Properties

Label 35T26
35T26 1 12 1->12 28 1->28 2 13 2->13 27 2->27 3 15 3->15 26 3->26 4 14 4->14 29 4->29 5 11 5->11 30 5->30 6 17 6->17 20 6->20 7 18 7->18 7->20 8 8->18 19 8->19 9 16 9->16 9->16 10 10->17 10->19 11->8 25 11->25 12->6 23 12->23 13->7 22 13->22 14->10 21 14->21 15->9 24 15->24 16->30 32 16->32 17->29 31 17->31 18->27 34 18->34 19->28 33 19->33 20->26 35 20->35 21->23 21->35 22->21 22->33 23->25 23->31 24->32 25->22 25->34 26->4 26->15 27->3 27->14 28->5 28->13 29->2 29->12 30->1 30->11 31->5 31->6 32->2 32->9 33->3 33->8 34->4 34->7 35->1 35->10
Degree $35$
Order $2520$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $A_5:F_7$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(35, 26);
 
Copy content sage:G = TransitiveGroup(35, 26)
 
Copy content oscar:G = transitive_group(35, 26)
 
Copy content gap:G := TransitiveGroup(35, 26);
 

Group invariants

Abstract group:  $A_5:F_7$
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:  $2520=2^{3} \cdot 3^{2} \cdot 5 \cdot 7$
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 $t$:  $26$
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:  $(1,12,23,31,6,20,26,4,14,21,35,10,17,29,2,13,22,33,8,19,28,5,11,25,34,7,18,27,3,15,24,32,9,16,30)$, $(1,28,13,7,20,35)(2,27,14,10,19,33,3,26,15,9,16,32)(4,29,12,6,17,31,5,30,11,8,18,34)(21,23,25,22)$
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)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$42$:  $F_7$
$120$:  $S_5$
$360$:  $S_5 \times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $S_5$

Degree 7: $F_7$

Low degree siblings

42T298

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 $2^{14},1^{7}$ $15$ $2$ $14$ $( 1, 3)( 4, 5)( 7, 8)( 9,10)(11,12)(13,14)(16,20)(17,18)(22,23)(24,25)(26,27)(28,29)(31,35)(32,33)$
2B $2^{16},1^{3}$ $70$ $2$ $16$ $( 1,13)( 2,14)( 3,15)( 4,11)( 5,12)( 6, 8)(16,34)(17,33)(18,32)(19,31)(20,35)(21,29)(22,26)(23,27)(24,28)(25,30)$
3A1 $3^{10},1^{5}$ $7$ $3$ $20$ $( 6,15,21)( 7,13,24)( 8,14,25)( 9,11,23)(10,12,22)(16,31,29)(17,33,27)(18,32,26)(19,34,30)(20,35,28)$
3A-1 $3^{10},1^{5}$ $7$ $3$ $20$ $( 6,21,15)( 7,24,13)( 8,25,14)( 9,23,11)(10,22,12)(16,29,31)(17,27,33)(18,26,32)(19,30,34)(20,28,35)$
3B $3^{7},1^{14}$ $20$ $3$ $14$ $( 2, 5, 4)( 6,10, 9)(11,15,12)(17,19,18)(21,22,23)(26,27,30)(32,33,34)$
3C1 $3^{11},1^{2}$ $140$ $3$ $22$ $( 1,32,23)( 2,34,21)( 3,31,25)( 4,35,22)( 5,33,24)( 7,10, 9)(11,20,26)(12,17,28)(13,18,27)(14,16,29)(15,19,30)$
3C-1 $3^{11},1^{2}$ $140$ $3$ $22$ $( 1,23,32)( 2,21,34)( 3,25,31)( 4,22,35)( 5,24,33)( 7, 9,10)(11,26,20)(12,28,17)(13,27,18)(14,29,16)(15,30,19)$
4A $4^{7},2^{3},1$ $210$ $4$ $24$ $( 1, 2, 5, 3)( 6,32, 8,35)( 7,34,10,31)( 9,33)(11,27)(12,29,13,30)(14,28,15,26)(16,24,19,22)(17,23)(18,25,20,21)$
5A $5^{7}$ $24$ $5$ $28$ $( 1, 4, 2, 3, 5)( 6, 8,10, 7, 9)(11,15,14,12,13)(16,18,20,17,19)(21,25,22,24,23)(26,28,27,30,29)(31,32,35,33,34)$
6A1 $6^{5},2,1^{3}$ $70$ $6$ $26$ $( 2, 4)( 6,17,15,33,21,27)( 7,20,13,35,24,28)( 8,16,14,31,25,29)( 9,19,11,34,23,30)(10,18,12,32,22,26)$
6A-1 $6^{5},2,1^{3}$ $70$ $6$ $26$ $( 2, 4)( 6,27,21,33,15,17)( 7,28,24,35,13,20)( 8,29,25,31,14,16)( 9,30,23,34,11,19)(10,26,22,32,12,18)$
6B1 $6^{4},3^{2},2^{2},1$ $105$ $6$ $26$ $( 1, 5)( 2, 3)( 6,14,21, 8,15,25)( 7,12,24,10,13,22)( 9,11,23)(16,34,29,19,31,30)(17,33,27)(18,35,26,20,32,28)$
6B-1 $6^{4},3^{2},2^{2},1$ $105$ $6$ $26$ $( 1, 5)( 2, 3)( 6,25,15, 8,21,14)( 7,22,13,10,24,12)( 9,23,11)(16,30,31,19,29,34)(17,27,33)(18,28,32,20,26,35)$
6C $6^{3},3,2^{7}$ $140$ $6$ $24$ $( 1,16)( 2,17, 5,19, 4,18)( 3,20)( 6,11,10,15, 9,12)( 7,14)( 8,13)(21,33,22,34,23,32)(24,31)(25,35)(26,30,27)(28,29)$
6D1 $6^{5},3,2$ $140$ $6$ $28$ $( 1,27,32,13,23,18)( 2,29,34,14,21,16)( 3,30,31,15,25,19)( 4,26,35,11,22,20)( 5,28,33,12,24,17)( 6, 8)( 7, 9,10)$
6D-1 $6^{5},3,2$ $140$ $6$ $28$ $( 1,18,23,13,32,27)( 2,16,21,14,34,29)( 3,19,25,15,31,30)( 4,20,22,11,35,26)( 5,17,24,12,33,28)( 6, 8)( 7,10, 9)$
7A $7^{5}$ $6$ $7$ $30$ $( 1,13,24,35, 7,20,28)( 2,15,21,34, 6,19,30)( 3,14,25,31, 8,16,29)( 4,11,23,33, 9,17,27)( 5,12,22,32,10,18,26)$
12A1 $12^{2},6,4,1$ $210$ $12$ $30$ $( 1, 3, 5, 2)( 6,20,14,32,21,28, 8,18,15,35,25,26)( 7,16,12,34,24,29,10,19,13,31,22,30)( 9,17,11,33,23,27)$
12A-1 $12^{2},6,4,1$ $210$ $12$ $30$ $( 1, 2, 5, 3)( 6,26,25,35,15,18, 8,28,21,32,14,20)( 7,30,22,31,13,19,10,29,24,34,12,16)( 9,27,23,33,11,17)$
14A $14^{2},7$ $90$ $14$ $32$ $( 1, 8,13,16,24,29,35, 3, 7,14,20,25,28,31)( 2, 6,15,19,21,30,34)( 4,10,11,18,23,26,33, 5, 9,12,17,22,27,32)$
15A1 $15^{2},5$ $168$ $15$ $32$ $( 1,14,34, 4,12,35, 3,15,33, 5,13,31, 2,11,32)( 6,23,18, 7,25,19, 9,22,20, 8,21,17,10,24,16)(26,28,29,30,27)$
15A-1 $15^{2},5$ $168$ $15$ $32$ $( 1,32,11, 2,31,13, 5,33,15, 3,35,12, 4,34,14)( 6,16,24,10,17,21, 8,20,22, 9,19,25, 7,18,23)(26,27,30,29,28)$
21A $21,7^{2}$ $120$ $21$ $32$ $( 1,23, 6,28,11,34,20, 4,21, 7,27,15,35,17, 2,24, 9,30,13,33,19)( 3,25, 8,29,14,31,16)( 5,22,10,26,12,32,18)$
35A1 $35$ $72$ $35$ $34$ $( 1,29,17,10,34,24,14, 4,26,19, 7,31,23,12, 2,28,16, 9,32,21,13, 3,27,18, 6,35,25,11, 5,30,20, 8,33,22,15)$
35A-1 $35$ $72$ $35$ $34$ $( 1,15,22,33, 8,20,30, 5,11,25,35, 6,18,27, 3,13,21,32, 9,16,28, 2,12,23,31, 7,19,26, 4,14,24,34,10,17,29)$

Malle's constant $a(G)$:     $1/14$

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

1A 2A 2B 3A1 3A-1 3B 3C1 3C-1 4A 5A 6A1 6A-1 6B1 6B-1 6C 6D1 6D-1 7A 12A1 12A-1 14A 15A1 15A-1 21A 35A1 35A-1
Size 1 15 70 7 7 20 140 140 210 24 70 70 105 105 140 140 140 6 210 210 90 168 168 120 72 72
2 P 1A 1A 1A 3A-1 3A1 3B 3C-1 3C1 2A 5A 3A1 3A-1 3A-1 3A1 3B 3C1 3C-1 7A 6B1 6B-1 7A 15A-1 15A1 21A 35A-1 35A1
3 P 1A 2A 2B 1A 1A 1A 1A 1A 4A 5A 2B 2B 2A 2A 2B 2B 2B 7A 4A 4A 14A 5A 5A 7A 35A1 35A-1
5 P 1A 2A 2B 3A-1 3A1 3B 3C-1 3C1 4A 1A 6A-1 6A1 6B-1 6B1 6C 6D-1 6D1 7A 12A-1 12A1 14A 3A-1 3A1 21A 7A 7A
7 P 1A 2A 2B 3A1 3A-1 3B 3C1 3C-1 4A 5A 6A1 6A-1 6B1 6B-1 6C 6D1 6D-1 1A 12A1 12A-1 2A 15A1 15A-1 3B 5A 5A
Type
2520.bk.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2520.bk.1b R 1 1 −1 1 1 1 1 1 −1 1 −1 −1 1 1 −1 −1 −1 1 −1 −1 1 1 1 1 1 1
2520.bk.1c1 C 1 1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 1 1
2520.bk.1c2 C 1 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 ζ3 ζ3−1 1 1 1
2520.bk.1d1 C 1 1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 −1 1 −ζ3 −ζ3−1 ζ3−1 ζ3 −1 −ζ3 −ζ3−1 1 −ζ3 −ζ3−1 1 ζ3−1 ζ3 1 1 1
2520.bk.1d2 C 1 1 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 −1 1 −ζ3−1 −ζ3 ζ3 ζ3−1 −1 −ζ3−1 −ζ3 1 −ζ3−1 −ζ3 1 ζ3 ζ3−1 1 1 1
2520.bk.4a R 4 0 2 4 4 1 1 1 0 −1 2 2 0 0 −1 −1 −1 4 0 0 0 −1 −1 1 −1 −1
2520.bk.4b R 4 0 −2 4 4 1 1 1 0 −1 −2 −2 0 0 1 1 1 4 0 0 0 −1 −1 1 −1 −1
2520.bk.4c1 C 4 0 2 4ζ3−1 4ζ3 1 ζ3−1 ζ3 0 −1 2ζ3 2ζ3−1 0 0 −1 −ζ3 −ζ3−1 4 0 0 0 −ζ3−1 −ζ3 1 −1 −1
2520.bk.4c2 C 4 0 2 4ζ3 4ζ3−1 1 ζ3 ζ3−1 0 −1 2ζ3−1 2ζ3 0 0 −1 −ζ3−1 −ζ3 4 0 0 0 −ζ3 −ζ3−1 1 −1 −1
2520.bk.4d1 C 4 0 −2 4ζ3−1 4ζ3 1 ζ3−1 ζ3 0 −1 −2ζ3 −2ζ3−1 0 0 1 ζ3 ζ3−1 4 0 0 0 −ζ3−1 −ζ3 1 −1 −1
2520.bk.4d2 C 4 0 −2 4ζ3 4ζ3−1 1 ζ3 ζ3−1 0 −1 −2ζ3−1 −2ζ3 0 0 1 ζ3−1 ζ3 4 0 0 0 −ζ3 −ζ3−1 1 −1 −1
2520.bk.5a R 5 1 −1 5 5 −1 −1 −1 1 0 −1 −1 1 1 −1 −1 −1 5 1 1 1 0 0 −1 0 0
2520.bk.5b R 5 1 1 5 5 −1 −1 −1 −1 0 1 1 1 1 1 1 1 5 −1 −1 1 0 0 −1 0 0
2520.bk.5c1 C 5 1 −1 5ζ3−1 5ζ3 −1 −ζ3−1 −ζ3 1 0 −ζ3 −ζ3−1 ζ3−1 ζ3 −1 −ζ3 −ζ3−1 5 ζ3 ζ3−1 1 0 0 −1 0 0
2520.bk.5c2 C 5 1 −1 5ζ3 5ζ3−1 −1 −ζ3 −ζ3−1 1 0 −ζ3−1 −ζ3 ζ3 ζ3−1 −1 −ζ3−1 −ζ3 5 ζ3−1 ζ3 1 0 0 −1 0 0
2520.bk.5d1 C 5 1 1 5ζ3−1 5ζ3 −1 −ζ3−1 −ζ3 −1 0 ζ3 ζ3−1 ζ3−1 ζ3 1 ζ3 ζ3−1 5 −ζ3 −ζ3−1 1 0 0 −1 0 0
2520.bk.5d2 C 5 1 1 5ζ3 5ζ3−1 −1 −ζ3 −ζ3−1 −1 0 ζ3−1 ζ3 ζ3 ζ3−1 1 ζ3−1 ζ3 5 −ζ3−1 −ζ3 1 0 0 −1 0 0
2520.bk.6a R 6 6 0 0 0 6 0 0 0 6 0 0 0 0 0 0 0 −1 0 0 −1 0 0 −1 −1 −1
2520.bk.6b R 6 −2 0 6 6 0 0 0 0 1 0 0 −2 −2 0 0 0 6 0 0 −2 1 1 0 1 1
2520.bk.6c1 C 6 −2 0 6ζ3−1 6ζ3 0 0 0 0 1 0 0 −2ζ3−1 −2ζ3 0 0 0 6 0 0 −2 ζ3−1 ζ3 0 1 1
2520.bk.6c2 C 6 −2 0 6ζ3 6ζ3−1 0 0 0 0 1 0 0 −2ζ3 −2ζ3−1 0 0 0 6 0 0 −2 ζ3 ζ3−1 0 1 1
2520.bk.18a1 C 18 −6 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 −3 0 0 1 0 0 0 ζ35−15−ζ35−14+2ζ35−13−2ζ354+ζ355+2ζ358−2ζ359+ζ3510−2ζ3511−ζ3514+2ζ3515−2ζ3516 −ζ35−15+ζ35−14−2ζ35−13−1+2ζ354−ζ355−2ζ358+2ζ359−ζ3510+2ζ3511+ζ3514−2ζ3515+2ζ3516
2520.bk.18a2 C 18 −6 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 −3 0 0 1 0 0 0 −ζ35−15+ζ35−14−2ζ35−13−1+2ζ354−ζ355−2ζ358+2ζ359−ζ3510+2ζ3511+ζ3514−2ζ3515+2ζ3516 ζ35−15−ζ35−14+2ζ35−13−2ζ354+ζ355+2ζ358−2ζ359+ζ3510−2ζ3511−ζ3514+2ζ3515−2ζ3516
2520.bk.24a R 24 0 0 0 0 6 0 0 0 −6 0 0 0 0 0 0 0 −4 0 0 0 0 0 −1 1 1
2520.bk.30a R 30 6 0 0 0 −6 0 0 0 0 0 0 0 0 0 0 0 −5 0 0 −1 0 0 1 0 0

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