Properties

Label 32T8
32T8 1 14 1->14 18 1->18 2 13 2->13 17 2->17 3 15 3->15 19 3->19 4 16 4->16 20 4->20 5 5->3 24 5->24 6 6->4 23 6->23 7 7->2 22 7->22 8 8->1 21 8->21 9 9->5 26 9->26 10 10->6 25 10->25 11 11->7 27 11->27 12 12->8 28 12->28 13->10 30 13->30 14->9 29 14->29 15->12 31 15->31 16->11 32 16->32 17->5 17->32 18->6 18->31 19->7 19->29 20->8 20->30 21->10 21->19 22->9 22->20 23->12 23->17 24->11 24->18 25->15 25->22 26->16 26->21 27->14 27->23 28->13 28->24 29->4 29->28 30->3 30->27 31->2 31->26 32->1 32->25
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $\OD_{32}$

Related objects

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(32, 8);
 
Copy content sage:G = TransitiveGroup(32, 8)
 
Copy content oscar:G = transitive_group(32, 8)
 
Copy content gap:G := TransitiveGroup(32, 8);
 

Group invariants

Abstract group:  $\OD_{32}$
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 $t$:  $8$
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:  $(1,18,6,23,12,28,13,30,3,19,7,22,9,26,16,32)(2,17,5,24,11,27,14,29,4,20,8,21,10,25,15,31)$, $(1,14,9,5,3,15,12,8)(2,13,10,6,4,16,11,7)(17,32,25,22,20,30,27,23)(18,31,26,21,19,29,28,24)$
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 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_8$ x 2, $C_4\times C_2$
$16$:  $C_8\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $C_8$ x 2, $C_4\times C_2$

Degree 16: $C_8\times C_2$, $C_{16} : C_2$

Low degree siblings

16T22

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 $2^{16}$ $1$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,27)(26,28)(29,31)(30,32)$
2B $2^{16}$ $2$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27)(29,32)(30,31)$
4A1 $4^{8}$ $1$ $4$ $24$ $( 1,12, 3, 9)( 2,11, 4,10)( 5,14, 8,15)( 6,13, 7,16)(17,27,20,25)(18,28,19,26)(21,31,24,29)(22,32,23,30)$
4A-1 $4^{8}$ $1$ $4$ $24$ $( 1, 9, 3,12)( 2,10, 4,11)( 5,15, 8,14)( 6,16, 7,13)(17,25,20,27)(18,26,19,28)(21,29,24,31)(22,30,23,32)$
4B $4^{8}$ $2$ $4$ $24$ $( 1,11, 3,10)( 2,12, 4, 9)( 5,13, 8,16)( 6,14, 7,15)(17,26,20,28)(18,25,19,27)(21,30,24,32)(22,29,23,31)$
8A1 $8^{4}$ $1$ $8$ $28$ $( 1, 6,12,13, 3, 7, 9,16)( 2, 5,11,14, 4, 8,10,15)(17,24,27,29,20,21,25,31)(18,23,28,30,19,22,26,32)$
8A-1 $8^{4}$ $1$ $8$ $28$ $( 1,16, 9, 7, 3,13,12, 6)( 2,15,10, 8, 4,14,11, 5)(17,31,25,21,20,29,27,24)(18,32,26,22,19,30,28,23)$
8A3 $8^{4}$ $1$ $8$ $28$ $( 1,13, 9, 6, 3,16,12, 7)( 2,14,10, 5, 4,15,11, 8)(17,29,25,24,20,31,27,21)(18,30,26,23,19,32,28,22)$
8A-3 $8^{4}$ $1$ $8$ $28$ $( 1, 7,12,16, 3, 6, 9,13)( 2, 8,11,15, 4, 5,10,14)(17,21,27,31,20,24,25,29)(18,22,28,32,19,23,26,30)$
8B1 $8^{4}$ $2$ $8$ $28$ $( 1, 5,12,14, 3, 8, 9,15)( 2, 6,11,13, 4, 7,10,16)(17,22,27,32,20,23,25,30)(18,21,28,31,19,24,26,29)$
8B-1 $8^{4}$ $2$ $8$ $28$ $( 1,14, 9, 5, 3,15,12, 8)( 2,13,10, 6, 4,16,11, 7)(17,32,25,22,20,30,27,23)(18,31,26,21,19,29,28,24)$
16A1 $16^{2}$ $2$ $16$ $30$ $( 1,18, 6,23,12,28,13,30, 3,19, 7,22, 9,26,16,32)( 2,17, 5,24,11,27,14,29, 4,20, 8,21,10,25,15,31)$
16A-1 $16^{2}$ $2$ $16$ $30$ $( 1,30,16,28, 9,23, 7,18, 3,32,13,26,12,22, 6,19)( 2,29,15,27,10,24, 8,17, 4,31,14,25,11,21, 5,20)$
16A3 $16^{2}$ $2$ $16$ $30$ $( 1,23,13,19, 9,32, 6,28, 3,22,16,18,12,30, 7,26)( 2,24,14,20,10,31, 5,27, 4,21,15,17,11,29, 8,25)$
16A-3 $16^{2}$ $2$ $16$ $30$ $( 1,28, 7,32,12,19,16,23, 3,26, 6,30, 9,18,13,22)( 2,27, 8,31,11,20,15,24, 4,25, 5,29,10,17,14,21)$
16B1 $16^{2}$ $2$ $16$ $30$ $( 1,17, 7,21,12,27,16,31, 3,20, 6,24, 9,25,13,29)( 2,18, 8,22,11,28,15,32, 4,19, 5,23,10,26,14,30)$
16B-1 $16^{2}$ $2$ $16$ $30$ $( 1,29,13,25, 9,24, 6,20, 3,31,16,27,12,21, 7,17)( 2,30,14,26,10,23, 5,19, 4,32,15,28,11,22, 8,18)$
16B3 $16^{2}$ $2$ $16$ $30$ $( 1,24,16,17, 9,31, 7,25, 3,21,13,20,12,29, 6,27)( 2,23,15,18,10,32, 8,26, 4,22,14,19,11,30, 5,28)$
16B-3 $16^{2}$ $2$ $16$ $30$ $( 1,27, 6,29,12,20,13,21, 3,25, 7,31, 9,17,16,24)( 2,28, 5,30,11,19,14,22, 4,26, 8,32,10,18,15,23)$

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 4A1 4A-1 4B 8A1 8A-1 8A3 8A-3 8B1 8B-1 16A1 16A-1 16A3 16A-3 16B1 16B-1 16B3 16B-3
Size 1 1 2 1 1 2 1 1 1 1 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 2A 2A 2A 4A1 4A-1 4A-1 4A1 4A1 4A-1 8A1 8A-1 8A3 8A-3 8A-3 8A3 8A-1 8A1
Type
32.17.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.17.1b R 1 1 −1 1 1 −1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 1 1
32.17.1c R 1 1 −1 1 1 −1 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
32.17.1d R 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
32.17.1e1 C 1 1 −1 1 1 −1 −1 −1 −1 −1 1 1 −i i i −i i −i −i i
32.17.1e2 C 1 1 −1 1 1 −1 −1 −1 −1 −1 1 1 i −i −i i −i i i −i
32.17.1f1 C 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −i i i −i −i i i −i
32.17.1f2 C 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 i −i −i i i −i −i i
32.17.1g1 C 1 1 −1 −1 −1 1 −ζ82 ζ82 ζ82 −ζ82 ζ82 −ζ82 ζ83 −ζ8 ζ8 −ζ83 −ζ83 ζ8 −ζ8 ζ83
32.17.1g2 C 1 1 −1 −1 −1 1 ζ82 −ζ82 −ζ82 ζ82 −ζ82 ζ82 −ζ8 ζ83 −ζ83 ζ8 ζ8 −ζ83 ζ83 −ζ8
32.17.1g3 C 1 1 −1 −1 −1 1 −ζ82 ζ82 ζ82 −ζ82 ζ82 −ζ82 −ζ83 ζ8 −ζ8 ζ83 ζ83 −ζ8 ζ8 −ζ83
32.17.1g4 C 1 1 −1 −1 −1 1 ζ82 −ζ82 −ζ82 ζ82 −ζ82 ζ82 ζ8 −ζ83 ζ83 −ζ8 −ζ8 ζ83 −ζ83 ζ8
32.17.1h1 C 1 1 1 −1 −1 −1 −ζ82 ζ82 ζ82 −ζ82 −ζ82 ζ82 ζ83 −ζ8 ζ8 −ζ83 ζ83 −ζ8 ζ8 −ζ83
32.17.1h2 C 1 1 1 −1 −1 −1 ζ82 −ζ82 −ζ82 ζ82 ζ82 −ζ82 −ζ8 ζ83 −ζ83 ζ8 −ζ8 ζ83 −ζ83 ζ8
32.17.1h3 C 1 1 1 −1 −1 −1 −ζ82 ζ82 ζ82 −ζ82 −ζ82 ζ82 −ζ83 ζ8 −ζ8 ζ83 −ζ83 ζ8 −ζ8 ζ83
32.17.1h4 C 1 1 1 −1 −1 −1 ζ82 −ζ82 −ζ82 ζ82 ζ82 −ζ82 ζ8 −ζ83 ζ83 −ζ8 ζ8 −ζ83 ζ83 −ζ8
32.17.2a1 C 2 −2 0 −2ζ82 2ζ82 0 2ζ83 −2ζ8 2ζ8 −2ζ83 0 0 0 0 0 0 0 0 0 0
32.17.2a2 C 2 −2 0 2ζ82 −2ζ82 0 −2ζ8 2ζ83 −2ζ83 2ζ8 0 0 0 0 0 0 0 0 0 0
32.17.2a3 C 2 −2 0 −2ζ82 2ζ82 0 −2ζ83 2ζ8 −2ζ8 2ζ83 0 0 0 0 0 0 0 0 0 0
32.17.2a4 C 2 −2 0 2ζ82 −2ζ82 0 2ζ8 −2ζ83 2ζ83 −2ζ8 0 0 0 0 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