Properties

Label 35T35
35T35 1 21 1->21 29 1->29 2 22 2->22 27 2->27 3 24 3->24 30 3->30 4 23 4->23 26 4->26 5 25 5->25 28 5->28 6 12 6->12 20 6->20 7 11 7->11 17 7->17 8 15 8->15 18 8->18 9 14 9->14 16 9->16 10 13 10->13 19 10->19 11->25 11->30 12->21 12->27 13->24 13->29 14->23 14->28 15->22 15->26 16->19 33 16->33 17->20 31 17->31 18->17 34 18->34 35 19->35 20->18 32 20->32 21->8 21->11 22->6 22->14 23->9 23->13 24->10 24->12 25->7 25->15 26->8 26->33 27->6 27->31 28->9 28->32 29->7 29->35 30->10 30->34 31->2 31->4 32->1 32->3 33->3 33->4 34->2 34->5 35->1 35->5
Degree $35$
Order $20160$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $S_5\times \GL(3,2)$

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

Group invariants

Abstract group:  $S_5\times \GL(3,2)$
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:  $20160=2^{6} \cdot 3^{2} \cdot 5 \cdot 7$
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 $t$:  $35$
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:  $(1,21,11,30,10,19,35,5,25,15,26,8,18,34,2,22,14,28,9,16,33,3,24,12,27,6,20,32)(4,23,13,29,7,17,31)$, $(1,29,35)(2,27,31)(3,30,34,5,28,32)(4,26,33)(6,12,21,8,15,22)(7,11,25)(9,14,23)(10,13,24)(16,19)(17,20,18)$
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)
$2$:  $C_2$
$120$:  $S_5$
$168$:  $\GL(3,2)$
$336$:  14T17

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $S_5$

Degree 7: $\GL(3,2)$

Low degree siblings

35T35, 40T11250, 42T723 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{35}$ $1$ $1$ $0$ $()$
2A $2^{7},1^{21}$ $10$ $2$ $7$ $( 3, 5)( 6, 8)(12,15)(16,19)(21,22)(28,30)(32,34)$
2B $2^{14},1^{7}$ $15$ $2$ $14$ $( 1, 3)( 4, 5)( 6,10)( 7, 8)(12,13)(14,15)(16,17)(18,19)(21,25)(22,23)(27,28)(29,30)(31,32)(33,34)$
2C $2^{10},1^{15}$ $21$ $2$ $10$ $(11,35)(12,32)(13,31)(14,33)(15,34)(21,28)(22,30)(23,29)(24,26)(25,27)$
2D $2^{13},1^{9}$ $210$ $2$ $13$ $( 1,24)( 2,25)( 3,21)( 4,23)( 5,22)( 9,10)(11,18)(12,16)(13,17)(14,20)(15,19)(26,27)(33,35)$
2E $2^{16},1^{3}$ $315$ $2$ $16$ $( 1,13)( 2,15)( 3,11)( 4,14)( 5,12)( 6, 9)( 7,10)(16,22)(17,25)(18,23)(19,24)(20,21)(26,28)(27,29)(31,33)(34,35)$
3A $3^{7},1^{14}$ $20$ $3$ $14$ $( 3, 5, 4)( 6, 8, 7)(12,13,15)(16,17,19)(21,22,23)(28,30,29)(31,34,32)$
3B $3^{10},1^{5}$ $56$ $3$ $20$ $( 1,25,10)( 2,24, 9)( 3,21, 6)( 4,23, 7)( 5,22, 8)(11,20,26)(12,16,30)(13,17,29)(14,18,27)(15,19,28)$
3C $3^{11},1^{2}$ $1120$ $3$ $22$ $( 1,23, 9)( 2,25, 7)( 3,21, 6)( 4,24,10)( 5,22, 8)(11,18,29)(12,16,30)(13,20,27)(14,17,26)(15,19,28)(31,35,33)$
4A $4^{7},1^{7}$ $30$ $4$ $21$ $( 1, 4, 3, 5)( 6, 8,10, 7)(12,14,13,15)(16,18,17,19)(21,22,25,23)(27,29,28,30)(31,34,32,33)$
4B $4^{5},2^{5},1^{5}$ $42$ $4$ $20$ $( 6,19)( 7,17)( 8,16)( 9,20)(10,18)(11,24,35,26)(12,22,32,30)(13,23,31,29)(14,25,33,27)(15,21,34,28)$
4C $4^{5},2^{6},1^{3}$ $420$ $4$ $21$ $( 1,33,10,25)( 2,34, 9,21)( 3,35, 6,24)( 4,31, 7,23)( 5,32, 8,22)(11,28)(12,30)(13,29)(14,27)(15,26)(19,20)$
4D $4^{7},2^{2},1^{3}$ $630$ $4$ $23$ $( 1, 2, 3, 4)( 6,13,10,11)( 7,14, 9,15)( 8,12)(16,32)(17,33,20,34)(18,35,19,31)(21,23,25,24)(26,28,29,27)$
4E $4^{5},2^{7},1$ $630$ $4$ $22$ $( 1,23,10,31)( 2,22, 9,32)( 3,21, 6,34)( 4,25, 7,33)( 5,24, 8,35)(11,30)(12,26)(13,27)(14,29)(15,28)(16,20)(17,18)$
4F $4^{8},2,1$ $1260$ $4$ $25$ $( 1,19,13,24)( 2,18,15,23)( 3,17,11,25)( 4,20,14,21)( 5,16,12,22)( 6,29, 9,27)( 7,26,10,28)( 8,30)(31,35,33,34)$
5A $5^{7}$ $24$ $5$ $28$ $( 1, 4, 2, 3, 5)( 6, 8,10, 7, 9)(11,15,12,14,13)(16,18,17,20,19)(21,22,25,23,24)(26,28,30,27,29)(31,35,34,32,33)$
6A $3^{7},2^{7}$ $20$ $6$ $21$ $( 1, 4, 2)( 3, 5)( 6, 8)( 7, 9,10)(11,14,13)(12,15)(16,19)(17,20,18)(21,22)(23,24,25)(26,27,29)(28,30)(31,35,33)(32,34)$
6B $6^{2},3^{3},2^{7}$ $420$ $6$ $23$ $( 1,24)( 2,25)( 3,23, 5,21, 4,22)( 6, 7, 8)( 9,10)(11,18)(12,19,13,16,15,17)(14,20)(26,27)(28,29,30)(31,32,34)(33,35)$
6C $6^{2},3^{3},2^{4},1^{6}$ $420$ $6$ $20$ $( 1, 7, 5,10, 4, 8)( 2, 9)( 3, 6)(12,14,13)(16,18,17)(21,34)(22,33,23,32,25,31)(24,35)(27,29,30)$
6D $6^{2},3^{6},2,1^{3}$ $560$ $6$ $23$ $( 1,33,14)( 2,32,11, 5,35,12)( 3,34,15)( 4,31,13)( 6,19,21)( 7,17,23)( 8,20,22, 9,16,24)(10,18,25)(26,30)$
6E $6^{4},3^{2},2^{2},1$ $840$ $6$ $26$ $( 1, 6,25, 3,10,21)( 2, 9,24)( 4, 8,23, 5, 7,22)(11,26,20)(12,29,16,13,30,17)(14,28,18,15,27,19)(31,32)(33,34)$
6F $6^{2},3^{7},2$ $1120$ $6$ $25$ $( 1, 9,23)( 2, 7,25)( 3, 8,21, 5, 6,22)( 4,10,24)(11,29,18)(12,28,16,15,30,19)(13,27,20)(14,26,17)(31,33,35)(32,34)$
7A1 $7^{5}$ $24$ $7$ $30$ $( 1,18,25,10,33,14,27)( 2,20,24, 9,35,11,26)( 3,19,21, 6,34,15,28)( 4,17,23, 7,31,13,29)( 5,16,22, 8,32,12,30)$
7A-1 $7^{5}$ $24$ $7$ $30$ $( 1,27,14,33,10,25,18)( 2,26,11,35, 9,24,20)( 3,28,15,34, 6,21,19)( 4,29,13,31, 7,23,17)( 5,30,12,32, 8,22,16)$
10A $10^{2},5^{3}$ $504$ $10$ $30$ $( 1, 3, 4, 5, 2)( 6, 7, 8, 9,10)(11,33,15,31,12,35,14,34,13,32)(16,20,18,19,17)(21,29,22,26,25,28,23,30,24,27)$
12A $12,6,4^{2},3,2^{3}$ $840$ $12$ $27$ $( 1,22, 7,33, 5,23,10,32, 4,25, 8,31)( 2,21, 9,34)( 3,24, 6,35)(11,28)(12,29,14,30,13,27)(15,26)(16,17,18)(19,20)$
12B $12,6,4^{2},3,2^{2},1^{2}$ $840$ $12$ $26$ $( 1, 8, 4,10, 5, 7)( 2, 9)( 3, 6)(11,24,26,35)(12,23,27,32,13,25,30,31,14,22,29,33)(15,21,28,34)(16,17,18)$
12C $12^{2},4,3^{2},1$ $1680$ $12$ $29$ $( 1,22, 6, 4,25, 8, 3,23,10, 5,21, 7)( 2,24, 9)(11,20,26)(12,19,29,14,16,28,13,18,30,15,17,27)(31,33,32,34)$
14A1 $14,7^{3}$ $240$ $14$ $31$ $( 1,25,33,27,18,10,14)( 2,24,35,26,20, 9,11)( 3,22,34,30,19, 8,15, 5,21,32,28,16, 6,12)( 4,23,31,29,17, 7,13)$
14A-1 $14,7^{3}$ $240$ $14$ $31$ $( 1,14,10,18,27,33,25)( 2,11, 9,20,26,35,24)( 3,12, 6,16,28,32,21, 5,15, 8,19,30,34,22)( 4,13, 7,17,29,31,23)$
14B1 $14^{2},7$ $360$ $14$ $32$ $( 1,35,18,11,25,26,10, 2,33,20,14,24,27, 9)( 3,34,19,15,21,28, 6)( 4,32,17,12,23,30, 7, 5,31,16,13,22,29, 8)$
14B-1 $14^{2},7$ $360$ $14$ $32$ $( 1, 9,27,24,14,20,33, 2,10,26,25,11,18,35)( 3, 6,28,21,15,19,34)( 4, 8,29,22,13,16,31, 5, 7,30,23,12,17,32)$
15A $15^{2},5$ $1344$ $15$ $32$ $( 1, 6,17, 5, 9,18, 3, 7,16, 2,10,19, 4, 8,20)(11,25,34,13,22,35,14,21,31,12,24,33,15,23,32)(26,27,28,29,30)$
20A $20,10,5$ $1008$ $20$ $32$ $( 1, 5, 3, 2, 4)( 6,20, 7,18, 8,19, 9,17,10,16)(11,23,33,30,15,24,31,27,12,21,35,29,14,22,34,26,13,25,32,28)$
21A1 $21,7^{2}$ $480$ $21$ $32$ $( 1,26,13,33, 9,23,18, 2,29,14,35, 7,25,20, 4,27,11,31,10,24,17)( 3,28,15,34, 6,21,19)( 5,30,12,32, 8,22,16)$
21A-1 $21,7^{2}$ $480$ $21$ $32$ $( 1,17,24,10,31,11,27, 4,20,25, 7,35,14,29, 2,18,23, 9,33,13,26)( 3,19,21, 6,34,15,28)( 5,16,22, 8,32,12,30)$
28A1 $28,7$ $720$ $28$ $33$ $( 1,22,35,29,18, 8,11, 4,25,32,26,17,10,12, 2,23,33,30,20, 7,14, 5,24,31,27,16, 9,13)( 3,21,34,28,19, 6,15)$
28A-1 $28,7$ $720$ $28$ $33$ $( 1,13, 9,16,27,31,24, 5,14, 7,20,30,33,23, 2,12,10,17,26,32,25, 4,11, 8,18,29,35,22)( 3,15, 6,19,28,34,21)$
35A1 $35$ $576$ $35$ $34$ $( 1,35,22,29, 6,14,20, 5,31,21,27, 9,12,17, 3,33,24,30, 7,15,18, 2,32,23,28,10,11,16, 4,34,25,26, 8,13,19)$
35A-1 $35$ $576$ $35$ $34$ $( 1,19,13, 8,26,25,34, 4,16,11,10,28,23,32, 2,18,15, 7,30,24,33, 3,17,12, 9,27,21,31, 5,20,14, 6,29,22,35)$
42A1 $21,14$ $480$ $42$ $33$ $( 1, 7,26,25,13,20,33, 4, 9,27,23,11,18,31, 2,10,29,24,14,17,35)( 3, 8,28,22,15,16,34, 5, 6,30,21,12,19,32)$
42A-1 $21,14$ $480$ $42$ $33$ $( 1,35,17,14,24,29,10, 2,31,18,11,23,27, 9, 4,33,20,13,25,26, 7)( 3,32,19,12,21,30, 6, 5,34,16,15,22,28, 8)$

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

42 x 42 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