Properties

Label 35T48
35T48 1 6 1->6 25 1->25 2 10 2->10 22 2->22 3 9 3->9 24 3->24 4 8 4->8 21 4->21 5 7 5->7 23 5->23 30 6->30 27 7->27 29 8->29 26 9->26 28 10->28 11 11->5 16 11->16 12 12->2 20 12->20 13 13->4 19 13->19 14 14->1 18 14->18 15 15->3 17 15->17 16->10 16->26 17->7 17->30 18->9 18->29 19->6 19->28 20->8 20->27 21->20 22->17 22->25 23->19 23->24 24->16 25->18 31 26->31 35 26->35 32 27->32 27->35 34 28->34 28->34 29->31 33 29->33 30->32 30->33 31->11 31->15 32->12 32->15 33->14 33->14 34->11 34->13 35->12 35->13
Degree $35$
Order $50400$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $F_5\times A_7$

Related objects

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(35, 48);
 
Copy content sage:G = TransitiveGroup(35, 48)
 
Copy content oscar:G = transitive_group(35, 48)
 
Copy content gap:G := TransitiveGroup(35, 48);
 

Group invariants

Abstract group:  $F_5\times A_7$
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:  $50400=2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 7$
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 $t$:  $48$
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:  $(1,25,18,9,26,35,13,4,21,20,8,29,31,15,3,24,16,10,28,34,11,5,23,19,6,30,33,14)(2,22,17,7,27,32,12)$, $(1,6)(2,10)(3,9)(4,8)(5,7)(11,16,26,31)(12,20,27,35)(13,19,28,34)(14,18,29,33)(15,17,30,32)(22,25)(23,24)$
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)
$2$:  $C_2$
$4$:  $C_4$
$20$:  $F_5$
$2520$:  $A_7$
$5040$:  $A_7\times C_2$
$10080$:  28T362

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$

Degree 7: $A_7$

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 $2^{14},1^{7}$ $5$ $2$ $14$ $( 1, 5)( 2, 4)( 6,10)( 7, 9)(11,15)(12,14)(16,20)(17,19)(21,25)(22,24)(26,30)(27,29)(31,35)(32,34)$
2B $2^{10},1^{15}$ $105$ $2$ $10$ $( 1,11)( 2,12)( 3,13)( 4,14)( 5,15)(21,31)(22,32)(23,33)(24,34)(25,35)$
2C $2^{16},1^{3}$ $525$ $2$ $16$ $( 1,29)( 2,28)( 3,27)( 4,26)( 5,30)( 6, 9)( 7, 8)(11,14)(12,13)(16,34)(17,33)(18,32)(19,31)(20,35)(21,24)(22,23)$
3A $3^{5},1^{20}$ $70$ $3$ $10$ $( 6,26,31)( 7,27,32)( 8,28,33)( 9,29,34)(10,30,35)$
3B $3^{10},1^{5}$ $280$ $3$ $20$ $( 1,16,31)( 2,17,32)( 3,18,33)( 4,19,34)( 5,20,35)( 6,21,26)( 7,22,27)( 8,23,28)( 9,24,29)(10,25,30)$
4A1 $4^{7},1^{7}$ $5$ $4$ $21$ $( 1, 2, 4, 3)( 6, 7, 9, 8)(11,12,14,13)(16,17,19,18)(21,22,24,23)(26,27,29,28)(31,32,34,33)$
4A-1 $4^{7},1^{7}$ $5$ $4$ $21$ $( 1, 3, 4, 2)( 6, 8, 9, 7)(11,13,14,12)(16,18,19,17)(21,23,24,22)(26,28,29,27)(31,33,34,32)$
4B1 $4^{7},2^{2},1^{3}$ $525$ $4$ $23$ $( 1,22, 5,24)( 2,25, 4,21)( 3,23)( 6, 7,10, 9)(11,17,15,19)(12,20,14,16)(13,18)(26,27,30,29)(31,32,35,34)$
4B-1 $4^{7},2^{2},1^{3}$ $525$ $4$ $23$ $( 1,24, 5,22)( 2,21, 4,25)( 3,23)( 6, 9,10, 7)(11,19,15,17)(12,16,14,20)(13,18)(26,29,30,27)(31,34,35,32)$
4C $4^{5},2^{5},1^{5}$ $630$ $4$ $20$ $( 1,31,11,21)( 2,32,12,22)( 3,33,13,23)( 4,34,14,24)( 5,35,15,25)( 6,16)( 7,17)( 8,18)( 9,19)(10,20)$
4D $4^{5},2^{7},1$ $3150$ $4$ $22$ $( 1,32,26,12)( 2,31,27,11)( 3,35,28,15)( 4,34,29,14)( 5,33,30,13)( 6,17)( 7,16)( 8,20)( 9,19)(10,18)(21,22)(23,25)$
4E1 $4^{8},2,1$ $3150$ $4$ $25$ $( 1,32,29,18)( 2,34,28,16)( 3,31,27,19)( 4,33,26,17)( 5,35,30,20)( 6,22, 9,23)( 7,24, 8,21)(10,25)(11,12,14,13)$
4E-1 $4^{8},2,1$ $3150$ $4$ $25$ $( 1,18,29,32)( 2,16,28,34)( 3,19,27,31)( 4,17,26,33)( 5,20,30,35)( 6,23, 9,22)( 7,21, 8,24)(10,25)(11,13,14,12)$
5A $5^{7}$ $4$ $5$ $28$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17)(21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)$
5B $5^{5},1^{10}$ $504$ $5$ $20$ $( 1,21, 6,26,31)( 2,22, 7,27,32)( 3,23, 8,28,33)( 4,24, 9,29,34)( 5,25,10,30,35)$
5C $5^{7}$ $2016$ $5$ $28$ $( 1, 5, 4, 3, 2)( 6,30,24,13,32)( 7,26,25,14,33)( 8,27,21,15,34)( 9,28,22,11,35)(10,29,23,12,31)(16,20,19,18,17)$
6A $3^{5},2^{10}$ $210$ $6$ $20$ $( 1,11)( 2,12)( 3,13)( 4,14)( 5,15)( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(16,26,31)(17,27,32)(18,28,33)(19,29,34)(20,30,35)$
6B $6^{2},3,2^{8},1^{4}$ $350$ $6$ $20$ $( 1, 5)( 2, 4)( 6,35,26,10,31,30)( 7,34,27, 9,32,29)( 8,33,28)(11,15)(12,14)(16,20)(17,19)(21,25)(22,24)$
6C $6^{2},3,2^{10}$ $1050$ $6$ $22$ $( 1,11,16)( 2,15,17, 5,12,20)( 3,14,18, 4,13,19)( 6,26)( 7,30)( 8,29)( 9,28)(10,27)(21,31)(22,35)(23,34)(24,33)(25,32)$
6D $6^{4},3^{2},2^{2},1$ $1400$ $6$ $26$ $( 1,34,16, 4,31,19)( 2,33,17, 3,32,18)( 5,35,20)( 6,29,21, 9,26,24)( 7,28,22, 8,27,23)(10,30,25)(11,14)(12,13)$
7A1 $7^{5}$ $360$ $7$ $30$ $( 1,11,21,16,26,31, 6)( 2,12,22,17,27,32, 7)( 3,13,23,18,28,33, 8)( 4,14,24,19,29,34, 9)( 5,15,25,20,30,35,10)$
7A-1 $7^{5}$ $360$ $7$ $30$ $( 1, 6,31,26,16,21,11)( 2, 7,32,27,17,22,12)( 3, 8,33,28,18,23,13)( 4, 9,34,29,19,24,14)( 5,10,35,30,20,25,15)$
10A $10^{2},5^{3}$ $420$ $10$ $30$ $( 1,13, 5,12, 4,11, 3,15, 2,14)( 6, 8,10, 7, 9)(16,18,20,17,19)(21,33,25,32,24,31,23,35,22,34)(26,28,30,27,29)$
10B $10^{2},5,2^{4},1^{2}$ $2520$ $10$ $26$ $( 1,29,21,34, 6, 4,26,24,31, 9)( 2,28,22,33, 7, 3,27,23,32, 8)( 5,30,25,35,10)(11,14)(12,13)(16,19)(17,18)$
12A1 $12,4^{4},3,1^{4}$ $350$ $12$ $25$ $( 1, 7,24, 3, 6,22, 4, 8,21, 2, 9,23)( 5,10,25)(11,12,14,13)(16,17,19,18)(26,27,29,28)(31,32,34,33)$
12A-1 $12,4^{4},3,1^{4}$ $350$ $12$ $25$ $( 1,23, 9, 2,21, 8, 4,22, 6, 3,24, 7)( 5,25,10)(11,13,14,12)(16,18,19,17)(26,28,29,27)(31,33,34,32)$
12B1 $12,4^{4},3,2^{2}$ $1050$ $12$ $27$ $( 1,24, 5,22)( 2,21, 4,25)( 3,23)( 6,29,35, 7,26,34,10,27,31, 9,30,32)( 8,28,33)(11,19,15,17)(12,16,14,20)(13,18)$
12B-1 $12,4^{4},3,2^{2}$ $1050$ $12$ $27$ $( 1,22, 5,24)( 2,25, 4,21)( 3,23)( 6,32,30, 9,31,27,10,34,26, 7,35,29)( 8,33,28)(11,17,15,19)(12,20,14,16)(13,18)$
12C1 $12^{2},4,3^{2},1$ $1400$ $12$ $29$ $( 1,17,34, 3,16,32, 4,18,31, 2,19,33)( 5,20,35)( 6,22,29, 8,21,27, 9,23,26, 7,24,28)(10,25,30)(11,12,14,13)$
12C-1 $12^{2},4,3^{2},1$ $1400$ $12$ $29$ $( 1,33,19, 2,31,18, 4,32,16, 3,34,17)( 5,35,20)( 6,28,24, 7,26,23, 9,27,21, 8,29,22)(10,30,25)(11,13,14,12)$
14A1 $14^{2},7$ $1800$ $14$ $32$ $( 1,30,11,35,21,10,16, 5,26,15,31,25, 6,20)( 2,29,12,34,22, 9,17, 4,27,14,32,24, 7,19)( 3,28,13,33,23, 8,18)$
14A-1 $14^{2},7$ $1800$ $14$ $32$ $( 1,20, 6,25,31,15,26, 5,16,10,21,35,11,30)( 2,19, 7,24,32,14,27, 4,17, 9,22,34,12,29)( 3,18, 8,23,33,13,28)$
15A $15,5^{4}$ $280$ $15$ $30$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,30,34,18,27,31,20,29,33,17,26,35,19,28,32)(21,25,24,23,22)$
15B $15^{2},5$ $1120$ $15$ $32$ $( 1,14,27, 5,13,26, 4,12,30, 3,11,29, 2,15,28)( 6,19,22,10,18,21, 9,17,25, 8,16,24, 7,20,23)(31,34,32,35,33)$
20A $20,10,5$ $2520$ $20$ $32$ $( 1,32,13,24, 5,31,12,23, 4,35,11,22, 3,34,15,21, 2,33,14,25)( 6,17, 8,19,10,16, 7,18, 9,20)(26,27,28,29,30)$
20B1 $20,5,4^{2},1^{2}$ $2520$ $20$ $29$ $( 1,32,29, 8,21, 2,34,28, 6,22, 4,33,26, 7,24, 3,31,27, 9,23)( 5,35,30,10,25)(11,12,14,13)(16,17,19,18)$
20B-1 $20,5,4^{2},1^{2}$ $2520$ $20$ $29$ $( 1,23, 9,27,31, 3,24, 7,26,33, 4,22, 6,28,34, 2,21, 8,29,32)( 5,25,10,30,35)(11,13,14,12)(16,18,19,17)$
28A1 $28,7$ $1800$ $28$ $33$ $( 1,22,30, 9,11,17,35, 4,21,27,10,14,16,32, 5,24,26, 7,15,19,31, 2,25,29, 6,12,20,34)( 3,23,28, 8,13,18,33)$
28A-1 $28,7$ $1800$ $28$ $33$ $( 1,34,20,12, 6,29,25, 2,31,19,15, 7,26,24, 5,32,16,14,10,27,21, 4,35,17,11, 9,30,22)( 3,33,18,13, 8,28,23)$
28A5 $28,7$ $1800$ $28$ $33$ $( 1,17,10,24,31,12,30, 4,16, 7,25,34,11,27, 5,19, 6,22,35,14,26, 2,20, 9,21,32,15,29)( 3,18, 8,23,33,13,28)$
28A-5 $28,7$ $1800$ $28$ $33$ $( 1,29,15,32,21, 9,20, 2,26,14,35,22, 6,19, 5,27,11,34,25, 7,16, 4,30,12,31,24,10,17)( 3,28,13,33,23, 8,18)$
30A $15,10^{2}$ $840$ $30$ $32$ $( 1,13, 5,12, 4,11, 3,15, 2,14)( 6,23,10,22, 9,21, 8,25, 7,24)(16,33,30,17,34,26,18,35,27,19,31,28,20,32,29)$
35A1 $35$ $1440$ $35$ $34$ $( 1,18,30,12,34,21, 8, 5,17,29,11,33,25, 7, 4,16,28,15,32,24, 6, 3,20,27,14,31,23,10, 2,19,26,13,35,22, 9)$
35A-1 $35$ $1440$ $35$ $34$ $( 1, 9,22,35,13,26,19, 2,10,23,31,14,27,20, 3, 6,24,32,15,28,16, 4, 7,25,33,11,29,17, 5, 8,21,34,12,30,18)$

Malle's constant $a(G)$:     $1/10$

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

45 x 45 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

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