Properties

Label 35T11
35T11 1 7 1->7 16 1->16 2 6 2->6 20 2->20 3 10 3->10 19 3->19 4 9 4->9 18 4->18 5 8 5->8 17 5->17 11 6->11 22 6->22 15 7->15 21 7->21 14 8->14 25 8->25 13 9->13 24 9->24 12 10->12 23 10->23 11->2 12->1 13->5 14->4 15->3 16->17 18->20 31 21->31 32 21->32 22->31 35 22->35 34 23->34 23->35 33 24->33 24->34 25->32 25->33 26 26->12 27 27->11 30 27->30 28 28->15 29 28->29 29->14 30->13 31->27 32->26 33->30 34->29 35->28
Degree $35$
Order $210$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_{35}:C_6$

Related objects

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(35, 11);
 
Copy content sage:G = TransitiveGroup(35, 11)
 
Copy content oscar:G = transitive_group(35, 11)
 
Copy content gap:G := TransitiveGroup(35, 11);
 

Group invariants

Abstract group:  $C_{35}:C_6$
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:  $210=2 \cdot 3 \cdot 5 \cdot 7$
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 $t$:  $11$
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:  $(1,7,21,32,26,12)(2,6,22,31,27,11)(3,10,23,35,28,15)(4,9,24,34,29,14)(5,8,25,33,30,13)(16,17)(18,20)$, $(1,16)(2,20)(3,19)(4,18)(5,17)(6,11)(7,15)(8,14)(9,13)(10,12)(21,31)(22,35)(23,34)(24,33)(25,32)(27,30)(28,29)$
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)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$10$:  $D_{5}$
$30$:  $D_5\times C_3$
$42$:  $F_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

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 $2^{17},1$ $35$ $2$ $17$ $( 1,21)( 2,25)( 3,24)( 4,23)( 5,22)( 6,16)( 7,20)( 8,19)( 9,18)(10,17)(12,15)(13,14)(26,31)(27,35)(28,34)(29,33)(30,32)$
3A1 $3^{10},1^{5}$ $7$ $3$ $20$ $( 1,16,11)( 2,17,12)( 3,18,13)( 4,19,14)( 5,20,15)( 6,26,31)( 7,27,32)( 8,28,33)( 9,29,34)(10,30,35)$
3A-1 $3^{10},1^{5}$ $7$ $3$ $20$ $( 1,11,16)( 2,12,17)( 3,13,18)( 4,14,19)( 5,15,20)( 6,31,26)( 7,32,27)( 8,33,28)( 9,34,29)(10,35,30)$
5A1 $5^{7}$ $2$ $5$ $28$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18)(21,24,22,25,23)(26,29,27,30,28)(31,34,32,35,33)$
5A2 $5^{7}$ $2$ $5$ $28$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$
6A1 $6^{5},2^{2},1$ $35$ $6$ $27$ $( 1,31, 6,21,26,16)( 2,35, 7,25,27,20)( 3,34, 8,24,28,19)( 4,33, 9,23,29,18)( 5,32,10,22,30,17)(12,15)(13,14)$
6A-1 $6^{5},2^{2},1$ $35$ $6$ $27$ $( 1,16,26,21, 6,31)( 2,20,27,25, 7,35)( 3,19,28,24, 8,34)( 4,18,29,23, 9,33)( 5,17,30,22,10,32)(12,15)(13,14)$
7A $7^{5}$ $6$ $7$ $30$ $( 1,16,31,11,26, 6,21)( 2,17,32,12,27, 7,22)( 3,18,33,13,28, 8,23)( 4,19,34,14,29, 9,24)( 5,20,35,15,30,10,25)$
15A1 $15^{2},5$ $14$ $15$ $32$ $( 1,12,18, 4,15,16, 2,13,19, 5,11,17, 3,14,20)( 6,32,28, 9,35,26, 7,33,29,10,31,27, 8,34,30)(21,22,23,24,25)$
15A-1 $15^{2},5$ $14$ $15$ $32$ $( 1,12,33, 4,15,31, 2,13,34, 5,11,32, 3,14,35)( 6,22,18, 9,25,16, 7,23,19,10,21,17, 8,24,20)(26,27,28,29,30)$
15A2 $15^{2},5$ $14$ $15$ $32$ $( 1,23,30, 2,24,26, 3,25,27, 4,21,28, 5,22,29)( 6,33,15, 7,34,11, 8,35,12, 9,31,13,10,32,14)(16,18,20,17,19)$
15A-2 $15^{2},5$ $14$ $15$ $32$ $( 1,23,35, 2,24,31, 3,25,32, 4,21,33, 5,22,34)( 6, 8,10, 7, 9)(11,28,20,12,29,16,13,30,17,14,26,18,15,27,19)$
35A1 $35$ $6$ $35$ $34$ $( 1,12,23,34,10,16,27, 3,14,25,31, 7,18,29, 5,11,22,33, 9,20,26, 2,13,24,35, 6,17,28, 4,15,21,32, 8,19,30)$
35A2 $35$ $6$ $35$ $34$ $( 1,23,10,27,14,31,18, 5,22, 9,26,13,35,17, 4,21, 8,30,12,34,16, 3,25, 7,29,11,33,20, 2,24, 6,28,15,32,19)$
35A4 $35$ $6$ $35$ $34$ $( 1,10,14,18,22,26,35, 4, 8,12,16,25,29,33, 2, 6,15,19,23,27,31, 5, 9,13,17,21,30,34, 3, 7,11,20,24,28,32)$
35A8 $35$ $6$ $35$ $34$ $( 1,18,35,12,29, 6,23, 5,17,34,11,28,10,22, 4,16,33,15,27, 9,21, 3,20,32,14,26, 8,25, 2,19,31,13,30, 7,24)$

Malle's constant $a(G)$:     $1/17$

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 5A1 5A2 6A1 6A-1 7A 15A1 15A-1 15A2 15A-2 35A1 35A2 35A4 35A8
Size 1 35 7 7 2 2 35 35 6 14 14 14 14 6 6 6 6
2 P 1A 1A 3A-1 3A1 5A2 5A1 3A-1 3A1 7A 15A2 15A-2 15A1 15A-1 35A2 35A4 35A8 35A1
3 P 1A 2A 1A 1A 5A2 5A1 2A 2A 7A 5A1 5A1 5A2 5A2 35A2 35A4 35A8 35A1
5 P 1A 2A 3A-1 3A1 1A 1A 6A-1 6A1 7A 3A1 3A-1 3A-1 3A1 7A 7A 7A 7A
7 P 1A 2A 3A1 3A-1 5A2 5A1 6A1 6A-1 1A 15A-2 15A2 15A-1 15A1 5A1 5A2 5A1 5A2
Type
210.3.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
210.3.1b R 1 −1 1 1 1 1 −1 −1 1 1 1 1 1 1 1 1 1
210.3.1c1 C 1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1
210.3.1c2 C 1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1
210.3.1d1 C 1 −1 ζ3−1 ζ3 1 1 −ζ3−1 −ζ3 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1
210.3.1d2 C 1 −1 ζ3 ζ3−1 1 1 −ζ3 −ζ3−1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1
210.3.2a1 R 2 0 2 2 ζ5−2+ζ52 ζ5−1+ζ5 0 0 2 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52
210.3.2a2 R 2 0 2 2 ζ5−1+ζ5 ζ5−2+ζ52 0 0 2 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5
210.3.2b1 C 2 0 2ζ15−5 2ζ155 ζ15−6+ζ156 ζ15−3+ζ153 0 0 2 1−ζ15−ζ152+ζ153−ζ154−ζ157 ζ15+ζ154 1−ζ15−ζ154+ζ155 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156
210.3.2b2 C 2 0 2ζ155 2ζ15−5 ζ15−6+ζ156 ζ15−3+ζ153 0 0 2 ζ15+ζ154 1−ζ15−ζ152+ζ153−ζ154−ζ157 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157 1−ζ15−ζ154+ζ155 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156
210.3.2b3 C 2 0 2ζ15−5 2ζ155 ζ15−3+ζ153 ζ15−6+ζ156 0 0 2 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157 1−ζ15−ζ154+ζ155 ζ15+ζ154 1−ζ15−ζ152+ζ153−ζ154−ζ157 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153
210.3.2b4 C 2 0 2ζ155 2ζ15−5 ζ15−3+ζ153 ζ15−6+ζ156 0 0 2 1−ζ15−ζ154+ζ155 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157 1−ζ15−ζ152+ζ153−ζ154−ζ157 ζ15+ζ154 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153
210.3.6a R 6 0 0 0 6 6 0 0 −1 0 0 0 0 −1 −1 −1 −1
210.3.6b1 R 6 0 0 0 3ζ35−14+3ζ3514 3ζ35−7+3ζ357 0 0 −1 0 0 0 0 2ζ35−17−ζ35−13+2ζ35−12+ζ35−10+ζ35−5+ζ354−ζ357+ζ358+ζ359+ζ3511+ζ3516 ζ35−13+2ζ35−ζ354+ζ358−ζ359+ζ3511−ζ3514+ζ3515+ζ3516 −2ζ35−17+ζ35−13−2ζ35−12−ζ35−10−ζ35−7−ζ35−5−ζ354−ζ358−ζ359−ζ3511−ζ3516 −ζ35−14−ζ35−13−2ζ35+ζ354−ζ358+ζ359−ζ3511−ζ3515−ζ3516
210.3.6b2 R 6 0 0 0 3ζ35−14+3ζ3514 3ζ35−7+3ζ357 0 0 −1 0 0 0 0 −2ζ35−17+ζ35−13−2ζ35−12−ζ35−10−ζ35−7−ζ35−5−ζ354−ζ358−ζ359−ζ3511−ζ3516 −ζ35−14−ζ35−13−2ζ35+ζ354−ζ358+ζ359−ζ3511−ζ3515−ζ3516 2ζ35−17−ζ35−13+2ζ35−12+ζ35−10+ζ35−5+ζ354−ζ357+ζ358+ζ359+ζ3511+ζ3516 ζ35−13+2ζ35−ζ354+ζ358−ζ359+ζ3511−ζ3514+ζ3515+ζ3516
210.3.6b3 R 6 0 0 0 3ζ35−7+3ζ357 3ζ35−14+3ζ3514 0 0 −1 0 0 0 0 ζ35−13+2ζ35−ζ354+ζ358−ζ359+ζ3511−ζ3514+ζ3515+ζ3516 −2ζ35−17+ζ35−13−2ζ35−12−ζ35−10−ζ35−7−ζ35−5−ζ354−ζ358−ζ359−ζ3511−ζ3516 −ζ35−14−ζ35−13−2ζ35+ζ354−ζ358+ζ359−ζ3511−ζ3515−ζ3516 2ζ35−17−ζ35−13+2ζ35−12+ζ35−10+ζ35−5+ζ354−ζ357+ζ358+ζ359+ζ3511+ζ3516
210.3.6b4 R 6 0 0 0 3ζ35−7+3ζ357 3ζ35−14+3ζ3514 0 0 −1 0 0 0 0 −ζ35−14−ζ35−13−2ζ35+ζ354−ζ358+ζ359−ζ3511−ζ3515−ζ3516 2ζ35−17−ζ35−13+2ζ35−12+ζ35−10+ζ35−5+ζ354−ζ357+ζ358+ζ359+ζ3511+ζ3516 ζ35−13+2ζ35−ζ354+ζ358−ζ359+ζ3511−ζ3514+ζ3515+ζ3516 −2ζ35−17+ζ35−13−2ζ35−12−ζ35−10−ζ35−7−ζ35−5−ζ354−ζ358−ζ359−ζ3511−ζ3516

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