Properties

Label 32T2
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $\OD_{16}:C_2$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(32, 2);
 
Copy content sage:G = TransitiveGroup(32, 2)
 
Copy content oscar:G = transitive_group(32, 2)
 
Copy content gap:G := TransitiveGroup(32, 2);
 

Group invariants

Abstract group:  $\OD_{16}:C_2$
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:  $32=2^{5}$
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 $t$:  $2$
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:  $(1,12,4,9)(2,11,3,10)(5,16,7,14)(6,15,8,13)(17,28,20,25)(18,27,19,26)(21,31,23,29)(22,32,24,30)$, $(1,31)(2,32)(3,30)(4,29)(5,20)(6,19)(7,17)(8,18)(9,23)(10,24)(11,22)(12,21)(13,26)(14,25)(15,27)(16,28)$, $(1,20,10,26,4,17,11,27)(2,19,9,25,3,18,12,28)(5,22,13,31,7,24,15,29)(6,21,14,32,8,23,16,30)$
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
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $C_4\times C_2$ x 6, $C_2^3$
$16$:  $C_4\times C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_4$ x 4, $C_2^2$ x 7

Degree 8: $C_4\times C_2$ x 6, $C_2^3$

Degree 16: $C_4\times C_2^2$, $(C_8:C_2):C_2$ x 3

Low degree siblings

16T16 x 3

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 $2^{16}$ $1$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32)$
2B $2^{16}$ $2$ $2$ $16$ $( 1,31)( 2,32)( 3,30)( 4,29)( 5,20)( 6,19)( 7,17)( 8,18)( 9,23)(10,24)(11,22)(12,21)(13,26)(14,25)(15,27)(16,28)$
2C $2^{16}$ $2$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2D $2^{16}$ $2$ $2$ $16$ $( 1,21)( 2,22)( 3,24)( 4,23)( 5,28)( 6,27)( 7,25)( 8,26)( 9,31)(10,32)(11,30)(12,29)(13,19)(14,20)(15,18)(16,17)$
4A1 $4^{8}$ $1$ $4$ $24$ $( 1,11, 4,10)( 2,12, 3, 9)( 5,15, 7,13)( 6,16, 8,14)(17,26,20,27)(18,25,19,28)(21,30,23,32)(22,29,24,31)$
4A-1 $4^{8}$ $1$ $4$ $24$ $( 1,10, 4,11)( 2, 9, 3,12)( 5,13, 7,15)( 6,14, 8,16)(17,27,20,26)(18,28,19,25)(21,32,23,30)(22,31,24,29)$
4B $4^{8}$ $2$ $4$ $24$ $( 1,12, 4, 9)( 2,11, 3,10)( 5,16, 7,14)( 6,15, 8,13)(17,28,20,25)(18,27,19,26)(21,31,23,29)(22,32,24,30)$
4C $4^{8}$ $2$ $4$ $24$ $( 1,22, 4,24)( 2,21, 3,23)( 5,27, 7,26)( 6,28, 8,25)( 9,32,12,30)(10,31,11,29)(13,20,15,17)(14,19,16,18)$
4D $4^{8}$ $2$ $4$ $24$ $( 1,30, 4,32)( 2,29, 3,31)( 5,18, 7,19)( 6,17, 8,20)( 9,22,12,24)(10,21,11,23)(13,28,15,25)(14,27,16,26)$
8A1 $8^{4}$ $1$ $8$ $28$ $( 1, 8,11,14, 4, 6,10,16)( 2, 7,12,13, 3, 5, 9,15)(17,21,26,30,20,23,27,32)(18,22,25,29,19,24,28,31)$
8A-1 $8^{4}$ $1$ $8$ $28$ $( 1,16,10, 6, 4,14,11, 8)( 2,15, 9, 5, 3,13,12, 7)(17,32,27,23,20,30,26,21)(18,31,28,24,19,29,25,22)$
8A3 $8^{4}$ $1$ $8$ $28$ $( 1,14,10, 8, 4,16,11, 6)( 2,13, 9, 7, 3,15,12, 5)(17,30,27,21,20,32,26,23)(18,29,28,22,19,31,25,24)$
8A-3 $8^{4}$ $1$ $8$ $28$ $( 1, 6,11,16, 4, 8,10,14)( 2, 5,12,15, 3, 7, 9,13)(17,23,26,32,20,21,27,30)(18,24,25,31,19,22,28,29)$
8B1 $8^{4}$ $2$ $8$ $28$ $( 1,18,11,25, 4,19,10,28)( 2,17,12,26, 3,20, 9,27)( 5,23,15,32, 7,21,13,30)( 6,24,16,31, 8,22,14,29)$
8B-1 $8^{4}$ $2$ $8$ $28$ $( 1,28,10,19, 4,25,11,18)( 2,27, 9,20, 3,26,12,17)( 5,30,13,21, 7,32,15,23)( 6,29,14,22, 8,31,16,24)$
8C1 $8^{4}$ $2$ $8$ $28$ $( 1,13,10, 7, 4,15,11, 5)( 2,14, 9, 8, 3,16,12, 6)(17,31,27,24,20,29,26,22)(18,32,28,23,19,30,25,21)$
8C-1 $8^{4}$ $2$ $8$ $28$ $( 1, 7,11,13, 4, 5,10,15)( 2, 8,12,14, 3, 6, 9,16)(17,24,26,31,20,22,27,29)(18,23,25,32,19,21,28,30)$
8D1 $8^{4}$ $2$ $8$ $28$ $( 1,26,11,20, 4,27,10,17)( 2,25,12,19, 3,28, 9,18)( 5,31,15,22, 7,29,13,24)( 6,32,16,21, 8,30,14,23)$
8D-1 $8^{4}$ $2$ $8$ $28$ $( 1,17,10,27, 4,20,11,26)( 2,18, 9,28, 3,19,12,25)( 5,24,13,29, 7,22,15,31)( 6,23,14,30, 8,21,16,32)$

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 4A1 4A-1 4B 4C 4D 8A1 8A-1 8A3 8A-3 8B1 8B-1 8C1 8C-1 8D1 8D-1
Size 1 1 2 2 2 1 1 2 2 2 1 1 1 1 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 4A1 4A-1 4A-1 4A1 4A1 4A-1 4A-1 4A1 4A1 4A-1
Type
32.38.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.38.1b R 1 1 −1 −1 1 1 1 −1 −1 1 −1 −1 −1 −1 1 1 1 1 −1 −1
32.38.1c R 1 1 −1 −1 1 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 1 1
32.38.1d R 1 1 −1 1 −1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 −1 −1 1 1
32.38.1e R 1 1 −1 1 −1 1 1 1 −1 −1 1 1 1 1 −1 −1 1 1 −1 −1
32.38.1f R 1 1 1 −1 −1 1 1 −1 1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1
32.38.1g R 1 1 1 −1 −1 1 1 −1 1 −1 1 1 1 1 1 1 −1 −1 −1 −1
32.38.1h R 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1
32.38.1i1 C 1 1 −1 −1 −1 −1 −1 1 1 1 −i i i −i i −i −i i i −i
32.38.1i2 C 1 1 −1 −1 −1 −1 −1 1 1 1 i −i −i i −i i i −i −i i
32.38.1j1 C 1 1 −1 1 1 −1 −1 −1 1 −1 −i i i −i i −i i −i −i i
32.38.1j2 C 1 1 −1 1 1 −1 −1 −1 1 −1 i −i −i i −i i −i i i −i
32.38.1k1 C 1 1 1 −1 1 −1 −1 1 −1 −1 −i i i −i −i i −i i −i i
32.38.1k2 C 1 1 1 −1 1 −1 −1 1 −1 −1 i −i −i i i −i i −i i −i
32.38.1l1 C 1 1 1 1 −1 −1 −1 −1 −1 1 −i i i −i −i i i −i i −i
32.38.1l2 C 1 1 1 1 −1 −1 −1 −1 −1 1 i −i −i i i −i −i i −i i
32.38.2a1 C 2 −2 0 0 0 −2ζ82 2ζ82 0 0 0 2ζ83 −2ζ8 2ζ8 −2ζ83 0 0 0 0 0 0
32.38.2a2 C 2 −2 0 0 0 2ζ82 −2ζ82 0 0 0 −2ζ8 2ζ83 −2ζ83 2ζ8 0 0 0 0 0 0
32.38.2a3 C 2 −2 0 0 0 −2ζ82 2ζ82 0 0 0 −2ζ83 2ζ8 −2ζ8 2ζ83 0 0 0 0 0 0
32.38.2a4 C 2 −2 0 0 0 2ζ82 −2ζ82 0 0 0 2ζ8 −2ζ83 2ζ83 −2ζ8 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