Properties

Label 32T10
32T10 1 9 1->9 22 1->22 2 10 2->10 21 2->21 3 12 3->12 24 3->24 4 11 4->11 23 4->23 5 27 5->27 32 5->32 6 28 6->28 31 6->31 7 25 7->25 29 7->29 8 26 8->26 30 8->30 19 9->19 9->30 20 10->20 10->29 17 11->17 11->31 18 12->18 12->32 13 13->1 13->7 14 14->2 14->8 15 15->4 15->5 16 16->3 16->6 17->7 17->26 18->8 18->25 19->5 19->28 20->6 20->27 21->9 21->16 22->10 22->15 23->12 23->13 24->11 24->14 25->3 25->15 26->4 26->16 27->2 27->13 28->1 28->14 29->19 29->23 30->20 30->24 31->18 31->21 32->17 32->22
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_2^2:C_8$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(32, 10);
 
Copy content sage:G = TransitiveGroup(32, 10)
 
Copy content oscar:G = transitive_group(32, 10)
 
Copy content gap:G := TransitiveGroup(32, 10);
 

Group invariants

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 4

Degree 8: $C_8$ x 2, $C_4\times C_2$, $D_4$ x 2, $C_8:C_2$, $C_2^2:C_4$ x 2

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

Low degree siblings

16T24 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 $1^{32}$ $1$ $1$ $0$ $()$
2A $2^{16}$ $1$ $2$ $16$ $( 1,18)( 2,17)( 3,19)( 4,20)( 5,24)( 6,23)( 7,21)( 8,22)( 9,25)(10,26)(11,27)(12,28)(13,31)(14,32)(15,30)(16,29)$
2B $2^{16}$ $1$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32)$
2C $2^{16}$ $1$ $2$ $16$ $( 1,19)( 2,20)( 3,18)( 4,17)( 5,22)( 6,21)( 7,23)( 8,24)( 9,28)(10,27)(11,26)(12,25)(13,29)(14,30)(15,32)(16,31)$
2D $2^{16}$ $2$ $2$ $16$ $( 1, 2)( 3, 4)( 5,23)( 6,24)( 7,22)( 8,21)( 9,10)(11,12)(13,32)(14,31)(15,29)(16,30)(17,18)(19,20)(25,26)(27,28)$
2E $2^{16}$ $2$ $2$ $16$ $( 1,20)( 2,19)( 3,17)( 4,18)( 5, 7)( 6, 8)( 9,27)(10,28)(11,25)(12,26)(13,15)(14,16)(21,24)(22,23)(29,32)(30,31)$
4A1 $4^{8}$ $1$ $4$ $24$ $( 1,10,19,27)( 2, 9,20,28)( 3,11,18,26)( 4,12,17,25)( 5,13,22,29)( 6,14,21,30)( 7,15,23,32)( 8,16,24,31)$
4A-1 $4^{8}$ $1$ $4$ $24$ $( 1,27,19,10)( 2,28,20, 9)( 3,26,18,11)( 4,25,17,12)( 5,29,22,13)( 6,30,21,14)( 7,32,23,15)( 8,31,24,16)$
4B1 $4^{8}$ $1$ $4$ $24$ $( 1,26,19,11)( 2,25,20,12)( 3,27,18,10)( 4,28,17, 9)( 5,31,22,16)( 6,32,21,15)( 7,30,23,14)( 8,29,24,13)$
4B-1 $4^{8}$ $1$ $4$ $24$ $( 1,11,19,26)( 2,12,20,25)( 3,10,18,27)( 4, 9,17,28)( 5,16,22,31)( 6,15,21,32)( 7,14,23,30)( 8,13,24,29)$
4C1 $4^{8}$ $2$ $4$ $24$ $( 1, 9,19,28)( 2,10,20,27)( 3,12,18,25)( 4,11,17,26)( 5,32,22,15)( 6,31,21,16)( 7,29,23,13)( 8,30,24,14)$
4C-1 $4^{8}$ $2$ $4$ $24$ $( 1,28,19, 9)( 2,27,20,10)( 3,25,18,12)( 4,26,17,11)( 5,15,22,32)( 6,16,21,31)( 7,13,23,29)( 8,14,24,30)$
8A1 $8^{4}$ $2$ $8$ $28$ $( 1,22,10,29,19, 5,27,13)( 2,21, 9,30,20, 6,28,14)( 3,24,11,31,18, 8,26,16)( 4,23,12,32,17, 7,25,15)$
8A-1 $8^{4}$ $2$ $8$ $28$ $( 1,13,27, 5,19,29,10,22)( 2,14,28, 6,20,30, 9,21)( 3,16,26, 8,18,31,11,24)( 4,15,25, 7,17,32,12,23)$
8A3 $8^{4}$ $2$ $8$ $28$ $( 1,29,27,22,19,13,10, 5)( 2,30,28,21,20,14, 9, 6)( 3,31,26,24,18,16,11, 8)( 4,32,25,23,17,15,12, 7)$
8A-3 $8^{4}$ $2$ $8$ $28$ $( 1, 5,10,13,19,22,27,29)( 2, 6, 9,14,20,21,28,30)( 3, 8,11,16,18,24,26,31)( 4, 7,12,15,17,23,25,32)$
8B1 $8^{4}$ $2$ $8$ $28$ $( 1,21,26,15,19, 6,11,32)( 2,22,25,16,20, 5,12,31)( 3,23,27,14,18, 7,10,30)( 4,24,28,13,17, 8, 9,29)$
8B-1 $8^{4}$ $2$ $8$ $28$ $( 1,14,11,23,19,30,26, 7)( 2,13,12,24,20,29,25, 8)( 3,15,10,21,18,32,27, 6)( 4,16, 9,22,17,31,28, 5)$
8B3 $8^{4}$ $2$ $8$ $28$ $( 1,30,11, 7,19,14,26,23)( 2,29,12, 8,20,13,25,24)( 3,32,10, 6,18,15,27,21)( 4,31, 9, 5,17,16,28,22)$
8B-3 $8^{4}$ $2$ $8$ $28$ $( 1, 6,26,32,19,21,11,15)( 2, 5,25,31,20,22,12,16)( 3, 7,27,30,18,23,10,14)( 4, 8,28,29,17,24, 9,13)$

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 4A1 4A-1 4B1 4B-1 4C1 4C-1 8A1 8A-1 8A3 8A-3 8B1 8B-1 8B3 8B-3
Size 1 1 1 1 2 2 1 1 1 1 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 2C 2C 2C 2C 2C 2C 4A1 4A-1 4A-1 4A1 4B1 4B-1 4B-1 4B1
Type
32.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.5.1b R 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 1 1
32.5.1c R 1 1 1 1 −1 −1 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
32.5.1d R 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
32.5.1e1 C 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 −i i i −i i −i −i i
32.5.1e2 C 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 i −i −i i −i i i −i
32.5.1f1 C 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −i i i −i −i i i −i
32.5.1f2 C 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 i −i −i i i −i −i i
32.5.1g1 C 1 1 −1 −1 −1 1 −ζ82 ζ82 −ζ82 ζ82 ζ82 −ζ82 ζ83 −ζ8 ζ8 −ζ83 −ζ83 ζ8 −ζ8 ζ83
32.5.1g2 C 1 1 −1 −1 −1 1 ζ82 −ζ82 ζ82 −ζ82 −ζ82 ζ82 −ζ8 ζ83 −ζ83 ζ8 ζ8 −ζ83 ζ83 −ζ8
32.5.1g3 C 1 1 −1 −1 −1 1 −ζ82 ζ82 −ζ82 ζ82 ζ82 −ζ82 −ζ83 ζ8 −ζ8 ζ83 ζ83 −ζ8 ζ8 −ζ83
32.5.1g4 C 1 1 −1 −1 −1 1 ζ82 −ζ82 ζ82 −ζ82 −ζ82 ζ82 ζ8 −ζ83 ζ83 −ζ8 −ζ8 ζ83 −ζ83 ζ8
32.5.1h1 C 1 1 −1 −1 1 −1 −ζ82 ζ82 −ζ82 ζ82 −ζ82 ζ82 ζ83 −ζ8 ζ8 −ζ83 ζ83 −ζ8 ζ8 −ζ83
32.5.1h2 C 1 1 −1 −1 1 −1 ζ82 −ζ82 ζ82 −ζ82 ζ82 −ζ82 −ζ8 ζ83 −ζ83 ζ8 −ζ8 ζ83 −ζ83 ζ8
32.5.1h3 C 1 1 −1 −1 1 −1 −ζ82 ζ82 −ζ82 ζ82 −ζ82 ζ82 −ζ83 ζ8 −ζ8 ζ83 −ζ83 ζ8 −ζ8 ζ83
32.5.1h4 C 1 1 −1 −1 1 −1 ζ82 −ζ82 ζ82 −ζ82 ζ82 −ζ82 ζ8 −ζ83 ζ83 −ζ8 ζ8 −ζ83 ζ83 −ζ8
32.5.2a R 2 −2 −2 2 0 0 −2 −2 2 2 0 0 0 0 0 0 0 0 0 0
32.5.2b R 2 −2 −2 2 0 0 2 2 −2 −2 0 0 0 0 0 0 0 0 0 0
32.5.2c1 C 2 −2 2 −2 0 0 −2i 2i 2i −2i 0 0 0 0 0 0 0 0 0 0
32.5.2c2 C 2 −2 2 −2 0 0 2i −2i −2i 2i 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