Properties

Label 35T34
35T34 1 12 1->12 35 1->35 2 13 2->13 31 2->31 3 14 3->14 32 3->32 4 15 4->15 33 4->33 5 11 5->11 34 5->34 6 6->5 27 6->27 7 7->1 28 7->28 8 8->2 29 8->29 9 9->3 30 9->30 10 10->4 26 10->26 22 11->22 25 11->25 21 12->21 23 12->23 13->22 24 13->24 14->23 14->25 15->21 15->24 16 16->2 16->15 17 17->3 17->11 18 18->4 18->12 19 19->5 19->13 20 20->1 20->14 21->17 21->20 22->16 22->18 23->17 23->19 24->18 24->20 25->16 25->19 26->7 26->30 27->8 27->26 28->9 28->27 29->10 29->28 30->6 30->29 31->10 31->32 32->6 32->33 33->7 33->34 34->8 34->35 35->9 35->31
Degree $35$
Order $12600$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $C_5\times A_7$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(35, 34);
 
Copy content sage:G = TransitiveGroup(35, 34)
 
Copy content oscar:G = transitive_group(35, 34)
 
Copy content gap:G := TransitiveGroup(35, 34);
 

Group invariants

Abstract group:  $C_5\times A_7$
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:  $12600=2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7$
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:  no
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 $t$:  $34$
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);
 
$\card{\Aut(F/K)}$:  $5$
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:  $(1,12,23,19,5,11,22,18,4,15,21,17,3,14,25,16,2,13,24,20)(6,27,8,29,10,26,7,28,9,30)(31,32,33,34,35)$, $(1,35,9,3,32,6,5,34,8,2,31,10,4,33,7)(11,25,19,13,22,16,15,24,18,12,21,20,14,23,17)(26,30,29,28,27)$
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)
$5$:  $C_5$
$2520$:  $A_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $C_5$

Degree 7: $A_7$

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 $2^{10},1^{15}$ $105$ $2$ $10$ $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,31)(12,32)(13,33)(14,34)(15,35)$
3A $3^{5},1^{20}$ $70$ $3$ $10$ $(11,21,26)(12,22,27)(13,23,28)(14,24,29)(15,25,30)$
3B $3^{10},1^{5}$ $280$ $3$ $20$ $( 1,16,26)( 2,17,27)( 3,18,28)( 4,19,29)( 5,20,30)( 6,21,31)( 7,22,32)( 8,23,33)( 9,24,34)(10,25,35)$
4A $4^{5},2^{5},1^{5}$ $630$ $4$ $20$ $( 6,31,21,11)( 7,32,22,12)( 8,33,23,13)( 9,34,24,14)(10,35,25,15)(16,26)(17,27)(18,28)(19,29)(20,30)$
5A1 $5^{7}$ $1$ $5$ $28$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$
5A-1 $5^{7}$ $1$ $5$ $28$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17)(21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)$
5A2 $5^{7}$ $1$ $5$ $28$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19)(21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)$
5A-2 $5^{7}$ $1$ $5$ $28$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18)(21,24,22,25,23)(26,29,27,30,28)(31,34,32,35,33)$
5B $5^{5},1^{10}$ $504$ $5$ $20$ $( 1,16,31,21,26)( 2,17,32,22,27)( 3,18,33,23,28)( 4,19,34,24,29)( 5,20,35,25,30)$
5C1 $5^{7}$ $504$ $5$ $28$ $( 1, 7,18,34,30)( 2, 8,19,35,26)( 3, 9,20,31,27)( 4,10,16,32,28)( 5, 6,17,33,29)(11,12,13,14,15)(21,22,23,24,25)$
5C-1 $5^{7}$ $504$ $5$ $28$ $( 1,30,34,18, 7)( 2,26,35,19, 8)( 3,27,31,20, 9)( 4,28,32,16,10)( 5,29,33,17, 6)(11,15,14,13,12)(21,25,24,23,22)$
5C2 $5^{7}$ $504$ $5$ $28$ $( 1,18,30, 7,34)( 2,19,26, 8,35)( 3,20,27, 9,31)( 4,16,28,10,32)( 5,17,29, 6,33)(11,13,15,12,14)(21,23,25,22,24)$
5C-2 $5^{7}$ $504$ $5$ $28$ $( 1,34, 7,30,18)( 2,35, 8,26,19)( 3,31, 9,27,20)( 4,32,10,28,16)( 5,33, 6,29,17)(11,14,12,15,13)(21,24,22,25,23)$
6A $3^{5},2^{10}$ $210$ $6$ $20$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)(11,26,21)(12,27,22)(13,28,23)(14,29,24)(15,30,25)(16,31)(17,32)(18,33)(19,34)(20,35)$
7A1 $7^{5}$ $360$ $7$ $30$ $( 1,21,11,16,26, 6,31)( 2,22,12,17,27, 7,32)( 3,23,13,18,28, 8,33)( 4,24,14,19,29, 9,34)( 5,25,15,20,30,10,35)$
7A-1 $7^{5}$ $360$ $7$ $30$ $( 1,31, 6,26,16,11,21)( 2,32, 7,27,17,12,22)( 3,33, 8,28,18,13,23)( 4,34, 9,29,19,14,24)( 5,35,10,30,20,15,25)$
10A1 $10^{2},5^{3}$ $105$ $10$ $30$ $( 1, 3, 5, 2, 4)( 6,23,10,22, 9,21, 8,25, 7,24)(11,33,15,32,14,31,13,35,12,34)(16,18,20,17,19)(26,28,30,27,29)$
10A-1 $10^{2},5^{3}$ $105$ $10$ $30$ $( 1, 4, 2, 5, 3)( 6,24, 7,25, 8,21, 9,22,10,23)(11,34,12,35,13,31,14,32,15,33)(16,19,17,20,18)(26,29,27,30,28)$
10A3 $10^{2},5^{3}$ $105$ $10$ $30$ $( 1, 2, 3, 4, 5)( 6,22, 8,24,10,21, 7,23, 9,25)(11,32,13,34,15,31,12,33,14,35)(16,17,18,19,20)(26,27,28,29,30)$
10A-3 $10^{2},5^{3}$ $105$ $10$ $30$ $( 1, 5, 4, 3, 2)( 6,25, 9,23, 7,21,10,24, 8,22)(11,35,14,33,12,31,15,34,13,32)(16,20,19,18,17)(26,30,29,28,27)$
15A1 $15,5^{4}$ $70$ $15$ $30$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,28,25,12,29,21,13,30,22,14,26,23,15,27,24)(16,18,20,17,19)(31,33,35,32,34)$
15A-1 $15,5^{4}$ $70$ $15$ $30$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,24,27,15,23,26,14,22,30,13,21,29,12,25,28)(16,19,17,20,18)(31,34,32,35,33)$
15A2 $15,5^{4}$ $70$ $15$ $30$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,25,29,13,22,26,15,24,28,12,21,30,14,23,27)(16,20,19,18,17)(31,35,34,33,32)$
15A-2 $15,5^{4}$ $70$ $15$ $30$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,27,23,14,30,21,12,28,24,15,26,22,13,29,25)(16,17,18,19,20)(31,32,33,34,35)$
15B1 $15^{2},5$ $280$ $15$ $32$ $( 1,28,20, 2,29,16, 3,30,17, 4,26,18, 5,27,19)( 6,33,25, 7,34,21, 8,35,22, 9,31,23,10,32,24)(11,13,15,12,14)$
15B-1 $15^{2},5$ $280$ $15$ $32$ $( 1,19,27, 5,18,26, 4,17,30, 3,16,29, 2,20,28)( 6,24,32,10,23,31, 9,22,35, 8,21,34, 7,25,33)(11,14,12,15,13)$
15B2 $15^{2},5$ $280$ $15$ $32$ $( 1,20,29, 3,17,26, 5,19,28, 2,16,30, 4,18,27)( 6,25,34, 8,22,31,10,24,33, 7,21,35, 9,23,32)(11,15,14,13,12)$
15B-2 $15^{2},5$ $280$ $15$ $32$ $( 1,27,18, 4,30,16, 2,28,19, 5,26,17, 3,29,20)( 6,32,23, 9,35,21, 7,33,24,10,31,22, 8,34,25)(11,12,13,14,15)$
20A1 $20,10,5$ $630$ $20$ $32$ $( 1, 2, 3, 4, 5)( 6,32,23,14,10,31,22,13, 9,35,21,12, 8,34,25,11, 7,33,24,15)(16,27,18,29,20,26,17,28,19,30)$
20A-1 $20,10,5$ $630$ $20$ $32$ $( 1, 5, 4, 3, 2)( 6,15,24,33, 7,11,25,34, 8,12,21,35, 9,13,22,31,10,14,23,32)(16,30,19,28,17,26,20,29,18,27)$
20A3 $20,10,5$ $630$ $20$ $32$ $( 1, 4, 2, 5, 3)( 6,14,22,35, 8,11,24,32,10,13,21,34, 7,15,23,31, 9,12,25,33)(16,29,17,30,18,26,19,27,20,28)$
20A-3 $20,10,5$ $630$ $20$ $32$ $( 1, 3, 5, 2, 4)( 6,33,25,12, 9,31,23,15, 7,34,21,13,10,32,24,11, 8,35,22,14)(16,28,20,27,19,26,18,30,17,29)$
30A1 $15,10^{2}$ $210$ $30$ $32$ $( 1, 7, 3, 9, 5, 6, 2, 8, 4,10)(11,22,28,14,25,26,12,23,29,15,21,27,13,24,30)(16,32,18,34,20,31,17,33,19,35)$
30A-1 $15,10^{2}$ $210$ $30$ $32$ $( 1,10, 4, 8, 2, 6, 5, 9, 3, 7)(11,30,24,13,27,21,15,29,23,12,26,25,14,28,22)(16,35,19,33,17,31,20,34,18,32)$
30A7 $15,10^{2}$ $210$ $30$ $32$ $( 1, 8, 5, 7, 4, 6, 3,10, 2, 9)(11,23,30,12,24,26,13,25,27,14,21,28,15,22,29)(16,33,20,32,19,31,18,35,17,34)$
30A-7 $15,10^{2}$ $210$ $30$ $32$ $( 1, 9, 2,10, 3, 6, 4, 7, 5, 8)(11,29,22,15,28,21,14,27,25,13,26,24,12,30,23)(16,34,17,35,18,31,19,32,20,33)$
35A1 $35$ $360$ $35$ $34$ $( 1,19,32,15, 8,21,29, 2,20,33,11, 9,22,30, 3,16,34,12,10,23,26, 4,17,35,13, 6,24,27, 5,18,31,14, 7,25,28)$
35A-1 $35$ $360$ $35$ $34$ $( 1,28,25, 7,14,31,18, 5,27,24, 6,13,35,17, 4,26,23,10,12,34,16, 3,30,22, 9,11,33,20, 2,29,21, 8,15,32,19)$
35A2 $35$ $360$ $35$ $34$ $( 1,32, 8,29,20,11,22, 3,34,10,26,17,13,24, 5,31, 7,28,19,15,21, 2,33, 9,30,16,12,23, 4,35, 6,27,18,14,25)$
35A-2 $35$ $360$ $35$ $34$ $( 1,25,14,18,27, 6,35, 4,23,12,16,30, 9,33, 2,21,15,19,28, 7,31, 5,24,13,17,26,10,34, 3,22,11,20,29, 8,32)$
35A4 $35$ $360$ $35$ $34$ $( 1, 8,20,22,34,26,13, 5, 7,19,21,33,30,12, 4, 6,18,25,32,29,11, 3,10,17,24,31,28,15, 2, 9,16,23,35,27,14)$
35A-4 $35$ $360$ $35$ $34$ $( 1,14,27,35,23,16, 9, 2,15,28,31,24,17,10, 3,11,29,32,25,18, 6, 4,12,30,33,21,19, 7, 5,13,26,34,22,20, 8)$
35A8 $35$ $360$ $35$ $34$ $( 1,20,34,13, 7,21,30, 4,18,32,11,10,24,28, 2,16,35,14, 8,22,26, 5,19,33,12, 6,25,29, 3,17,31,15, 9,23,27)$
35A-8 $35$ $360$ $35$ $34$ $( 1,27,23, 9,15,31,17, 3,29,25, 6,12,33,19, 5,26,22, 8,14,35,16, 2,28,24,10,11,32,18, 4,30,21, 7,13,34,20)$

Malle's constant $a(G)$:     $1/10$

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

45 x 45 character table

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