Properties

Label 35T7
35T7 1 6 1->6 20 1->20 2 10 2->10 16 2->16 3 9 3->9 17 3->17 4 8 4->8 18 4->18 5 7 5->7 19 5->19 11 6->11 15 6->15 7->11 7->15 12 8->12 14 8->14 13 9->13 9->13 10->12 10->14 11->10 11->16 12->6 12->20 13->7 13->19 14->8 14->18 15->9 15->17 16->5 21 16->21 17->1 25 17->25 18->2 24 18->24 19->3 23 19->23 20->4 22 20->22 26 21->26 35 21->35 30 22->30 31 22->31 29 23->29 32 23->32 28 24->28 33 24->33 27 25->27 34 25->34 26->30 26->31 27->26 27->35 28->27 28->34 29->28 29->33 30->29 30->32 31->1 31->25 32->5 32->21 33->4 33->22 34->3 34->23 35->2 35->24
Degree $35$
Order $140$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $D_5\times D_7$

Related objects

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(35, 7);
 
Copy content sage:G = TransitiveGroup(35, 7)
 
Copy content oscar:G = transitive_group(35, 7)
 
Copy content gap:G := TransitiveGroup(35, 7);
 

Group invariants

Abstract group:  $D_5\times D_7$
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:  $140=2^{2} \cdot 5 \cdot 7$
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:  yes
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 $t$:  $7$
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:  $(1,20,4,18,2,16,5,19,3,17)(6,15,9,13,7,11,10,14,8,12)(21,35,24,33,22,31,25,34,23,32)(26,30,29,28,27)$, $(1,6,11,16,21,26,31)(2,10,12,20,22,30,32,5,7,15,17,25,27,35)(3,9,13,19,23,29,33,4,8,14,18,24,28,34)$
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
$4$:  $C_2^2$
$10$:  $D_{5}$
$14$:  $D_{7}$
$20$:  $D_{10}$
$28$:  $D_{14}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

Degree 7: $D_{7}$

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 $2^{14},1^{7}$ $5$ $2$ $14$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)$
2B $2^{15},1^{5}$ $7$ $2$ $15$ $( 1,31)( 2,32)( 3,33)( 4,34)( 5,35)( 6,26)( 7,27)( 8,28)( 9,29)(10,30)(11,21)(12,22)(13,23)(14,24)(15,25)$
2C $2^{17},1$ $35$ $2$ $17$ $( 1, 8)( 2, 7)( 3, 6)( 4,10)( 5, 9)(11,33)(12,32)(13,31)(14,35)(15,34)(16,28)(17,27)(18,26)(19,30)(20,29)(21,23)(24,25)$
5A1 $5^{7}$ $2$ $5$ $28$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$
5A2 $5^{7}$ $2$ $5$ $28$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19)(21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)$
7A1 $7^{5}$ $2$ $7$ $30$ $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33)( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$
7A2 $7^{5}$ $2$ $7$ $30$ $( 1,11,21,31, 6,16,26)( 2,12,22,32, 7,17,27)( 3,13,23,33, 8,18,28)( 4,14,24,34, 9,19,29)( 5,15,25,35,10,20,30)$
7A3 $7^{5}$ $2$ $7$ $30$ $( 1,16,31,11,26, 6,21)( 2,17,32,12,27, 7,22)( 3,18,33,13,28, 8,23)( 4,19,34,14,29, 9,24)( 5,20,35,15,30,10,25)$
10A1 $10^{3},5$ $14$ $10$ $31$ $( 1,35, 4,33, 2,31, 5,34, 3,32)( 6,30, 9,28, 7,26,10,29, 8,27)(11,25,14,23,12,21,15,24,13,22)(16,20,19,18,17)$
10A3 $10^{3},5$ $14$ $10$ $31$ $( 1,33, 5,32, 4,31, 3,35, 2,34)( 6,28,10,27, 9,26, 8,30, 7,29)(11,23,15,22,14,21,13,25,12,24)(16,18,20,17,19)$
14A1 $14^{2},7$ $10$ $14$ $32$ $( 1,21, 6,26,11,31,16)( 2,25, 7,30,12,35,17, 5,22,10,27,15,32,20)( 3,24, 8,29,13,34,18, 4,23, 9,28,14,33,19)$
14A3 $14^{2},7$ $10$ $14$ $32$ $( 1,29,16, 9,31,24,11, 4,26,19, 6,34,21,14)( 2,28,17, 8,32,23,12, 3,27,18, 7,33,22,13)( 5,30,20,10,35,25,15)$
14A5 $14^{2},7$ $10$ $14$ $32$ $( 1, 6,11,16,21,26,31)( 2,10,12,20,22,30,32, 5, 7,15,17,25,27,35)( 3, 9,13,19,23,29,33, 4, 8,14,18,24,28,34)$
35A1 $35$ $4$ $35$ $34$ $( 1,29,17,10,33,21,14, 2,30,18, 6,34,22,15, 3,26,19, 7,35,23,11, 4,27,20, 8,31,24,12, 5,28,16, 9,32,25,13)$
35A2 $35$ $4$ $35$ $34$ $( 1,17,33,14,30, 6,22, 3,19,35,11,27, 8,24, 5,16,32,13,29,10,21, 2,18,34,15,26, 7,23, 4,20,31,12,28, 9,25)$
35A3 $35$ $4$ $35$ $34$ $( 1,10,14,18,22,26,35, 4, 8,12,16,25,29,33, 2, 6,15,19,23,27,31, 5, 9,13,17,21,30,34, 3, 7,11,20,24,28,32)$
35A4 $35$ $4$ $35$ $34$ $( 1,33,30,22,19,11, 8, 5,32,29,21,18,15, 7, 4,31,28,25,17,14, 6, 3,35,27,24,16,13,10, 2,34,26,23,20,12, 9)$
35A8 $35$ $4$ $35$ $34$ $( 1,30,19, 8,32,21,15, 4,28,17, 6,35,24,13, 2,26,20, 9,33,22,11, 5,29,18, 7,31,25,14, 3,27,16,10,34,23,12)$
35A9 $35$ $4$ $35$ $34$ $( 1,19,32,15,28, 6,24, 2,20,33,11,29, 7,25, 3,16,34,12,30, 8,21, 4,17,35,13,26, 9,22, 5,18,31,14,27,10,23)$

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 2C 5A1 5A2 7A1 7A2 7A3 10A1 10A3 14A1 14A3 14A5 35A1 35A2 35A3 35A4 35A8 35A9
Size 1 5 7 35 2 2 2 2 2 14 14 10 10 10 4 4 4 4 4 4
2 P 1A 1A 1A 1A 5A2 5A1 7A2 7A3 7A1 5A2 5A1 7A1 7A3 7A2 35A2 35A4 35A1 35A8 35A9 35A3
5 P 1A 2A 2B 2C 5A2 5A1 7A3 7A1 7A2 10A3 10A1 14A3 14A5 14A1 35A3 35A1 35A9 35A2 35A4 35A8
7 P 1A 2A 2B 2C 1A 1A 7A2 7A3 7A1 2B 2B 14A5 14A1 14A3 7A3 7A1 7A2 7A2 7A3 7A1
Type
140.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
140.7.1b R 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 1 1 1 1 1 1
140.7.1c R 1 −1 1 −1 1 1 1 1 1 1 1 −1 −1 −1 1 1 1 1 1 1
140.7.1d R 1 1 −1 −1 1 1 1 1 1 −1 −1 1 1 1 1 1 1 1 1 1
140.7.2a1 R 2 0 2 0 ζ5−2+ζ52 ζ5−1+ζ5 2 2 2 ζ5−2+ζ52 ζ5−1+ζ5 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5
140.7.2a2 R 2 0 2 0 ζ5−1+ζ5 ζ5−2+ζ52 2 2 2 ζ5−1+ζ5 ζ5−2+ζ52 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52
140.7.2b1 R 2 0 −2 0 ζ5−2+ζ52 ζ5−1+ζ5 2 2 2 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5
140.7.2b2 R 2 0 −2 0 ζ5−1+ζ5 ζ5−2+ζ52 2 2 2 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52
140.7.2c1 R 2 2 0 0 2 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 0 0 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73
140.7.2c2 R 2 2 0 0 2 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 0 0 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72
140.7.2c3 R 2 2 0 0 2 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 0 0 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7
140.7.2d1 R 2 −2 0 0 2 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 0 0 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73
140.7.2d2 R 2 −2 0 0 2 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 0 0 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72
140.7.2d3 R 2 −2 0 0 2 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 0 0 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7
140.7.4a1 R 4 0 0 0 2ζ35−14+2ζ3514 2ζ35−7+2ζ357 2ζ35−15+2ζ3515 2ζ35−5+2ζ355 2ζ35−10+2ζ3510 0 0 0 0 0 −ζ35−17−ζ35−7+ζ352−ζ353−ζ358+ζ3512−ζ3513 −ζ353+ζ354−ζ3510+ζ3511−ζ3517 ζ35−17+ζ35−12+ζ35−7+ζ353−ζ355+ζ358+ζ359−ζ3512+ζ3513+ζ3516 −ζ35−12−ζ35−10−ζ35−5−ζ352+ζ353−ζ354−ζ359−ζ3511−ζ3516+ζ3517 −ζ35−17−ζ35−12+1−ζ354+ζ355+ζ357−ζ358−ζ359+ζ3510−ζ3511+ζ3512−ζ3513−ζ3516+ζ3517 ζ35−17+ζ35−12+ζ35−10+ζ35−5+ζ354−ζ357+ζ358+ζ359+ζ3511−ζ3512+ζ3513+ζ3516−ζ3517
140.7.4a2 R 4 0 0 0 2ζ35−14+2ζ3514 2ζ35−7+2ζ357 2ζ35−10+2ζ3510 2ζ35−15+2ζ3515 2ζ35−5+2ζ355 0 0 0 0 0 ζ35−17+ζ35−12+ζ35−10+ζ35−5+ζ354−ζ357+ζ358+ζ359+ζ3511−ζ3512+ζ3513+ζ3516−ζ3517 ζ35−17+ζ35−12+ζ35−7+ζ353−ζ355+ζ358+ζ359−ζ3512+ζ3513+ζ3516 −ζ35−17−ζ35−12+1−ζ354+ζ355+ζ357−ζ358−ζ359+ζ3510−ζ3511+ζ3512−ζ3513−ζ3516+ζ3517 −ζ35−17−ζ35−7+ζ352−ζ353−ζ358+ζ3512−ζ3513 −ζ353+ζ354−ζ3510+ζ3511−ζ3517 −ζ35−12−ζ35−10−ζ35−5−ζ352+ζ353−ζ354−ζ359−ζ3511−ζ3516+ζ3517
140.7.4a3 R 4 0 0 0 2ζ35−14+2ζ3514 2ζ35−7+2ζ357 2ζ35−5+2ζ355 2ζ35−10+2ζ3510 2ζ35−15+2ζ3515 0 0 0 0 0 −ζ35−12−ζ35−10−ζ35−5−ζ352+ζ353−ζ354−ζ359−ζ3511−ζ3516+ζ3517 −ζ35−17−ζ35−12+1−ζ354+ζ355+ζ357−ζ358−ζ359+ζ3510−ζ3511+ζ3512−ζ3513−ζ3516+ζ3517 −ζ353+ζ354−ζ3510+ζ3511−ζ3517 ζ35−17+ζ35−12+ζ35−10+ζ35−5+ζ354−ζ357+ζ358+ζ359+ζ3511−ζ3512+ζ3513+ζ3516−ζ3517 ζ35−17+ζ35−12+ζ35−7+ζ353−ζ355+ζ358+ζ359−ζ3512+ζ3513+ζ3516 −ζ35−17−ζ35−7+ζ352−ζ353−ζ358+ζ3512−ζ3513
140.7.4a4 R 4 0 0 0 2ζ35−7+2ζ357 2ζ35−14+2ζ3514 2ζ35−15+2ζ3515 2ζ35−5+2ζ355 2ζ35−10+2ζ3510 0 0 0 0 0 ζ35−17+ζ35−12+ζ35−7+ζ353−ζ355+ζ358+ζ359−ζ3512+ζ3513+ζ3516 −ζ35−12−ζ35−10−ζ35−5−ζ352+ζ353−ζ354−ζ359−ζ3511−ζ3516+ζ3517 −ζ35−17−ζ35−7+ζ352−ζ353−ζ358+ζ3512−ζ3513 −ζ353+ζ354−ζ3510+ζ3511−ζ3517 ζ35−17+ζ35−12+ζ35−10+ζ35−5+ζ354−ζ357+ζ358+ζ359+ζ3511−ζ3512+ζ3513+ζ3516−ζ3517 −ζ35−17−ζ35−12+1−ζ354+ζ355+ζ357−ζ358−ζ359+ζ3510−ζ3511+ζ3512−ζ3513−ζ3516+ζ3517
140.7.4a5 R 4 0 0 0 2ζ35−7+2ζ357 2ζ35−14+2ζ3514 2ζ35−10+2ζ3510 2ζ35−15+2ζ3515 2ζ35−5+2ζ355 0 0 0 0 0 −ζ35−17−ζ35−12+1−ζ354+ζ355+ζ357−ζ358−ζ359+ζ3510−ζ3511+ζ3512−ζ3513−ζ3516+ζ3517 −ζ35−17−ζ35−7+ζ352−ζ353−ζ358+ζ3512−ζ3513 ζ35−17+ζ35−12+ζ35−10+ζ35−5+ζ354−ζ357+ζ358+ζ359+ζ3511−ζ3512+ζ3513+ζ3516−ζ3517 ζ35−17+ζ35−12+ζ35−7+ζ353−ζ355+ζ358+ζ359−ζ3512+ζ3513+ζ3516 −ζ35−12−ζ35−10−ζ35−5−ζ352+ζ353−ζ354−ζ359−ζ3511−ζ3516+ζ3517 −ζ353+ζ354−ζ3510+ζ3511−ζ3517
140.7.4a6 R 4 0 0 0 2ζ35−7+2ζ357 2ζ35−14+2ζ3514 2ζ35−5+2ζ355 2ζ35−10+2ζ3510 2ζ35−15+2ζ3515 0 0 0 0 0 −ζ353+ζ354−ζ3510+ζ3511−ζ3517 ζ35−17+ζ35−12+ζ35−10+ζ35−5+ζ354−ζ357+ζ358+ζ359+ζ3511−ζ3512+ζ3513+ζ3516−ζ3517 −ζ35−12−ζ35−10−ζ35−5−ζ352+ζ353−ζ354−ζ359−ζ3511−ζ3516+ζ3517 −ζ35−17−ζ35−12+1−ζ354+ζ355+ζ357−ζ358−ζ359+ζ3510−ζ3511+ζ3512−ζ3513−ζ3516+ζ3517 −ζ35−17−ζ35−7+ζ352−ζ353−ζ358+ζ3512−ζ3513 ζ35−17+ζ35−12+ζ35−7+ζ353−ζ355+ζ358+ζ359−ζ3512+ζ3513+ζ3516

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