Properties

Label 35T30
35T30 1 7 1->7 21 1->21 2 9 2->9 22 2->22 3 8 3->8 23 3->23 4 6 4->6 25 4->25 5 10 5->10 24 5->24 32 6->32 33 6->33 34 7->34 35 7->35 31 8->31 8->33 9->31 9->34 10->32 10->35 11 11->8 11->21 12 12->7 12->22 13 13->6 13->24 14 14->9 14->25 15 15->10 15->23 16 16->14 17 16->17 17->15 18 18->12 20 18->20 19 19->11 19->18 20->13 20->19 21->4 26 21->26 22->5 28 22->28 23->2 29 23->29 24->1 30 24->30 25->3 27 25->27 26->1 27->3 27->30 28->2 29->4 30->5 31->11 31->16 32->13 32->18 33->14 33->19 34->12 34->17 35->15 35->20
Degree $35$
Order $5040$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $F_7\times S_5$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(35, 30);
 
Copy content sage:G = TransitiveGroup(35, 30)
 
Copy content oscar:G = transitive_group(35, 30)
 
Copy content gap:G := TransitiveGroup(35, 30);
 

Group invariants

Abstract group:  $F_7\times S_5$
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:  $5040=2^{4} \cdot 3^{2} \cdot 5 \cdot 7$
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:  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 $t$:  $30$
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:  $(1,21,26)(2,22,28)(3,23,29,4,25,27)(5,24,30)(6,32,13)(7,34,12)(8,33,14,9,31,11)(10,35,15)(16,17)(18,20,19)$, $(1,7,35,20,13,24)(2,9,34,17,15,23)(3,8,31,16,14,25)(4,6,33,19,11,21)(5,10,32,18,12,22)(27,30)$
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)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$12$:  $C_6\times C_2$
$42$:  $F_7$
$84$:  $F_7 \times C_2$
$120$:  $S_5$
$240$:  $S_5\times C_2$
$360$:  $S_5 \times C_3$
$720$:  30T180

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $S_5$

Degree 7: $F_7$

Low degree siblings

42T417

Siblings are shown 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 $2^{15},1^{5}$ $7$ $2$ $15$ $( 1,35)( 2,34)( 3,31)( 4,33)( 5,32)( 6,30)( 7,28)( 8,29)( 9,27)(10,26)(11,23)(12,22)(13,24)(14,25)(15,21)$
2B $2^{7},1^{21}$ $10$ $2$ $7$ $( 1, 5)( 7,10)(12,13)(18,20)(22,24)(26,28)(32,35)$
2C $2^{14},1^{7}$ $15$ $2$ $14$ $( 1, 3)( 2, 4)( 6, 9)( 7, 8)(11,15)(13,14)(16,20)(17,19)(21,23)(24,25)(27,30)(28,29)(31,35)(33,34)$
2D $2^{16},1^{3}$ $70$ $2$ $16$ $( 1,18)( 2,19)( 3,16)( 4,17)( 5,20)( 6,15)( 7,12)( 8,14)( 9,11)(10,13)(21,34)(22,35)(23,33)(24,32)(25,31)(26,28)$
2E $2^{17},1$ $105$ $2$ $17$ $( 1,35)( 2,31)( 3,34)( 4,32)( 5,33)( 6,29)( 7,28)( 8,30)( 9,26)(10,27)(11,22)(12,23)(13,24)(14,21)(15,25)(16,19)(17,18)$
3A1 $3^{10},1^{5}$ $7$ $3$ $20$ $( 1,13,35)( 2,15,34)( 3,14,31)( 4,11,33)( 5,12,32)( 6,21,19)( 7,24,20)( 8,25,16)( 9,23,17)(10,22,18)$
3A-1 $3^{10},1^{5}$ $7$ $3$ $20$ $( 1,35,13)( 2,34,15)( 3,31,14)( 4,33,11)( 5,32,12)( 6,19,21)( 7,20,24)( 8,16,25)( 9,17,23)(10,18,22)$
3B $3^{7},1^{14}$ $20$ $3$ $14$ $( 1, 4, 5)( 7, 9,10)(11,12,13)(17,18,20)(22,24,23)(26,28,27)(32,35,33)$
3C1 $3^{11},1^{2}$ $140$ $3$ $22$ $( 1,28, 7)( 2,30, 6)( 3,26, 9)( 4,29,10)( 5,27, 8)(11,14,12)(16,22,33)(17,25,32)(18,23,31)(19,21,34)(20,24,35)$
3C-1 $3^{11},1^{2}$ $140$ $3$ $22$ $( 1, 7,28)( 2, 6,30)( 3, 9,26)( 4,10,29)( 5, 8,27)(11,12,14)(16,33,22)(17,32,25)(18,31,23)(19,34,21)(20,35,24)$
4A $4^{7},1^{7}$ $30$ $4$ $21$ $( 1, 5, 3, 4)( 7,10, 8, 9)(11,13,12,14)(16,17,20,18)(22,25,23,24)(26,29,27,28)(31,33,35,32)$
4B $4^{7},2^{3},1$ $210$ $4$ $24$ $( 1, 2, 3, 4)( 6,31, 9,35)( 7,34, 8,33)(10,32)(11,28,15,29)(12,26)(13,30,14,27)(16,23,20,21)(17,24,19,25)(18,22)$
5A $5^{7}$ $24$ $5$ $28$ $( 1, 5, 2, 3, 4)( 6, 8, 9, 7,10)(11,13,12,15,14)(16,17,20,18,19)(21,25,23,24,22)(26,30,29,27,28)(31,33,35,32,34)$
6A1 $6^{5},1^{5}$ $7$ $6$ $25$ $( 6,19,15,34,21,30)( 7,20,13,35,24,28)( 8,16,14,31,25,29)( 9,17,11,33,23,27)(10,18,12,32,22,26)$
6A-1 $6^{5},1^{5}$ $7$ $6$ $25$ $( 6,30,21,34,15,19)( 7,28,24,35,13,20)( 8,29,25,31,14,16)( 9,27,23,33,11,17)(10,26,22,32,12,18)$
6B $3^{7},2^{7}$ $20$ $6$ $21$ $( 1, 3)( 2, 4, 5)( 6, 9,10)( 7, 8)(11,12,15)(13,14)(16,20)(17,18,19)(21,23,22)(24,25)(26,30,27)(28,29)(31,35)(32,34,33)$
6C1 $6^{5},2,1^{3}$ $70$ $6$ $26$ $( 1,22,13,18,35,10)( 2,21,15,19,34, 6)( 3,25,14,16,31, 8)( 4,23,11,17,33, 9)( 5,24,12,20,32, 7)(26,28)$
6C-1 $6^{5},2,1^{3}$ $70$ $6$ $26$ $( 1,10,35,18,13,22)( 2, 6,34,19,15,21)( 3, 8,31,16,14,25)( 4, 9,33,17,11,23)( 5, 7,32,20,12,24)(26,28)$
6D1 $6^{2},3^{6},2,1^{3}$ $70$ $6$ $23$ $( 1,12,35, 5,13,32)( 2,15,34)( 3,14,31)( 4,11,33)( 6,21,19)( 7,22,20,10,24,18)( 8,25,16)( 9,23,17)(26,28)$
6D-1 $6^{2},3^{6},2,1^{3}$ $70$ $6$ $23$ $( 1,32,13, 5,35,12)( 2,34,15)( 3,31,14)( 4,33,11)( 6,19,21)( 7,18,24,10,20,22)( 8,16,25)( 9,17,23)(26,28)$
6E1 $6^{4},3^{2},2^{2},1$ $105$ $6$ $26$ $( 1, 3)( 2, 4)( 6,23,15, 9,21,11)( 7,25,13, 8,24,14)(10,22,12)(16,28,31,20,29,35)(17,30,33,19,27,34)(18,26,32)$
6E-1 $6^{4},3^{2},2^{2},1$ $105$ $6$ $26$ $( 1, 3)( 2, 4)( 6,11,21, 9,15,23)( 7,14,24, 8,13,25)(10,12,22)(16,35,29,20,31,28)(17,34,27,19,33,30)(18,32,26)$
6F1 $6^{5},2^{2},1$ $105$ $6$ $27$ $( 1,13,28,35,24, 7)( 2,14,30,31,21, 8)( 3,15,29,34,25, 6)( 4,12,27,32,23,10)( 5,11,26,33,22, 9)(16,19)(17,18)$
6F-1 $6^{5},2^{2},1$ $105$ $6$ $27$ $( 1, 7,24,35,28,13)( 2, 8,21,31,30,14)( 3, 6,25,34,29,15)( 4,10,23,32,27,12)( 5, 9,22,33,26,11)(16,19)(17,18)$
6G $6^{3},3,2^{6},1^{2}$ $140$ $6$ $23$ $( 1,32, 4,35, 5,33)( 2,34)( 3,31)( 6,30)( 7,26, 9,28,10,27)( 8,29)(11,24,12,23,13,22)(14,25)(15,21)(17,20,18)$
6H $6^{3},3,2^{7}$ $140$ $6$ $24$ $( 1,25, 5,24, 3,22)( 2,23)( 4,21)( 6,17)( 7,16,10,20, 8,18)( 9,19)(11,15)(12,13,14)(26,35,29,32,28,31)(27,34)(30,33)$
6I1 $6^{5},3,2$ $140$ $6$ $28$ $( 1,19,28,21, 7,34)( 2,20,30,24, 6,35)( 3,17,26,25, 9,32)( 4,18,29,23,10,31)( 5,16,27,22, 8,33)(11,12,14)(13,15)$
6I-1 $6^{5},3,2$ $140$ $6$ $28$ $( 1,34, 7,21,28,19)( 2,35, 6,24,30,20)( 3,32, 9,25,26,17)( 4,31,10,23,29,18)( 5,33, 8,22,27,16)(11,14,12)(13,15)$
6J1 $6^{2},3^{7},2$ $140$ $6$ $25$ $( 1,30,23)( 2,27,24)( 3,26,25, 5,29,22)( 4,28,21)( 6,11,35)( 7,15,33)( 8,12,31,10,14,32)( 9,13,34)(16,18)(17,20,19)$
6J-1 $6^{2},3^{7},2$ $140$ $6$ $25$ $( 1,23,30)( 2,24,27)( 3,22,29, 5,25,26)( 4,21,28)( 6,35,11)( 7,33,15)( 8,32,14,10,31,12)( 9,34,13)(16,18)(17,19,20)$
6K1 $6^{5},3,1^{2}$ $140$ $6$ $27$ $( 1,35, 7,24,28,20)( 2,33,10,21,27,18)( 3,31, 8,25,29,16)( 4,32, 6,23,26,19)( 5,34, 9,22,30,17)(11,12,15)$
6K-1 $6^{5},3,1^{2}$ $140$ $6$ $27$ $( 1,20,28,24, 7,35)( 2,18,27,21,10,33)( 3,16,29,25, 8,31)( 4,19,26,23, 6,32)( 5,17,30,22, 9,34)(11,15,12)$
7A $7^{5}$ $6$ $7$ $30$ $( 1,20,35,13,28, 7,24)( 2,19,34,15,30, 6,21)( 3,16,31,14,29, 8,25)( 4,17,33,11,27, 9,23)( 5,18,32,12,26,10,22)$
10A $10^{3},5$ $168$ $10$ $31$ $( 1, 3, 5, 4, 2)( 6,35, 8,32, 9,34, 7,31,10,33)(11,30,13,29,12,27,15,28,14,26)(16,22,17,21,20,25,18,23,19,24)$
12A1 $12^{2},6,4,1$ $210$ $12$ $30$ $( 1, 4, 3, 2)( 6,28,23,31,15,20, 9,29,21,35,11,16)( 7,27,25,34,13,17, 8,30,24,33,14,19)(10,26,22,32,12,18)$
12A-1 $12^{2},6,4,1$ $210$ $12$ $30$ $( 1, 2, 3, 4)( 6,16,11,35,21,29, 9,20,15,31,23,28)( 7,19,14,33,24,30, 8,17,13,34,25,27)(10,18,12,32,22,26)$
12B1 $12^{2},4,3^{2},1$ $210$ $12$ $29$ $( 1,23,31, 5,24,33, 3,22,35, 4,25,32)( 2,21,34)( 7, 9, 8,10)(11,29,18,13,27,16,12,28,17,14,26,20)(15,30,19)$
12B-1 $12^{2},4,3^{2},1$ $210$ $12$ $29$ $( 1,32,25, 4,35,22, 3,33,24, 5,31,23)( 2,34,21)( 7,10, 8, 9)(11,20,26,14,17,28,12,16,27,13,18,29)(15,19,30)$
14A $14,7^{3}$ $60$ $14$ $31$ $( 1, 8,13,16,24,29,35, 3, 7,14,20,25,28,31)( 2, 6,15,19,21,30,34)( 4, 9,11,17,23,27,33)( 5,10,12,18,22,26,32)$
14B $14^{2},7$ $90$ $14$ $32$ $( 1,21, 7,30,13,34,20, 2,24, 6,28,15,35,19)( 3,22, 8,26,14,32,16, 5,25,10,29,12,31,18)( 4,23, 9,27,11,33,17)$
15A1 $15^{2},5$ $168$ $15$ $32$ $( 1, 2, 4, 5, 3)( 6,23,12, 8,24,15, 9,22,14, 7,21,11,10,25,13)(16,28,34,17,26,31,20,30,33,18,29,35,19,27,32)$
15A-1 $15^{2},5$ $168$ $15$ $32$ $( 1, 3, 5, 4, 2)( 6,13,25,10,11,21, 7,14,22, 9,15,24, 8,12,23)(16,32,27,19,35,29,18,33,30,20,31,26,17,34,28)$
21A $21,7^{2}$ $120$ $21$ $32$ $( 1,20,35,13,28, 7,24)( 2,18,33,15,26, 9,21, 5,17,34,12,27, 6,22, 4,19,32,11,30,10,23)( 3,16,31,14,29, 8,25)$
28A $28,7$ $180$ $28$ $33$ $( 1,12,21,31, 7,18,30, 3,13,22,34, 8,20,26, 2,14,24,32, 6,16,28, 5,15,25,35,10,19,29)( 4,11,23,33, 9,17,27)$
30A1 $30,5$ $168$ $30$ $33$ $( 1, 5, 2, 3, 4)( 6,29,23,35,12,19, 8,27,24,32,15,16, 9,28,22,34,14,17, 7,26,21,31,11,20,10,30,25,33,13,18)$
30A-1 $30,5$ $168$ $30$ $33$ $( 1, 4, 3, 2, 5)( 6,18,13,33,25,30,10,20,11,31,21,26, 7,17,14,34,22,28, 9,16,15,32,24,27, 8,19,12,35,23,29)$
35A $35$ $144$ $35$ $34$ $( 1,12,25,34, 9,20,26, 3,15,23,35,10,16,30, 4,13,22,31, 6,17,28, 5,14,21,33, 7,18,29, 2,11,24,32, 8,19,27)$
42A $21,14$ $120$ $42$ $33$ $( 1,29,20, 8,35,25,13, 3,28,16, 7,31,24,14)( 2,27,18, 6,33,22,15, 4,26,19, 9,32,21,11, 5,30,17,10,34,23,12)$

Malle's constant $a(G)$:     $1/7$

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

49 x 49 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