Properties

Label 35T16
35T16 1 16 1->16 33 1->33 2 18 2->18 34 2->34 3 20 3->20 35 3->35 4 17 4->17 31 4->31 5 19 5->19 32 5->32 6 6->3 7 7->4 8 7->8 8->5 10 8->10 9 9->1 9->7 10->2 10->9 11 11->8 11->31 12 12->9 12->33 13 13->10 13->35 14 14->6 14->32 15 15->7 15->34 16->13 21 16->21 17->14 23 17->23 18->15 25 18->25 19->11 22 19->22 20->12 24 20->24 21->11 21->18 22->13 22->19 23->15 23->20 24->12 24->16 25->14 25->17 26 26->1 26->23 27 27->3 27->24 28 28->5 28->25 29 29->2 29->21 30 30->4 30->22 31->26 31->28 32->28 32->29 33->30 33->30 34->26 34->27 35->27 35->29
Degree $35$
Order $420$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_{35}:C_{12}$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(35, 16);
 
Copy content sage:G = TransitiveGroup(35, 16)
 
Copy content oscar:G = transitive_group(35, 16)
 
Copy content gap:G := TransitiveGroup(35, 16);
 

Group invariants

Abstract group:  $C_{35}:C_{12}$
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:  $420=2^{2} \cdot 3 \cdot 5 \cdot 7$
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 $n$:  $35$
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 $t$:  $16$
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);
 
$\card{\Aut(F/K)}$:  $1$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(35).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(35), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(35), G));
 
Generators:  $(1,33,30,22,19,11,8,5,32,29,21,18,15,7,4,31,28,25,17,14,6,3,35,27,24,16,13,10,2,34,26,23,20,12,9)$, $(1,16,21,11,31,26)(2,18,25,14,32,28,5,19,22,13,35,29)(3,20,24,12,33,30,4,17,23,15,34,27)(7,8,10,9)$
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)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$12$:  $C_{12}$
$20$:  $F_5$
$42$:  $F_7$
$60$:  $F_5\times C_3$
$84$:  28T12

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$

Degree 7: $F_7$

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 $1^{35}$ $1$ $1$ $0$ $()$
2A $2^{14},1^{7}$ $5$ $2$ $14$ $( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,22)(23,25)(26,27)(28,30)(31,32)(33,35)$
3A1 $3^{10},1^{5}$ $7$ $3$ $20$ $( 1, 6,16)( 2, 7,17)( 3, 8,18)( 4, 9,19)( 5,10,20)(11,26,21)(12,27,22)(13,28,23)(14,29,24)(15,30,25)$
3A-1 $3^{10},1^{5}$ $7$ $3$ $20$ $( 1,16, 6)( 2,17, 7)( 3,18, 8)( 4,19, 9)( 5,20,10)(11,21,26)(12,22,27)(13,23,28)(14,24,29)(15,25,30)$
4A1 $4^{7},2^{3},1$ $35$ $4$ $24$ $( 1,10, 2, 8)( 3, 6, 5, 7)( 4, 9)(11,35,12,33)(13,31,15,32)(14,34)(16,30,17,28)(18,26,20,27)(19,29)(21,25,22,23)$
4A-1 $4^{7},2^{3},1$ $35$ $4$ $24$ $( 1, 8, 2,10)( 3, 7, 5, 6)( 4, 9)(11,33,12,35)(13,32,15,31)(14,34)(16,28,17,30)(18,27,20,26)(19,29)(21,23,22,25)$
5A $5^{7}$ $4$ $5$ $28$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17)(21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)$
6A1 $6^{4},3^{2},2^{2},1$ $35$ $6$ $26$ $( 1,17,11, 2,16,12)( 3,20,13, 5,18,15)( 4,19,14)( 6,27,31, 7,26,32)( 8,30,33,10,28,35)( 9,29,34)(21,22)(23,25)$
6A-1 $6^{4},3^{2},2^{2},1$ $35$ $6$ $26$ $( 1,12,16, 2,11,17)( 3,15,18, 5,13,20)( 4,14,19)( 6,32,26, 7,31,27)( 8,35,28,10,33,30)( 9,34,29)(21,22)(23,25)$
7A $7^{5}$ $6$ $7$ $30$ $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33)( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$
12A1 $12^{2},6,4,1$ $35$ $12$ $30$ $( 1,33,17,10,11,28, 2,35,16, 8,12,30)( 3,32,20, 6,13,27, 5,31,18, 7,15,26)( 4,34,19, 9,14,29)(21,23,22,25)$
12A-1 $12^{2},6,4,1$ $35$ $12$ $30$ $( 1,30,12, 8,16,35, 2,28,11,10,17,33)( 3,26,15, 7,18,31, 5,27,13, 6,20,32)( 4,29,14, 9,19,34)(21,25,22,23)$
12A5 $12^{2},6,4,1$ $35$ $12$ $30$ $( 1,28,12,10,16,33, 2,30,11, 8,17,35)( 3,27,15, 6,18,32, 5,26,13, 7,20,31)( 4,29,14, 9,19,34)(21,23,22,25)$
12A-5 $12^{2},6,4,1$ $35$ $12$ $30$ $( 1,35,17, 8,11,30, 2,33,16,10,12,28)( 3,31,20, 7,13,26, 5,32,18, 6,15,27)( 4,34,19, 9,14,29)(21,25,22,23)$
14A $14^{2},7$ $30$ $14$ $32$ $( 1,16,31,11,26, 6,21)( 2,20,32,15,27,10,22, 5,17,35,12,30, 7,25)( 3,19,33,14,28, 9,23, 4,18,34,13,29, 8,24)$
15A1 $15^{2},5$ $28$ $15$ $32$ $( 1,20, 9, 3,17, 6, 5,19, 8, 2,16,10, 4,18, 7)(11,25,29,13,22,26,15,24,28,12,21,30,14,23,27)(31,35,34,33,32)$
15A-1 $15^{2},5$ $28$ $15$ $32$ $( 1, 7,18, 4,10,16, 2, 8,19, 5, 6,17, 3, 9,20)(11,27,23,14,30,21,12,28,24,15,26,22,13,29,25)(31,32,33,34,35)$
35A1 $35$ $12$ $35$ $34$ $( 1,18,35,12,29, 6,23, 5,17,34,11,28,10,22, 4,16,33,15,27, 9,21, 3,20,32,14,26, 8,25, 2,19,31,13,30, 7,24)$
35A-1 $35$ $12$ $35$ $34$ $( 1,35,29,23,17,11,10, 4,33,27,21,20,14, 8, 2,31,30,24,18,12, 6, 5,34,28,22,16,15, 9, 3,32,26,25,19,13, 7)$

Malle's constant $a(G)$:     $1/14$

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 3A1 3A-1 4A1 4A-1 5A 6A1 6A-1 7A 12A1 12A-1 12A5 12A-5 14A 15A1 15A-1 35A1 35A-1
Size 1 5 7 7 35 35 4 35 35 6 35 35 35 35 30 28 28 12 12
2 P 1A 1A 3A-1 3A1 2A 2A 5A 3A-1 3A1 7A 6A1 6A-1 6A-1 6A1 7A 15A-1 15A1 35A-1 35A1
3 P 1A 2A 1A 1A 4A-1 4A1 5A 2A 2A 7A 4A1 4A-1 4A1 4A-1 14A 5A 5A 35A1 35A-1
5 P 1A 2A 3A-1 3A1 4A1 4A-1 1A 6A-1 6A1 7A 12A5 12A-5 12A1 12A-1 14A 3A1 3A-1 7A 7A
7 P 1A 2A 3A1 3A-1 4A-1 4A1 5A 6A1 6A-1 1A 12A-5 12A5 12A-1 12A1 2A 15A1 15A-1 5A 5A
Type
420.15.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
420.15.1b R 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1
420.15.1c1 C 1 1 ζ3−1 ζ3 1 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3−1 ζ3 1 ζ3 ζ3−1 1 1
420.15.1c2 C 1 1 ζ3 ζ3−1 1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 ζ3−1 ζ3 1 1
420.15.1d1 C 1 −1 1 1 −i i 1 −1 −1 1 i −i i −i −1 1 1 1 1
420.15.1d2 C 1 −1 1 1 i −i 1 −1 −1 1 −i i −i i −1 1 1 1 1
420.15.1e1 C 1 1 ζ3−1 ζ3 −1 −1 1 ζ3−1 ζ3 1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 ζ3 ζ3−1 1 1
420.15.1e2 C 1 1 ζ3 ζ3−1 −1 −1 1 ζ3 ζ3−1 1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 ζ3−1 ζ3 1 1
420.15.1f1 C 1 −1 −ζ122 ζ124 −ζ123 ζ123 1 ζ122 −ζ124 1 −ζ12 ζ125 −ζ125 ζ12 −1 ζ124 −ζ122 1 1
420.15.1f2 C 1 −1 ζ124 −ζ122 ζ123 −ζ123 1 −ζ124 ζ122 1 ζ125 −ζ12 ζ12 −ζ125 −1 −ζ122 ζ124 1 1
420.15.1f3 C 1 −1 −ζ122 ζ124 ζ123 −ζ123 1 ζ122 −ζ124 1 ζ12 −ζ125 ζ125 −ζ12 −1 ζ124 −ζ122 1 1
420.15.1f4 C 1 −1 ζ124 −ζ122 −ζ123 ζ123 1 −ζ124 ζ122 1 −ζ125 ζ12 −ζ12 ζ125 −1 −ζ122 ζ124 1 1
420.15.4a R 4 0 4 4 0 0 −1 0 0 4 0 0 0 0 0 −1 −1 −1 −1
420.15.4b1 C 4 0 4ζ3−1 4ζ3 0 0 −1 0 0 4 0 0 0 0 0 −ζ3 −ζ3−1 −1 −1
420.15.4b2 C 4 0 4ζ3 4ζ3−1 0 0 −1 0 0 4 0 0 0 0 0 −ζ3−1 −ζ3 −1 −1
420.15.6a R 6 6 0 0 0 0 6 0 0 −1 0 0 0 0 −1 0 0 −1 −1
420.15.6b S 6 −6 0 0 0 0 6 0 0 −1 0 0 0 0 1 0 0 −1 −1
420.15.12a1 C 12 0 0 0 0 0 −3 0 0 −2 0 0 0 0 0 0 0 ζ35−15−ζ35−14+2ζ35−13+1−2ζ354+ζ355+2ζ358−2ζ359+ζ3510−2ζ3511−ζ3514+2ζ3515−2ζ3516 −ζ35−15+ζ35−14−2ζ35−13+2ζ354−ζ355−2ζ358+2ζ359−ζ3510+2ζ3511+ζ3514−2ζ3515+2ζ3516
420.15.12a2 C 12 0 0 0 0 0 −3 0 0 −2 0 0 0 0 0 0 0 −ζ35−15+ζ35−14−2ζ35−13+2ζ354−ζ355−2ζ358+2ζ359−ζ3510+2ζ3511+ζ3514−2ζ3515+2ζ3516 ζ35−15−ζ35−14+2ζ35−13+1−2ζ354+ζ355+2ζ358−2ζ359+ζ3510−2ζ3511−ζ3514+2ζ3515−2ζ3516

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