Properties

Label 35T42
35T42 1 18 1->18 29 1->29 2 17 2->17 28 2->28 3 16 3->16 27 3->27 4 20 4->20 26 4->26 5 19 5->19 30 5->30 6 6->4 13 6->13 7 7->3 12 7->12 8 8->2 11 8->11 9 9->1 15 9->15 10 10->5 14 10->14 11->9 12->8 13->7 14->6 15->10 24 16->24 16->28 23 17->23 17->27 22 18->22 18->26 21 19->21 19->30 25 20->25 20->29 33 21->33 32 22->32 31 23->31 35 24->35 34 25->34 26->23 26->34 27->22 27->33 28->21 28->32 29->25 29->31 30->24 30->35 31->3 31->14 32->2 32->13 33->1 33->12 34->5 34->11 35->4 35->15
Degree $35$
Order $25200$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $C_5:S_7$

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, 42);
 
Copy content sage:G = TransitiveGroup(35, 42)
 
Copy content oscar:G = transitive_group(35, 42)
 
Copy content gap:G := TransitiveGroup(35, 42);
 

Group invariants

Abstract group:  $C_5:S_7$
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:  $25200=2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7$
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 $t$:  $42$
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:  $(1,18,26,23,31,3,16,28,21,33)(2,17,27,22,32)(4,20,29,25,34,5,19,30,24,35)(6,13)(7,12)(8,11)(9,15)(10,14)$, $(1,29,31,14,6,4,26,34,11,9)(2,28,32,13,7,3,27,33,12,8)(5,30,35,15,10)(16,24)(17,23)(18,22)(19,21)(20,25)$
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)
$2$:  $C_2$
$10$:  $D_{5}$
$5040$:  $S_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

Degree 7: $S_7$

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 $2^{10},1^{15}$ $105$ $2$ $10$ $( 1,16)( 2,17)( 3,18)( 4,19)( 5,20)(11,21)(12,22)(13,23)(14,24)(15,25)$
2B $2^{15},1^{5}$ $105$ $2$ $15$ $( 1, 7)( 2, 6)( 3,10)( 4, 9)( 5, 8)(11,12)(13,15)(16,17)(18,20)(21,22)(23,25)(26,27)(28,30)(31,32)(33,35)$
2C $2^{17},1$ $525$ $2$ $17$ $( 1,18)( 2,17)( 3,16)( 4,20)( 5,19)( 6,28)( 7,27)( 8,26)( 9,30)(10,29)(11,13)(14,15)(21,33)(22,32)(23,31)(24,35)(25,34)$
3A $3^{5},1^{20}$ $70$ $3$ $10$ $(11,21,16)(12,22,17)(13,23,18)(14,24,19)(15,25,20)$
3B $3^{10},1^{5}$ $280$ $3$ $20$ $( 1, 6,31)( 2, 7,32)( 3, 8,33)( 4, 9,34)( 5,10,35)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)$
4A $4^{5},2^{5},1^{5}$ $630$ $4$ $20$ $( 1,11,16,21)( 2,12,17,22)( 3,13,18,23)( 4,14,19,24)( 5,15,20,25)(26,31)(27,32)(28,33)(29,34)(30,35)$
4B $4^{5},2^{6},1^{3}$ $1050$ $4$ $21$ $( 1,33,26,23)( 2,32,27,22)( 3,31,28,21)( 4,35,29,25)( 5,34,30,24)( 6, 8)( 9,10)(11,13)(14,15)(16,18)(19,20)$
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 $5^{7}$ $2$ $5$ $28$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$
5B $5^{5},1^{10}$ $504$ $5$ $20$ $( 1,11,16, 6,21)( 2,12,17, 7,22)( 3,13,18, 8,23)( 4,14,19, 9,24)( 5,15,20,10,25)$
5C1 $5^{7}$ $1008$ $5$ $28$ $( 1, 3, 5, 2, 4)( 6,18,30,22,14)( 7,19,26,23,15)( 8,20,27,24,11)( 9,16,28,25,12)(10,17,29,21,13)(31,33,35,32,34)$
5C2 $5^{7}$ $1008$ $5$ $28$ $( 1, 5, 4, 3, 2)( 6,30,14,18,22)( 7,26,15,19,23)( 8,27,11,20,24)( 9,28,12,16,25)(10,29,13,17,21)(31,35,34,33,32)$
6A $3^{5},2^{10}$ $210$ $6$ $20$ $( 1,26)( 2,27)( 3,28)( 4,29)( 5,30)( 6,16,11)( 7,17,12)( 8,18,13)( 9,19,14)(10,20,15)(21,31)(22,32)(23,33)(24,34)(25,35)$
6B $6^{2},3,2^{9},1^{2}$ $2100$ $6$ $21$ $( 1, 7)( 2, 6)( 3,10)( 4, 9)( 5, 8)(11,17,21,12,16,22)(13,20,23,15,18,25)(14,19,24)(26,27)(28,30)(31,32)(33,35)$
6C $6^{5},2^{2},1$ $4200$ $6$ $27$ $( 1,23, 6,18,31,28)( 2,22, 7,17,32,27)( 3,21, 8,16,33,26)( 4,25, 9,20,34,30)( 5,24,10,19,35,29)(11,13)(14,15)$
7A $7^{5}$ $720$ $7$ $30$ $( 1,11,16, 6,21,26,31)( 2,12,17, 7,22,27,32)( 3,13,18, 8,23,28,33)( 4,14,19, 9,24,29,34)( 5,15,20,10,25,30,35)$
10A1 $10^{2},5^{3}$ $210$ $10$ $30$ $( 1,20, 4,18, 2,16, 5,19, 3,17)( 6,10, 9, 8, 7)(11,25,14,23,12,21,15,24,13,22)(26,30,29,28,27)(31,35,34,33,32)$
10A3 $10^{2},5^{3}$ $210$ $10$ $30$ $( 1,18, 5,17, 4,16, 3,20, 2,19)( 6, 8,10, 7, 9)(11,23,15,22,14,21,13,25,12,24)(26,28,30,27,29)(31,33,35,32,34)$
10B $10^{2},5,2^{5}$ $2520$ $10$ $27$ $( 1, 9,11,24,16, 4, 6,14,21,19)( 2, 8,12,23,17, 3, 7,13,22,18)( 5,10,15,25,20)(26,34)(27,33)(28,32)(29,31)(30,35)$
12A $6^{2},4^{5},3$ $2100$ $12$ $27$ $( 1,23,26,33)( 2,22,27,32)( 3,21,28,31)( 4,25,29,35)( 5,24,30,34)( 6,13,16, 8,11,18)( 7,12,17)( 9,15,19,10,14,20)$
15A1 $15,5^{4}$ $140$ $15$ $30$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,35,19,13,32,16,15,34,18,12,31,20,14,33,17)(21,25,24,23,22)(26,30,29,28,27)$
15A2 $15,5^{4}$ $140$ $15$ $30$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,19,32,15,18,31,14,17,35,13,16,34,12,20,33)(21,24,22,25,23)(26,29,27,30,28)$
15B1 $15^{2},5$ $560$ $15$ $32$ $( 1,13,20, 2,14,16, 3,15,17, 4,11,18, 5,12,19)( 6,33,30, 7,34,26, 8,35,27, 9,31,28,10,32,29)(21,23,25,22,24)$
15B2 $15^{2},5$ $560$ $15$ $32$ $( 1,20,14, 3,17,11, 5,19,13, 2,16,15, 4,18,12)( 6,30,34, 8,27,31,10,29,33, 7,26,35, 9,28,32)(21,25,24,23,22)$
20A1 $20,10,5$ $1260$ $20$ $32$ $( 1,13,20,22, 4,11,18,25, 2,14,16,23, 5,12,19,21, 3,15,17,24)( 6, 8,10, 7, 9)(26,33,30,32,29,31,28,35,27,34)$
20A3 $20,10,5$ $1260$ $20$ $32$ $( 1,22,18,14, 5,21,17,13, 4,25,16,12, 3,24,20,11, 2,23,19,15)( 6, 7, 8, 9,10)(26,32,28,34,30,31,27,33,29,35)$
30A1 $15,10^{2}$ $420$ $30$ $32$ $( 1,23, 5,22, 4,21, 3,25, 2,24)( 6,28,10,27, 9,26, 8,30, 7,29)(11,18,35,12,19,31,13,20,32,14,16,33,15,17,34)$
30A7 $15,10^{2}$ $420$ $30$ $32$ $( 1,25, 4,23, 2,21, 5,24, 3,22)( 6,30, 9,28, 7,26,10,29, 8,27)(11,20,34,13,17,31,15,19,33,12,16,35,14,18,32)$
35A1 $35$ $720$ $35$ $34$ $( 1,10,34,18,27,11,25, 4, 8,32,16,30,14,23, 2, 6,35,19,28,12,21, 5, 9,33,17,26,15,24, 3, 7,31,20,29,13,22)$
35A2 $35$ $720$ $35$ $34$ $( 1,34,27,25, 8,16,14, 2,35,28,21, 9,17,15, 3,31,29,22,10,18,11, 4,32,30,23, 6,19,12, 5,33,26,24, 7,20,13)$
35A4 $35$ $720$ $35$ $34$ $( 1,27, 8,14,35,21,17, 3,29,10,11,32,23,19, 5,26, 7,13,34,25,16, 2,28, 9,15,31,22,18, 4,30, 6,12,33,24,20)$
35A8 $35$ $720$ $35$ $34$ $( 1, 8,35,17,29,11,23, 5, 7,34,16,28,15,22, 4, 6,33,20,27,14,21, 3,10,32,19,26,13,25, 2, 9,31,18,30,12,24)$

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

33 x 33 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