( 6,26,21, 9)( 7,28,24,11)( 8,27,23,12)$
4D $4^{8}$ $2$ $4$ $24$ $( 1, 5,19,22)( 2, 6,20,21)( 3, 7,17,24)( 4, 8,18,23)( 9,16,26,30)(10,15,25,29)(11,14,28,31)(12,13,27,32)$
4E $4^{8}$ $2$ $4$ $24$ $( 1,10,19,25)( 2, 9,20,26)( 3,11,17,28)( 4,12,18,27)( 5,29,22,15)( 6,30,21,16)( 7,31,24,14)( 8,32,23,13)$
4F $4^{8}$ $2$ $4$ $24$ $( 1,31,19,14)( 2,32,20,13)( 3,29,17,15)( 4,30,18,16)( 5,28,22,11)( 6,27,21,12)( 7,25,24,10)( 8,26,23, 9)$
4G $4^{8}$ $2$ $4$ $24$ $( 1,24,19, 7)( 2,23,20, 8)( 3,22,17, 5)( 4,21,18, 6)( 9,32,26,13)(10,31,25,14)(11,29,28,15)(12,30,27,16)$
4H $4^{8}$ $2$ $4$ $24$ $( 1,28,19,11)( 2,27,20,12)( 3,25,17,10)( 4,26,18, 9)( 5,14,22,31)( 6,13,21,32)( 7,15,24,29)( 8,16,23,30)$

Malle's constant $a(G)$:     $1/16$

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 4A1 4A-1 4B1 4B-1 4C 4D 4E 4F 4G 4H
Size 1 1 1 1 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2C 2C 2C 2C 2C 2C 2C 2C 2C 2C
Type
32.48.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.48.1b R 1 −1 −1 1 −1 1 −1 1 −1 1 −1 −1 1 1 1 1 1 −1 −1 −1
32.48.1c R 1 −1 −1 1 −1 1 −1 1 1 −1 1 1 −1 −1 1 −1 −1 −1 1 1
32.48.1d R 1 −1 −1 1 −1 1 1 −1 −1 1 1 1 −1 −1 −1 −1 1 1 1 −1
32.48.1e R 1 −1 −1 1 −1 1 1 −1 1 −1 −1 −1 1 1 −1 1 −1 1 −1 1
32.48.1f R 1 −1 −1 1 1 −1 −1 1 −1 1 1 1 −1 −1 −1 1 −1 1 −1 1
32.48.1g R 1 −1 −1 1 1 −1 −1 1 1 −1 −1 −1 1 1 −1 −1 1 1 1 −1
32.48.1h R 1 −1 −1 1 1 −1 1 −1 −1 1 −1 −1 1 1 1 −1 −1 −1 1 1
32.48.1i R 1 −1 −1 1 1 −1 1 −1 1 −1 1 1 −1 −1 1 1 1 −1 −1 −1
32.48.1j R 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1 1 1
32.48.1k R 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 −1 −1 1 −1 −1
32.48.1l R 1 1 1 1 −1 −1 1 1 −1 −1 1 1 1 1 −1 −1 1 −1 −1 1
32.48.1m R 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 −1 1 −1 −1 1 −1
32.48.1n R 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 1 −1 −1 1 −1
32.48.1o R 1 1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 −1 −1 1 −1 −1 1
32.48.1p R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 −1 −1 1 −1 −1
32.48.2a1 C 2 −2 2 −2 0 0 0 0 0 0 −2i 2i −2i 2i 0 0 0 0 0 0
32.48.2a2 C 2 −2 2 −2 0 0 0 0 0 0 2i −2i 2i −2i 0 0 0 0 0 0
32.48.2b1 C 2 2 −2 −2 0 0 0 0 0 0 −2i 2i 2i −2i 0 0 0 0 0 0
32.48.2b2 C 2 2 −2 −2 0 0 0 0 0 0 2i −2i −2i 2i 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

Data not computed