Properties

Label 35T23
35T23 1 2 1->2 18 1->18 5 2->5 17 2->17 3 19 3->19 4 16 4->16 5->1 20 5->20 6 23 6->23 32 6->32 7 22 7->22 34 7->34 8 21 8->21 31 8->31 9 25 9->25 33 9->33 10 24 10->24 35 10->35 11 27 11->27 29 11->29 12 28 12->28 12->28 13 26 13->26 30 13->30 14 14->29 14->30 15 15->26 15->27 16->25 16->34 17->23 17->31 18->24 18->35 19->22 19->33 20->21 20->32 21->4 21->18 22->1 22->20 23->3 24->5 24->19 25->2 26->7 26->13 27->8 28->10 28->15 29->6 30->9 30->12 31->15 32->7 32->13 33->14 34->10 34->11 35->6 35->12
Degree $35$
Order $1680$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $D_7\times S_5$

Related objects

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(35, 23);
 
Copy content sage:G = TransitiveGroup(35, 23)
 
Copy content oscar:G = transitive_group(35, 23)
 
Copy content gap:G := TransitiveGroup(35, 23);
 

Group invariants

Abstract group:  $D_7\times S_5$
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:  $1680=2^{4} \cdot 3 \cdot 5 \cdot 7$
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:  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 $t$:  $23$
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:  $(1,18,35,12,28,10,24,5,20,32,13,26,7,22)(2,17,31,15,27,8,21,4,16,34,11,29,6,23,3,19,33,14,30,9,25)$, $(1,2,5)(6,32,7,34,10,35)(8,31)(9,33)(11,27)(12,28,15,26,13,30)(14,29)(16,25)(17,23)(18,24,19,22,20,21)$
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
$4$:  $C_2^2$
$14$:  $D_{7}$
$28$:  $D_{14}$
$120$:  $S_5$
$240$:  $S_5\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $S_5$

Degree 7: $D_{7}$

Low degree siblings

42T218

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 $2^{7},1^{21}$ $10$ $2$ $7$ $( 1, 2)( 6, 7)(13,15)(19,20)(21,24)(28,30)(34,35)$
2C $2^{14},1^{7}$ $15$ $2$ $14$ $( 2, 5)( 3, 4)( 6,10)( 8, 9)(11,14)(12,15)(16,17)(18,19)(21,22)(23,25)(26,30)(27,29)(31,33)(32,34)$
2D $2^{16},1^{3}$ $70$ $2$ $16$ $( 1,35)( 2,34)( 3,31)( 4,32)( 5,33)( 6,30)( 7,28)( 8,29)( 9,26)(10,27)(11,22)(12,23)(13,24)(14,25)(15,21)(17,18)$
2E $2^{17},1$ $105$ $2$ $17$ $( 1,18)( 2,19)( 3,17)( 4,16)( 5,20)( 6,15)( 7,12)( 8,11)( 9,14)(10,13)(21,34)(22,35)(23,31)(24,32)(25,33)(26,28)(27,29)$
3A $3^{7},1^{14}$ $20$ $3$ $14$ $( 3, 5, 4)( 8,10, 9)(11,14,12)(16,18,17)(22,23,25)(26,27,29)(31,32,33)$
4A $4^{7},1^{7}$ $30$ $4$ $21$ $( 2, 3, 5, 4)( 6, 8,10, 9)(11,15,14,12)(16,18,17,19)(21,25,22,23)(26,27,30,29)(31,32,33,34)$
4B $4^{7},2^{3},1$ $210$ $4$ $24$ $( 1,32, 4,34)( 2,35, 5,33)( 3,31)( 6,28,10,27)( 7,26, 9,30)( 8,29)(11,21,13,22)(12,23,15,24)(14,25)(17,19,20,18)$
5A $5^{7}$ $24$ $5$ $28$ $( 1, 3, 5, 4, 2)( 6, 7, 8,10, 9)(11,15,13,14,12)(16,18,17,19,20)(21,24,25,22,23)(26,27,30,28,29)(31,32,33,34,35)$
6A $3^{7},2^{7}$ $20$ $6$ $21$ $( 1, 2)( 3, 4, 5)( 6, 7)( 8, 9,10)(11,12,14)(13,15)(16,17,18)(19,20)(21,24)(22,25,23)(26,29,27)(28,30)(31,33,32)(34,35)$
6B $6^{3},3,2^{7}$ $140$ $6$ $24$ $( 1,31, 2,35, 3,34)( 4,32)( 5,33)( 6,28, 8,30, 7,29)( 9,26)(10,27)(11,22)(12,23)(13,25,15,24,14,21)(16,19,20)(17,18)$
6C $6^{3},3,2^{6},1^{2}$ $140$ $6$ $23$ $( 1,24)( 2,23, 5,21, 4,22)( 3,25)( 6,17,10,19, 9,18)( 7,20)( 8,16)(11,12,15)(26,34,27,32,30,33)(28,35)(29,31)$
7A1 $7^{5}$ $2$ $7$ $30$ $( 1,24, 7,28,13,35,20)( 2,21, 6,30,15,34,19)( 3,25, 8,29,14,31,16)( 4,23, 9,27,11,33,17)( 5,22,10,26,12,32,18)$
7A2 $7^{5}$ $2$ $7$ $30$ $( 1, 7,13,20,24,28,35)( 2, 6,15,19,21,30,34)( 3, 8,14,16,25,29,31)( 4, 9,11,17,23,27,33)( 5,10,12,18,22,26,32)$
7A3 $7^{5}$ $2$ $7$ $30$ $( 1,28,20, 7,35,24,13)( 2,30,19, 6,34,21,15)( 3,29,16, 8,31,25,14)( 4,27,17, 9,33,23,11)( 5,26,18,10,32,22,12)$
10A $10^{3},5$ $168$ $10$ $31$ $( 1,33, 3,34, 5,35, 4,31, 2,32)( 6,26, 7,27, 8,30,10,28, 9,29)(11,25,15,22,13,23,14,21,12,24)(16,19,18,20,17)$
14A1 $14,7^{3}$ $20$ $14$ $31$ $( 1,15,24,34, 7,19,28, 2,13,21,35, 6,20,30)( 3,14,25,31, 8,16,29)( 4,11,23,33, 9,17,27)( 5,12,22,32,10,18,26)$
14A3 $14,7^{3}$ $20$ $14$ $31$ $( 1,34,28,21,20,15, 7, 2,35,30,24,19,13, 6)( 3,31,29,25,16,14, 8)( 4,33,27,23,17,11, 9)( 5,32,26,22,18,12,10)$
14A5 $14,7^{3}$ $20$ $14$ $31$ $( 1,19,35,15,28, 6,24, 2,20,34,13,30, 7,21)( 3,16,31,14,29, 8,25)( 4,17,33,11,27, 9,23)( 5,18,32,12,26,10,22)$
14B1 $14^{2},7$ $30$ $14$ $32$ $( 1, 7,13,20,24,28,35)( 2,10,15,18,21,26,34, 5, 6,12,19,22,30,32)( 3, 9,14,17,25,27,31, 4, 8,11,16,23,29,33)$
14B3 $14^{2},7$ $30$ $14$ $32$ $( 1,20,35,13,28, 7,24)( 2,18,34,12,30,10,21, 5,19,32,15,26, 6,22)( 3,17,31,11,29, 9,25, 4,16,33,14,27, 8,23)$
14B5 $14^{2},7$ $30$ $14$ $32$ $( 1,28,20, 7,35,24,13)( 2,26,19,10,34,22,15, 5,30,18, 6,32,21,12)( 3,27,16, 9,31,23,14, 4,29,17, 8,33,25,11)$
21A1 $21,7^{2}$ $40$ $21$ $32$ $( 1,35,28,24,20,13, 7)( 2,34,30,21,19,15, 6)( 3,32,27,25,18,11, 8, 5,33,29,22,17,14,10, 4,31,26,23,16,12, 9)$
21A2 $21,7^{2}$ $40$ $21$ $32$ $( 1,28,20, 7,35,24,13)( 2,30,19, 6,34,21,15)( 3,27,18, 8,33,22,14, 4,26,16, 9,32,25,11, 5,29,17,10,31,23,12)$
21A4 $21,7^{2}$ $40$ $21$ $32$ $( 1,20,35,13,28, 7,24)( 2,19,34,15,30, 6,21)( 3,18,33,14,26, 9,25, 5,17,31,12,27, 8,22, 4,16,32,11,29,10,23)$
28A1 $28,7$ $60$ $28$ $33$ $( 1,24, 7,28,13,35,20)( 2,23,10,29,15,33,18, 3,21, 9,26,14,34,17, 5,25, 6,27,12,31,19, 4,22, 8,30,11,32,16)$
28A3 $28,7$ $60$ $28$ $33$ $( 1,28,20, 7,35,24,13)( 2,29,18, 9,34,25,12, 4,30,16,10,33,21,14, 5,27,19, 8,32,23,15, 3,26,17, 6,31,22,11)$
28A5 $28,7$ $60$ $28$ $33$ $( 1,35,28,24,20,13, 7)( 2,33,26,25,19,11,10, 3,34,27,22,16,15, 9, 5,31,30,23,18,14, 6, 4,32,29,21,17,12, 8)$
35A1 $35$ $48$ $35$ $34$ $( 1, 9,14,18,21,28,33, 3,10,15,20,23,29,32, 2, 7,11,16,22,30,35, 4, 8,12,19,24,27,31, 5, 6,13,17,25,26,34)$
35A2 $35$ $48$ $35$ $34$ $( 1,14,21,33,10,20,29, 2,11,22,35, 8,19,27, 5,13,25,34, 9,18,28, 3,15,23,32, 7,16,30, 4,12,24,31, 6,17,26)$
35A3 $35$ $48$ $35$ $34$ $( 1,18,33,15,29, 7,22, 4,19,31,13,26, 9,21, 3,20,32,11,30, 8,24, 5,17,34,14,28,10,23, 2,16,35,12,27, 6,25)$
42A1 $21,14$ $40$ $42$ $33$ $( 1,19,35,15,28, 6,24, 2,20,34,13,30, 7,21)( 3,17,32,14,27,10,25, 4,18,31,11,26, 8,23, 5,16,33,12,29, 9,22)$
42A5 $21,14$ $40$ $42$ $33$ $( 1,34,28,21,20,15, 7, 2,35,30,24,19,13, 6)( 3,33,26,25,17,12, 8, 4,32,29,23,18,14, 9, 5,31,27,22,16,11,10)$
42A11 $21,14$ $40$ $42$ $33$ $( 1,15,24,34, 7,19,28, 2,13,21,35, 6,20,30)( 3,11,22,31, 9,18,29, 4,12,25,33,10,16,27, 5,14,23,32, 8,17,26)$

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

35 x 35 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