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Properties

Label 31T2
31T2 1 2 1->2 30 1->30 3 2->3 29 2->29 4 3->4 28 3->28 5 4->5 27 4->27 6 5->6 26 5->26 7 6->7 25 6->25 8 7->8 24 7->24 9 8->9 23 8->23 10 9->10 22 9->22 11 10->11 21 10->21 12 11->12 20 11->20 13 12->13 19 12->19 14 13->14 18 13->18 15 14->15 17 14->17 16 15->16 15->16 16->17 17->18 18->19 19->20 20->21 21->22 22->23 23->24 24->25 25->26 26->27 27->28 28->29 29->30 31 30->31 31->1
Degree $31$
Order $62$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive yes
$p$-group no
Group: $D_{31}$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(31, 2);
 
Copy content sage:G = TransitiveGroup(31, 2)
 
Copy content oscar:G = transitive_group(31, 2)
 
Copy content gap:G := TransitiveGroup(31, 2);
 

Group invariants

Abstract group:  $D_{31}$
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:  $62=2 \cdot 31$
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:   not nilpotent
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 $n$:  $31$
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:  yes
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);
 
$\card{\Aut(F/K)}$:  $1$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(31).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(31), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(31), G));
 
Generators:  $(1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)$, $(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31)$
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)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{31}$ $1$ $1$ $0$ $()$
2A $2^{15},1$ $31$ $2$ $15$ $( 1,20)( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(21,31)(22,30)(23,29)(24,28)(25,27)$
31A1 $31$ $2$ $31$ $30$ $( 1,28,24,20,16,12, 8, 4,31,27,23,19,15,11, 7, 3,30,26,22,18,14,10, 6, 2,29,25,21,17,13, 9, 5)$
31A2 $31$ $2$ $31$ $30$ $( 1,24,16, 8,31,23,15, 7,30,22,14, 6,29,21,13, 5,28,20,12, 4,27,19,11, 3,26,18,10, 2,25,17, 9)$
31A3 $31$ $2$ $31$ $30$ $( 1,20, 8,27,15, 3,22,10,29,17, 5,24,12,31,19, 7,26,14, 2,21, 9,28,16, 4,23,11,30,18, 6,25,13)$
31A4 $31$ $2$ $31$ $30$ $( 1,16,31,15,30,14,29,13,28,12,27,11,26,10,25, 9,24, 8,23, 7,22, 6,21, 5,20, 4,19, 3,18, 2,17)$
31A5 $31$ $2$ $31$ $30$ $( 1,12,23, 3,14,25, 5,16,27, 7,18,29, 9,20,31,11,22, 2,13,24, 4,15,26, 6,17,28, 8,19,30,10,21)$
31A6 $31$ $2$ $31$ $30$ $( 1, 8,15,22,29, 5,12,19,26, 2, 9,16,23,30, 6,13,20,27, 3,10,17,24,31, 7,14,21,28, 4,11,18,25)$
31A7 $31$ $2$ $31$ $30$ $( 1, 4, 7,10,13,16,19,22,25,28,31, 3, 6, 9,12,15,18,21,24,27,30, 2, 5, 8,11,14,17,20,23,26,29)$
31A8 $31$ $2$ $31$ $30$ $( 1,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
31A9 $31$ $2$ $31$ $30$ $( 1,27,22,17,12, 7, 2,28,23,18,13, 8, 3,29,24,19,14, 9, 4,30,25,20,15,10, 5,31,26,21,16,11, 6)$
31A10 $31$ $2$ $31$ $30$ $( 1,23,14, 5,27,18, 9,31,22,13, 4,26,17, 8,30,21,12, 3,25,16, 7,29,20,11, 2,24,15, 6,28,19,10)$
31A11 $31$ $2$ $31$ $30$ $( 1,19, 6,24,11,29,16, 3,21, 8,26,13,31,18, 5,23,10,28,15, 2,20, 7,25,12,30,17, 4,22, 9,27,14)$
31A12 $31$ $2$ $31$ $30$ $( 1,15,29,12,26, 9,23, 6,20, 3,17,31,14,28,11,25, 8,22, 5,19, 2,16,30,13,27,10,24, 7,21, 4,18)$
31A13 $31$ $2$ $31$ $30$ $( 1,11,21,31,10,20,30, 9,19,29, 8,18,28, 7,17,27, 6,16,26, 5,15,25, 4,14,24, 3,13,23, 2,12,22)$
31A14 $31$ $2$ $31$ $30$ $( 1, 7,13,19,25,31, 6,12,18,24,30, 5,11,17,23,29, 4,10,16,22,28, 3, 9,15,21,27, 2, 8,14,20,26)$
31A15 $31$ $2$ $31$ $30$ $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31, 2, 4, 6, 8,10,12,14,16,18,20,22,24,26,28,30)$

Malle's constant $a(G)$:     $1/15$

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 31A1 31A2 31A3 31A4 31A5 31A6 31A7 31A8 31A9 31A10 31A11 31A12 31A13 31A14 31A15
Size 1 31 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 31A2 31A4 31A6 31A8 31A10 31A12 31A14 31A15 31A13 31A11 31A9 31A7 31A5 31A3 31A1
31 P 1A 2A 31A3 31A6 31A9 31A12 31A15 31A13 31A10 31A7 31A4 31A1 31A2 31A5 31A8 31A11 31A14
Type
62.1.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
62.1.1b R 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
62.1.2a1 R 2 0 ζ31−15+ζ3115 ζ31−1+ζ31 ζ31−14+ζ3114 ζ31−2+ζ312 ζ31−13+ζ3113 ζ31−3+ζ313 ζ31−12+ζ3112 ζ31−4+ζ314 ζ31−11+ζ3111 ζ31−5+ζ315 ζ31−10+ζ3110 ζ31−6+ζ316 ζ31−9+ζ319 ζ31−7+ζ317 ζ31−8+ζ318
62.1.2a2 R 2 0 ζ31−14+ζ3114 ζ31−3+ζ313 ζ31−11+ζ3111 ζ31−6+ζ316 ζ31−8+ζ318 ζ31−9+ζ319 ζ31−5+ζ315 ζ31−12+ζ3112 ζ31−2+ζ312 ζ31−15+ζ3115 ζ31−1+ζ31 ζ31−13+ζ3113 ζ31−4+ζ314 ζ31−10+ζ3110 ζ31−7+ζ317
62.1.2a3 R 2 0 ζ31−13+ζ3113 ζ31−5+ζ315 ζ31−8+ζ318 ζ31−10+ζ3110 ζ31−3+ζ313 ζ31−15+ζ3115 ζ31−2+ζ312 ζ31−11+ζ3111 ζ31−7+ζ317 ζ31−6+ζ316 ζ31−12+ζ3112 ζ31−1+ζ31 ζ31−14+ζ3114 ζ31−4+ζ314 ζ31−9+ζ319
62.1.2a4 R 2 0 ζ31−12+ζ3112 ζ31−7+ζ317 ζ31−5+ζ315 ζ31−14+ζ3114 ζ31−2+ζ312 ζ31−10+ζ3110 ζ31−9+ζ319 ζ31−3+ζ313 ζ31−15+ζ3115 ζ31−4+ζ314 ζ31−8+ζ318 ζ31−11+ζ3111 ζ31−1+ζ31 ζ31−13+ζ3113 ζ31−6+ζ316
62.1.2a5 R 2 0 ζ31−11+ζ3111 ζ31−9+ζ319 ζ31−2+ζ312 ζ31−13+ζ3113 ζ31−7+ζ317 ζ31−4+ζ314 ζ31−15+ζ3115 ζ31−5+ζ315 ζ31−6+ζ316 ζ31−14+ζ3114 ζ31−3+ζ313 ζ31−8+ζ318 ζ31−12+ζ3112 ζ31−1+ζ31 ζ31−10+ζ3110
62.1.2a6 R 2 0 ζ31−10+ζ3110 ζ31−11+ζ3111 ζ31−1+ζ31 ζ31−9+ζ319 ζ31−12+ζ3112 ζ31−2+ζ312 ζ31−8+ζ318 ζ31−13+ζ3113 ζ31−3+ζ313 ζ31−7+ζ317 ζ31−14+ζ3114 ζ31−4+ζ314 ζ31−6+ζ316 ζ31−15+ζ3115 ζ31−5+ζ315
62.1.2a7 R 2 0 ζ31−9+ζ319 ζ31−13+ζ3113 ζ31−4+ζ314 ζ31−5+ζ315 ζ31−14+ζ3114 ζ31−8+ζ318 ζ31−1+ζ31 ζ31−10+ζ3110 ζ31−12+ζ3112 ζ31−3+ζ313 ζ31−6+ζ316 ζ31−15+ζ3115 ζ31−7+ζ317 ζ31−2+ζ312 ζ31−11+ζ3111
62.1.2a8 R 2 0 ζ31−8+ζ318 ζ31−15+ζ3115 ζ31−7+ζ317 ζ31−1+ζ31 ζ31−9+ζ319 ζ31−14+ζ3114 ζ31−6+ζ316 ζ31−2+ζ312 ζ31−10+ζ3110 ζ31−13+ζ3113 ζ31−5+ζ315 ζ31−3+ζ313 ζ31−11+ζ3111 ζ31−12+ζ3112 ζ31−4+ζ314
62.1.2a9 R 2 0 ζ31−7+ζ317 ζ31−14+ζ3114 ζ31−10+ζ3110 ζ31−3+ζ313 ζ31−4+ζ314 ζ31−11+ζ3111 ζ31−13+ζ3113 ζ31−6+ζ316 ζ31−1+ζ31 ζ31−8+ζ318 ζ31−15+ζ3115 ζ31−9+ζ319 ζ31−2+ζ312 ζ31−5+ζ315 ζ31−12+ζ3112
62.1.2a10 R 2 0 ζ31−6+ζ316 ζ31−12+ζ3112 ζ31−13+ζ3113 ζ31−7+ζ317 ζ31−1+ζ31 ζ31−5+ζ315 ζ31−11+ζ3111 ζ31−14+ζ3114 ζ31−8+ζ318 ζ31−2+ζ312 ζ31−4+ζ314 ζ31−10+ζ3110 ζ31−15+ζ3115 ζ31−9+ζ319 ζ31−3+ζ313
62.1.2a11 R 2 0 ζ31−5+ζ315 ζ31−10+ζ3110 ζ31−15+ζ3115 ζ31−11+ζ3111 ζ31−6+ζ316 ζ31−1+ζ31 ζ31−4+ζ314 ζ31−9+ζ319 ζ31−14+ζ3114 ζ31−12+ζ3112 ζ31−7+ζ317 ζ31−2+ζ312 ζ31−3+ζ313 ζ31−8+ζ318 ζ31−13+ζ3113
62.1.2a12 R 2 0 ζ31−4+ζ314 ζ31−8+ζ318 ζ31−12+ζ3112 ζ31−15+ζ3115 ζ31−11+ζ3111 ζ31−7+ζ317 ζ31−3+ζ313 ζ31−1+ζ31 ζ31−5+ζ315 ζ31−9+ζ319 ζ31−13+ζ3113 ζ31−14+ζ3114 ζ31−10+ζ3110 ζ31−6+ζ316 ζ31−2+ζ312
62.1.2a13 R 2 0 ζ31−3+ζ313 ζ31−6+ζ316 ζ31−9+ζ319 ζ31−12+ζ3112 ζ31−15+ζ3115 ζ31−13+ζ3113 ζ31−10+ζ3110 ζ31−7+ζ317 ζ31−4+ζ314 ζ31−1+ζ31 ζ31−2+ζ312 ζ31−5+ζ315 ζ31−8+ζ318 ζ31−11+ζ3111 ζ31−14+ζ3114
62.1.2a14 R 2 0 ζ31−2+ζ312 ζ31−4+ζ314 ζ31−6+ζ316 ζ31−8+ζ318 ζ31−10+ζ3110 ζ31−12+ζ3112 ζ31−14+ζ3114 ζ31−15+ζ3115 ζ31−13+ζ3113 ζ31−11+ζ3111 ζ31−9+ζ319 ζ31−7+ζ317 ζ31−5+ζ315 ζ31−3+ζ313 ζ31−1+ζ31
62.1.2a15 R 2 0 ζ31−1+ζ31 ζ31−2+ζ312 ζ31−3+ζ313 ζ31−4+ζ314 ζ31−5+ζ315 ζ31−6+ζ316 ζ31−7+ζ317 ζ31−8+ζ318 ζ31−9+ζ319 ζ31−10+ζ3110 ζ31−11+ζ3111 ζ31−12+ζ3112 ζ31−13+ζ3113 ζ31−14+ζ3114 ζ31−15+ζ3115

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