Properties

Label 35T4
35T4 1 29 1->29 2 5 2->5 28 2->28 3 4 3->4 27 3->27 26 4->26 30 5->30 6 24 6->24 31 6->31 7 23 7->23 35 7->35 8 22 8->22 34 8->34 9 21 9->21 33 9->33 10 25 10->25 32 10->32 11 19 11->19 11->26 12 18 12->18 12->30 13 17 13->17 13->29 14 16 14->16 14->28 15 20 15->20 15->27 16->21 17->25 18->24 19->23 20->22 31->34 32->33
Degree $35$
Order $70$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $D_{35}$

Related objects

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(35, 4);
 
Copy content sage:G = TransitiveGroup(35, 4)
 
Copy content oscar:G = transitive_group(35, 4)
 
Copy content gap:G := TransitiveGroup(35, 4);
 

Group invariants

Abstract group:  $D_{35}$
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:  $70=2 \cdot 5 \cdot 7$
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 $t$:  $4$
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:  $(2,5)(3,4)(6,31)(7,35)(8,34)(9,33)(10,32)(11,26)(12,30)(13,29)(14,28)(15,27)(16,21)(17,25)(18,24)(19,23)(20,22)$, $(1,29)(2,28)(3,27)(4,26)(5,30)(6,24)(7,23)(8,22)(9,21)(10,25)(11,19)(12,18)(13,17)(14,16)(15,20)(31,34)(32,33)$
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)
$2$:  $C_2$
$10$:  $D_{5}$
$14$:  $D_{7}$

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 $2^{17},1$ $35$ $2$ $17$ $( 2, 5)( 3, 4)( 6,31)( 7,35)( 8,34)( 9,33)(10,32)(11,26)(12,30)(13,29)(14,28)(15,27)(16,21)(17,25)(18,24)(19,23)(20,22)$
5A1 $5^{7}$ $2$ $5$ $28$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17)(21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)$
5A2 $5^{7}$ $2$ $5$ $28$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18)(21,24,22,25,23)(26,29,27,30,28)(31,34,32,35,33)$
7A1 $7^{5}$ $2$ $7$ $30$ $( 1,11,21,31, 6,16,26)( 2,12,22,32, 7,17,27)( 3,13,23,33, 8,18,28)( 4,14,24,34, 9,19,29)( 5,15,25,35,10,20,30)$
7A2 $7^{5}$ $2$ $7$ $30$ $( 1,21, 6,26,11,31,16)( 2,22, 7,27,12,32,17)( 3,23, 8,28,13,33,18)( 4,24, 9,29,14,34,19)( 5,25,10,30,15,35,20)$
7A3 $7^{5}$ $2$ $7$ $30$ $( 1,31,26,21,16,11, 6)( 2,32,27,22,17,12, 7)( 3,33,28,23,18,13, 8)( 4,34,29,24,19,14, 9)( 5,35,30,25,20,15,10)$
35A1 $35$ $2$ $35$ $34$ $( 1,33,30,22,19,11, 8, 5,32,29,21,18,15, 7, 4,31,28,25,17,14, 6, 3,35,27,24,16,13,10, 2,34,26,23,20,12, 9)$
35A2 $35$ $2$ $35$ $34$ $( 1,30,19, 8,32,21,15, 4,28,17, 6,35,24,13, 2,26,20, 9,33,22,11, 5,29,18, 7,31,25,14, 3,27,16,10,34,23,12)$
35A3 $35$ $2$ $35$ $34$ $( 1,22, 8,29,15,31,17, 3,24,10,26,12,33,19, 5,21, 7,28,14,35,16, 2,23, 9,30,11,32,18, 4,25, 6,27,13,34,20)$
35A4 $35$ $2$ $35$ $34$ $( 1,19,32,15,28, 6,24, 2,20,33,11,29, 7,25, 3,16,34,12,30, 8,21, 4,17,35,13,26, 9,22, 5,18,31,14,27,10,23)$
35A6 $35$ $2$ $35$ $34$ $( 1, 8,15,17,24,26,33, 5, 7,14,16,23,30,32, 4, 6,13,20,22,29,31, 3,10,12,19,21,28,35, 2, 9,11,18,25,27,34)$
35A8 $35$ $2$ $35$ $34$ $( 1,32,28,24,20,11, 7, 3,34,30,21,17,13, 9, 5,31,27,23,19,15, 6, 2,33,29,25,16,12, 8, 4,35,26,22,18,14,10)$
35A9 $35$ $2$ $35$ $34$ $( 1,29,17,10,33,21,14, 2,30,18, 6,34,22,15, 3,26,19, 7,35,23,11, 4,27,20, 8,31,24,12, 5,28,16, 9,32,25,13)$
35A11 $35$ $2$ $35$ $34$ $( 1,18,35,12,29, 6,23, 5,17,34,11,28,10,22, 4,16,33,15,27, 9,21, 3,20,32,14,26, 8,25, 2,19,31,13,30, 7,24)$
35A12 $35$ $2$ $35$ $34$ $( 1,15,24,33, 7,16,30, 4,13,22,31,10,19,28, 2,11,25,34, 8,17,26, 5,14,23,32, 6,20,29, 3,12,21,35, 9,18,27)$
35A13 $35$ $2$ $35$ $34$ $( 1, 7,13,19,25,26,32, 3, 9,15,16,22,28,34, 5, 6,12,18,24,30,31, 2, 8,14,20,21,27,33, 4,10,11,17,23,29,35)$
35A16 $35$ $2$ $35$ $34$ $( 1,28,20, 7,34,21,13, 5,27,19, 6,33,25,12, 4,26,18,10,32,24,11, 3,30,17, 9,31,23,15, 2,29,16, 8,35,22,14)$
35A17 $35$ $2$ $35$ $34$ $( 1,25, 9,28,12,31,20, 4,23, 7,26,15,34,18, 2,21,10,29,13,32,16, 5,24, 8,27,11,35,19, 3,22, 6,30,14,33,17)$

Malle's constant $a(G)$:     $1/17$

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 5A1 5A2 7A1 7A2 7A3 35A1 35A2 35A3 35A4 35A6 35A8 35A9 35A11 35A12 35A13 35A16 35A17
Size 1 35 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 5A2 5A1 7A2 7A3 7A1 35A2 35A4 35A6 35A8 35A12 35A16 35A17 35A13 35A11 35A9 35A3 35A1
5 P 1A 2A 5A2 5A1 7A3 7A1 7A2 35A3 35A6 35A9 35A12 35A17 35A11 35A8 35A2 35A1 35A4 35A13 35A16
7 P 1A 2A 1A 1A 7A2 7A3 7A1 7A1 7A2 7A3 7A3 7A1 7A1 7A2 7A3 7A2 7A1 7A2 7A3
Type
70.3.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
70.3.1b R 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
70.3.2a1 R 2 0 ζ5−2+ζ52 ζ5−1+ζ5 2 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
70.3.2a2 R 2 0 ζ5−1+ζ5 ζ5−2+ζ52 2 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
70.3.2b1 R 2 0 2 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72
70.3.2b2 R 2 0 2 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7
70.3.2b3 R 2 0 2 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73
70.3.2c1 R 2 0 ζ35−14+ζ3514 ζ35−7+ζ357 ζ35−15+ζ3515 ζ35−5+ζ355 ζ35−10+ζ3510 ζ35−11+ζ3511 ζ35−12+ζ3512 ζ35−9+ζ359 ζ35−3+ζ353 ζ35−8+ζ358 ζ35−16+ζ3516 ζ35−1+ζ35 ζ35−17+ζ3517 ζ35−6+ζ356 ζ35−13+ζ3513 ζ35−2+ζ352 ζ35−4+ζ354
70.3.2c2 R 2 0 ζ35−14+ζ3514 ζ35−7+ζ357 ζ35−15+ζ3515 ζ35−5+ζ355 ζ35−10+ζ3510 ζ35−4+ζ354 ζ35−2+ζ352 ζ35−16+ζ3516 ζ35−17+ζ3517 ζ35−13+ζ3513 ζ35−9+ζ359 ζ35−6+ζ356 ζ35−3+ζ353 ζ35−1+ζ35 ζ35−8+ζ358 ζ35−12+ζ3512 ζ35−11+ζ3511
70.3.2c3 R 2 0 ζ35−14+ζ3514 ζ35−7+ζ357 ζ35−10+ζ3510 ζ35−15+ζ3515 ζ35−5+ζ355 ζ35−16+ζ3516 ζ35−8+ζ358 ζ35−6+ζ356 ζ35−2+ζ352 ζ35−17+ζ3517 ζ35−1+ζ35 ζ35−11+ζ3511 ζ35−12+ζ3512 ζ35−4+ζ354 ζ35−3+ζ353 ζ35−13+ζ3513 ζ35−9+ζ359
70.3.2c4 R 2 0 ζ35−14+ζ3514 ζ35−7+ζ357 ζ35−10+ζ3510 ζ35−15+ζ3515 ζ35−5+ζ355 ζ35−9+ζ359 ζ35−13+ζ3513 ζ35−1+ζ35 ζ35−12+ζ3512 ζ35−3+ζ353 ζ35−6+ζ356 ζ35−4+ζ354 ζ35−2+ζ352 ζ35−11+ζ3511 ζ35−17+ζ3517 ζ35−8+ζ358 ζ35−16+ζ3516
70.3.2c5 R 2 0 ζ35−14+ζ3514 ζ35−7+ζ357 ζ35−5+ζ355 ζ35−10+ζ3510 ζ35−15+ζ3515 ζ35−6+ζ356 ζ35−3+ζ353 ζ35−11+ζ3511 ζ35−8+ζ358 ζ35−2+ζ352 ζ35−4+ζ354 ζ35−9+ζ359 ζ35−13+ζ3513 ζ35−16+ζ3516 ζ35−12+ζ3512 ζ35−17+ζ3517 ζ35−1+ζ35
70.3.2c6 R 2 0 ζ35−14+ζ3514 ζ35−7+ζ357 ζ35−5+ζ355 ζ35−10+ζ3510 ζ35−15+ζ3515 ζ35−1+ζ35 ζ35−17+ζ3517 ζ35−4+ζ354 ζ35−13+ζ3513 ζ35−12+ζ3512 ζ35−11+ζ3511 ζ35−16+ζ3516 ζ35−8+ζ358 ζ35−9+ζ359 ζ35−2+ζ352 ζ35−3+ζ353 ζ35−6+ζ356
70.3.2c7 R 2 0 ζ35−7+ζ357 ζ35−14+ζ3514 ζ35−15+ζ3515 ζ35−5+ζ355 ζ35−10+ζ3510 ζ35−17+ζ3517 ζ35−9+ζ359 ζ35−2+ζ352 ζ35−11+ζ3511 ζ35−6+ζ356 ζ35−12+ζ3512 ζ35−8+ζ358 ζ35−4+ζ354 ζ35−13+ζ3513 ζ35−1+ζ35 ζ35−16+ζ3516 ζ35−3+ζ353
70.3.2c8 R 2 0 ζ35−7+ζ357 ζ35−14+ζ3514 ζ35−15+ζ3515 ζ35−5+ζ355 ζ35−10+ζ3510 ζ35−3+ζ353 ζ35−16+ζ3516 ζ35−12+ζ3512 ζ35−4+ζ354 ζ35−1+ζ35 ζ35−2+ζ352 ζ35−13+ζ3513 ζ35−11+ζ3511 ζ35−8+ζ358 ζ35−6+ζ356 ζ35−9+ζ359 ζ35−17+ζ3517
70.3.2c9 R 2 0 ζ35−7+ζ357 ζ35−14+ζ3514 ζ35−10+ζ3510 ζ35−15+ζ3515 ζ35−5+ζ355 ζ35−12+ζ3512 ζ35−6+ζ356 ζ35−13+ζ3513 ζ35−16+ζ3516 ζ35−4+ζ354 ζ35−8+ζ358 ζ35−17+ζ3517 ζ35−9+ζ359 ζ35−3+ζ353 ζ35−11+ζ3511 ζ35−1+ζ35 ζ35−2+ζ352
70.3.2c10 R 2 0 ζ35−7+ζ357 ζ35−14+ζ3514 ζ35−10+ζ3510 ζ35−15+ζ3515 ζ35−5+ζ355 ζ35−2+ζ352 ζ35−1+ζ35 ζ35−8+ζ358 ζ35−9+ζ359 ζ35−11+ζ3511 ζ35−13+ζ3513 ζ35−3+ζ353 ζ35−16+ζ3516 ζ35−17+ζ3517 ζ35−4+ζ354 ζ35−6+ζ356 ζ35−12+ζ3512
70.3.2c11 R 2 0 ζ35−7+ζ357 ζ35−14+ζ3514 ζ35−5+ζ355 ζ35−10+ζ3510 ζ35−15+ζ3515 ζ35−13+ζ3513 ζ35−11+ζ3511 ζ35−17+ζ3517 ζ35−6+ζ356 ζ35−16+ζ3516 ζ35−3+ζ353 ζ35−2+ζ352 ζ35−1+ζ35 ζ35−12+ζ3512 ζ35−9+ζ359 ζ35−4+ζ354 ζ35−8+ζ358
70.3.2c12 R 2 0 ζ35−7+ζ357 ζ35−14+ζ3514 ζ35−5+ζ355 ζ35−10+ζ3510 ζ35−15+ζ3515 ζ35−8+ζ358 ζ35−4+ζ354 ζ35−3+ζ353 ζ35−1+ζ35 ζ35−9+ζ359 ζ35−17+ζ3517 ζ35−12+ζ3512 ζ35−6+ζ356 ζ35−2+ζ352 ζ35−16+ζ3516 ζ35−11+ζ3511 ζ35−13+ζ3513

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