Properties

Label 35T14
35T14 1 8 1->8 22 1->22 2 7 2->7 24 2->24 3 6 3->6 21 3->21 4 10 4->10 23 4->23 5 9 5->9 25 5->25 18 6->18 27 6->27 17 7->17 29 7->29 16 8->16 26 8->26 20 9->20 28 9->28 19 10->19 30 10->30 11 11->28 32 11->32 12 12->27 34 12->34 13 13->26 31 13->31 14 14->30 33 14->33 15 15->29 35 15->35 16->2 16->3 17->2 17->4 18->1 18->1 19->3 19->5 20->4 20->5 21->7 21->13 22->9 22->12 23->6 23->11 24->8 24->15 25->10 25->14 26->12 26->23 27->14 27->22 28->11 28->21 29->13 29->25 30->15 30->24 31->17 31->33 32->19 33->16 34->18 34->35 35->20
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, 14);
 
Copy content sage:G = TransitiveGroup(35, 14)
 
Copy content oscar:G = transitive_group(35, 14)
 
Copy content gap:G := TransitiveGroup(35, 14);
 

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 $t$:  $14$
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:  $(1,22,9,28,11,32,19,3,21,7,29,13,31,17,4,23,6,27,14,33,16,2,24,8,26,12,34,18)(5,25,10,30,15,35,20)$, $(1,8,16,3,6,18)(2,7,17)(4,10,19,5,9,20)(11,28,21,13,26,23)(12,27,22)(14,30,24,15,29,25)(31,33)(34,35)$
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)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$12$:  $C_{12}$
$20$:  $F_5$
$21$:  $C_7:C_3$
$42$:  $(C_7:C_3) \times C_2$
$60$:  $F_5\times C_3$
$84$:  28T13

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$

Degree 7: $C_7:C_3$

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

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 7A1 7A-1 12A1 12A-1 12A5 12A-5 14A1 14A-1 15A1 15A-1 28A1 28A-1 28A5 28A-5 35A1 35A-1
Size 1 5 7 7 5 5 4 35 35 3 3 35 35 35 35 15 15 28 28 15 15 15 15 12 12
2 P 1A 1A 3A-1 3A1 2A 2A 5A 3A-1 3A1 7A1 7A-1 6A1 6A-1 6A-1 6A1 7A1 7A-1 15A-1 15A1 14A1 14A-1 14A-1 14A1 35A1 35A-1
3 P 1A 2A 1A 1A 4A-1 4A1 5A 2A 2A 7A-1 7A1 4A1 4A-1 4A1 4A-1 14A-1 14A1 5A 5A 28A-1 28A1 28A-5 28A5 35A-1 35A1
5 P 1A 2A 3A-1 3A1 4A1 4A-1 1A 6A-1 6A1 7A-1 7A1 12A5 12A-5 12A1 12A-1 14A-1 14A1 3A1 3A-1 28A5 28A-5 28A1 28A-1 7A1 7A-1
7 P 1A 2A 3A1 3A-1 4A-1 4A1 5A 6A1 6A-1 1A 1A 12A-5 12A5 12A-1 12A1 2A 2A 15A1 15A-1 4A1 4A-1 4A1 4A-1 5A 5A
Type
420.14.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
420.14.1b R 1 1 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1
420.14.1c1 C 1 1 ζ3−1 ζ3 1 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 1 1 1 1 1 1
420.14.1c2 C 1 1 ζ3 ζ3−1 1 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3−1 ζ3 1 1 1 1 1 1
420.14.1d1 C 1 −1 1 1 −i i 1 −1 −1 1 1 i −i i −i −1 −1 1 1 i −i i −i 1 1
420.14.1d2 C 1 −1 1 1 i −i 1 −1 −1 1 1 −i i −i i −1 −1 1 1 −i i −i i 1 1
420.14.1e1 C 1 1 ζ3−1 ζ3 −1 −1 1 ζ3−1 ζ3 1 1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 1 ζ3 ζ3−1 −1 −1 −1 −1 1 1
420.14.1e2 C 1 1 ζ3 ζ3−1 −1 −1 1 ζ3 ζ3−1 1 1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 1 ζ3−1 ζ3 −1 −1 −1 −1 1 1
420.14.1f1 C 1 −1 −ζ122 ζ124 −ζ123 ζ123 1 ζ122 −ζ124 1 1 −ζ12 ζ125 −ζ125 ζ12 −1 −1 ζ124 −ζ122 ζ123 −ζ123 ζ123 −ζ123 1 1
420.14.1f2 C 1 −1 ζ124 −ζ122 ζ123 −ζ123 1 −ζ124 ζ122 1 1 ζ125 −ζ12 ζ12 −ζ125 −1 −1 −ζ122 ζ124 −ζ123 ζ123 −ζ123 ζ123 1 1
420.14.1f3 C 1 −1 −ζ122 ζ124 ζ123 −ζ123 1 ζ122 −ζ124 1 1 ζ12 −ζ125 ζ125 −ζ12 −1 −1 ζ124 −ζ122 −ζ123 ζ123 −ζ123 ζ123 1 1
420.14.1f4 C 1 −1 ζ124 −ζ122 −ζ123 ζ123 1 −ζ124 ζ122 1 1 −ζ125 ζ12 −ζ12 ζ125 −1 −1 −ζ122 ζ124 ζ123 −ζ123 ζ123 −ζ123 1 1
420.14.3a1 C 3 3 0 0 3 3 3 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
420.14.3a2 C 3 3 0 0 3 3 3 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
420.14.3b1 C 3 3 0 0 −3 −3 3 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 0 0 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
420.14.3b2 C 3 3 0 0 −3 −3 3 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 0 0 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
420.14.3c1 C 3 −3 0 0 −3ζ287 3ζ287 3 0 0 −1+ζ282−ζ284−ζ288 −ζ282+ζ284+ζ288 0 0 0 0 1−ζ282+ζ284+ζ288 ζ282−ζ284−ζ288 0 0 ζ28−ζ287+ζ289−ζ2811 ζ28+ζ289−ζ2811 −ζ28−ζ289+ζ2811 −ζ28+ζ287−ζ289+ζ2811 −ζ282+ζ284+ζ288 −1+ζ282−ζ284−ζ288
420.14.3c2 C 3 −3 0 0 3ζ287 −3ζ287 3 0 0 −ζ282+ζ284+ζ288 −1+ζ282−ζ284−ζ288 0 0 0 0 ζ282−ζ284−ζ288 1−ζ282+ζ284+ζ288 0 0 ζ28+ζ289−ζ2811 ζ28−ζ287+ζ289−ζ2811 −ζ28+ζ287−ζ289+ζ2811 −ζ28−ζ289+ζ2811 −1+ζ282−ζ284−ζ288 −ζ282+ζ284+ζ288
420.14.3c3 C 3 −3 0 0 −3ζ287 3ζ287 3 0 0 −ζ282+ζ284+ζ288 −1+ζ282−ζ284−ζ288 0 0 0 0 ζ282−ζ284−ζ288 1−ζ282+ζ284+ζ288 0 0 −ζ28−ζ289+ζ2811 −ζ28+ζ287−ζ289+ζ2811 ζ28−ζ287+ζ289−ζ2811 ζ28+ζ289−ζ2811 −1+ζ282−ζ284−ζ288 −ζ282+ζ284+ζ288
420.14.3c4 C 3 −3 0 0 3ζ287 −3ζ287 3 0 0 −1+ζ282−ζ284−ζ288 −ζ282+ζ284+ζ288 0 0 0 0 1−ζ282+ζ284+ζ288 ζ282−ζ284−ζ288 0 0 −ζ28+ζ287−ζ289+ζ2811 −ζ28−ζ289+ζ2811 ζ28+ζ289−ζ2811 ζ28−ζ287+ζ289−ζ2811 −ζ282+ζ284+ζ288 −1+ζ282−ζ284−ζ288
420.14.4a R 4 0 4 4 0 0 −1 0 0 4 4 0 0 0 0 0 0 −1 −1 0 0 0 0 −1 −1
420.14.4b1 C 4 0 4ζ3−1 4ζ3 0 0 −1 0 0 4 4 0 0 0 0 0 0 −ζ3 −ζ3−1 0 0 0 0 −1 −1
420.14.4b2 C 4 0 4ζ3 4ζ3−1 0 0 −1 0 0 4 4 0 0 0 0 0 0 −ζ3−1 −ζ3 0 0 0 0 −1 −1
420.14.12a1 C 12 0 0 0 0 0 −3 0 0 −4ζ7−3−4−4ζ7−4ζ72 4ζ7−3+4ζ7+4ζ72 0 0 0 0 0 0 0 0 0 0 0 0 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72
420.14.12a2 C 12 0 0 0 0 0 −3 0 0 4ζ7−3+4ζ7+4ζ72 −4ζ7−3−4−4ζ7−4ζ72 0 0 0 0 0 0 0 0 0 0 0 0 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72

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