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Properties

Label 31T7
31T7 1 2 1->2 9 1->9 3 2->3 18 2->18 4 3->4 27 3->27 5 4->5 4->5 6 5->6 14 5->14 7 6->7 23 6->23 7->1 8 7->8 8->9 10 8->10 9->10 19 9->19 11 10->11 28 10->28 11->6 12 11->12 13 12->13 15 12->15 13->14 24 13->24 14->2 14->15 15->11 16 15->16 17 16->17 20 16->20 17->18 29 17->29 18->7 18->19 19->16 19->20 21 20->21 25 20->25 21->3 22 21->22 22->12 22->23 23->21 23->24 24->25 30 24->30 25->8 26 25->26 26->17 26->27 27->26 27->28 28->4 28->29 29->13 29->30 30->22 31 30->31 31->1
Degree $31$
Order $465$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive yes
$p$-group no
Group: $C_{31}:C_{15}$

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, 7);
 
Copy content sage:G = TransitiveGroup(31, 7)
 
Copy content oscar:G = transitive_group(31, 7)
 
Copy content gap:G := TransitiveGroup(31, 7);
 

Group invariants

Abstract group:  $C_{31}:C_{15}$
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:  $465=3 \cdot 5 \cdot 31$
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 $t$:  $7$
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:  $(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31)$, $(1,9,19,16,20,25,8,10,28,4,5,14,2,18,7)(3,27,26,17,29,13,24,30,22,12,15,11,6,23,21)$
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)
$3$:  $C_3$
$5$:  $C_5$
$15$:  $C_{15}$

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 $1^{31}$ $1$ $1$ $0$ $()$
3A1 $3^{10},1$ $31$ $3$ $20$ $( 1,27, 2)( 3, 6,21)( 4,11,15)( 5,16, 9)( 7,26,28)( 8,31,22)(12,20,29)(13,25,23)(14,30,17)(18,19,24)$
3A-1 $3^{10},1$ $31$ $3$ $20$ $( 1, 2,27)( 3,21, 6)( 4,15,11)( 5, 9,16)( 7,28,26)( 8,22,31)(12,29,20)(13,23,25)(14,17,30)(18,24,19)$
5A1 $5^{6},1$ $31$ $5$ $24$ $( 1,31,23,21, 5)( 2, 8,25, 6, 9)( 3,16,27,22,13)( 4,24,29, 7,17)(11,18,12,26,14)(15,19,20,28,30)$
5A-1 $5^{6},1$ $31$ $5$ $24$ $( 1, 5,21,23,31)( 2, 9, 6,25, 8)( 3,13,22,27,16)( 4,17, 7,29,24)(11,14,26,12,18)(15,30,28,20,19)$
5A2 $5^{6},1$ $31$ $5$ $24$ $( 1,23, 5,31,21)( 2,25, 9, 8, 6)( 3,27,13,16,22)( 4,29,17,24, 7)(11,12,14,18,26)(15,20,30,19,28)$
5A-2 $5^{6},1$ $31$ $5$ $24$ $( 1,21,31, 5,23)( 2, 6, 8, 9,25)( 3,22,16,13,27)( 4, 7,24,17,29)(11,26,18,14,12)(15,28,19,30,20)$
15A1 $15^{2},1$ $31$ $15$ $28$ $( 1,25,16,31, 6,27,23, 9,22,21, 2,13, 5, 8, 3)( 4,20,14,24,28,11,29,30,18, 7,15,12,17,19,26)$
15A-1 $15^{2},1$ $31$ $15$ $28$ $( 1, 3, 8, 5,13, 2,21,22, 9,23,27, 6,31,16,25)( 4,26,19,17,12,15, 7,18,30,29,11,28,24,14,20)$
15A2 $15^{2},1$ $31$ $15$ $28$ $( 1,16, 6,23,22, 2, 5, 3,25,31,27, 9,21,13, 8)( 4,14,28,29,18,15,17,26,20,24,11,30, 7,12,19)$
15A-2 $15^{2},1$ $31$ $15$ $28$ $( 1, 8,13,21, 9,27,31,25, 3, 5, 2,22,23, 6,16)( 4,19,12, 7,30,11,24,20,26,17,15,18,29,28,14)$
15A4 $15^{2},1$ $31$ $15$ $28$ $( 1, 6,22, 5,25,27,21, 8,16,23, 2, 3,31, 9,13)( 4,28,18,17,20,11, 7,19,14,29,15,26,24,30,12)$
15A-4 $15^{2},1$ $31$ $15$ $28$ $( 1,13, 9,31, 3, 2,23,16, 8,21,27,25, 5,22, 6)( 4,12,30,24,26,15,29,14,19, 7,11,20,17,18,28)$
15A7 $15^{2},1$ $31$ $15$ $28$ $( 1, 9, 3,23, 8,27, 5, 6,13,31, 2,16,21,25,22)( 4,30,26,29,19,11,17,28,12,24,15,14, 7,20,18)$
15A-7 $15^{2},1$ $31$ $15$ $28$ $( 1,22,25,21,16, 2,31,13, 6, 5,27, 8,23, 3, 9)( 4,18,20, 7,14,15,24,12,28,17,11,19,29,26,30)$
31A1 $31$ $15$ $31$ $30$ $( 1,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
31A-1 $31$ $15$ $31$ $30$ $( 1,29,26,23,20,17,14,11, 8, 5, 2,30,27,24,21,18,15,12, 9, 6, 3,31,28,25,22,19,16,13,10, 7, 4)$

Malle's constant $a(G)$:     $1/20$

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 3A1 3A-1 5A1 5A-1 5A2 5A-2 15A1 15A-1 15A2 15A-2 15A4 15A-4 15A7 15A-7 31A1 31A-1
Size 1 31 31 31 31 31 31 31 31 31 31 31 31 31 31 15 15
3 P 1A 3A-1 3A1 5A2 5A-2 5A-1 5A1 15A2 15A-2 15A4 15A-4 15A-7 15A7 15A-1 15A1 31A1 31A-1
5 P 1A 1A 1A 5A-2 5A2 5A1 5A-1 5A1 5A-1 5A2 5A-2 5A-1 5A1 5A2 5A-2 31A-1 31A1
31 P 1A 3A-1 3A1 1A 1A 1A 1A 3A1 3A-1 3A-1 3A1 3A1 3A-1 3A1 3A-1 31A1 31A-1
Type
465.1.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
465.1.1b1 C 1 ζ3−1 ζ3 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3 ζ3−1 ζ3−1 1 1
465.1.1b2 C 1 ζ3 ζ3−1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3−1 ζ3 ζ3 1 1
465.1.1c1 C 1 1 1 ζ5−2 ζ5−1 ζ52 ζ5 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 1 1
465.1.1c2 C 1 1 1 ζ52 ζ5 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 1 1
465.1.1c3 C 1 1 1 ζ5−1 ζ52 ζ5 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 1 1
465.1.1c4 C 1 1 1 ζ5 ζ5−2 ζ5−1 ζ52 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 1 1
465.1.1d1 C 1 ζ15−5 ζ155 ζ15−6 ζ15−3 ζ156 ζ153 ζ152 ζ15−2 ζ154 ζ15−4 ζ15−1 ζ15−7 ζ157 ζ15 1 1
465.1.1d2 C 1 ζ155 ζ15−5 ζ156 ζ153 ζ15−6 ζ15−3 ζ15−2 ζ152 ζ15−4 ζ154 ζ15 ζ157 ζ15−7 ζ15−1 1 1
465.1.1d3 C 1 ζ15−5 ζ155 ζ156 ζ153 ζ15−6 ζ15−3 ζ15−7 ζ157 ζ15 ζ15−1 ζ15−4 ζ152 ζ15−2 ζ154 1 1
465.1.1d4 C 1 ζ155 ζ15−5 ζ15−6 ζ15−3 ζ156 ζ153 ζ157 ζ15−7 ζ15−1 ζ15 ζ154 ζ15−2 ζ152 ζ15−4 1 1
465.1.1d5 C 1 ζ15−5 ζ155 ζ15−3 ζ156 ζ153 ζ15−6 ζ15−4 ζ154 ζ157 ζ15−7 ζ152 ζ15−1 ζ15 ζ15−2 1 1
465.1.1d6 C 1 ζ155 ζ15−5 ζ153 ζ15−6 ζ15−3 ζ156 ζ154 ζ15−4 ζ15−7 ζ157 ζ15−2 ζ15 ζ15−1 ζ152 1 1
465.1.1d7 C 1 ζ15−5 ζ155 ζ153 ζ15−6 ζ15−3 ζ156 ζ15−1 ζ15 ζ15−2 ζ152 ζ15−7 ζ15−4 ζ154 ζ157 1 1
465.1.1d8 C 1 ζ155 ζ15−5 ζ15−3 ζ156 ζ153 ζ15−6 ζ15 ζ15−1 ζ152 ζ15−2 ζ157 ζ154 ζ15−4 ζ15−7 1 1
465.1.15a1 C 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −ζ31−15−ζ31−13−ζ31−12−ζ31−11−ζ31−6−ζ31−3−1−ζ31−ζ312−ζ314−ζ315−ζ317−ζ318−ζ319−ζ3110−ζ3114 ζ31−15+ζ31−13+ζ31−12+ζ31−11+ζ31−6+ζ31−3+ζ31+ζ312+ζ314+ζ315+ζ317+ζ318+ζ319+ζ3110+ζ3114
465.1.15a2 C 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ζ31−15+ζ31−13+ζ31−12+ζ31−11+ζ31−6+ζ31−3+ζ31+ζ312+ζ314+ζ315+ζ317+ζ318+ζ319+ζ3110+ζ3114 −ζ31−15−ζ31−13−ζ31−12−ζ31−11−ζ31−6−ζ31−3−1−ζ31−ζ312−ζ314−ζ315−ζ317−ζ318−ζ319−ζ3110−ζ3114

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

$f_{ 1 } =$ $1073741824 x^{31} - 8321499136 \left(s^{2}+31 t^{2}\right) x^{29} + 29125246976 \left(s^{2}+31 t^{2}\right)^{2} x^{27} - 60850962432 \left(s^{2}+31 t^{2}\right)^{3} x^{25} + 84515225600 \left(s^{2}+31 t^{2}\right)^{4} x^{23} - 82239815680 \left(s^{2}+31 t^{2}\right)^{5} x^{21} + 57567870976 \left(s^{2}+31 t^{2}\right)^{6} x^{19} - 29297934336 \left(s^{2}+31 t^{2}\right)^{7} x^{17} + 10827497472 \left(s^{2}+31 t^{2}\right)^{8} x^{15} - 2870927360 \left(s^{2}+31 t^{2}\right)^{9} x^{13} + 533172224 \left(s^{2}+31 t^{2}\right)^{10} x^{11} - 66646528 \left(s^{2}+31 t^{2}\right)^{11} x^{9} + 5261568 \left(s^{2}+31 t^{2}\right)^{12} x^{7} - 236096 \left(s^{2}+31 t^{2}\right)^{13} x^{5} + 4960 \left(s^{2}+31 t^{2}\right)^{14} x^{3} - 31 \left(s^{2}+31 t^{2}\right)^{15} x - s \left(s^{2}+31 t^{2}\right)^{15}$ Copy content Toggle raw display