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Properties

Label 42T37
42T37 1 10 1->10 28 1->28 2 9 2->9 27 2->27 3 11 3->11 30 3->30 4 12 4->12 29 4->29 5 7 5->7 25 5->25 6 8 6->8 26 6->26 7->10 7->28 8->9 8->27 9->12 9->29 10->11 10->30 11->7 11->25 12->8 12->26 13 39 13->39 41 13->41 14 40 14->40 42 14->42 15 15->40 15->42 16 16->39 16->41 17 37 17->37 38 17->38 18 18->37 18->38 19 19->15 19->16 20 20->15 20->16 21 21->13 21->13 22 22->14 22->14 23 23->17 23->18 24 24->17 24->18 25->19 31 25->31 26->20 32 26->32 27->24 34 27->34 28->23 33 28->33 29->21 36 29->36 30->22 35 30->35 31->1 31->5 32->2 32->6 33->1 33->5 34->2 34->6 35->3 35->4 36->3 36->4 37->23 37->33 38->24 38->34 39->20 39->32 40->19 40->31 41->21 41->36 42->22 42->35
Degree $42$
Order $168$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $\PSL(2,7)$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(42, 37);
 
Copy content sage:G = TransitiveGroup(42, 37)
 
Copy content oscar:G = transitive_group(42, 37)
 
Copy content gap:G := TransitiveGroup(42, 37);
 

Group invariants

Abstract group:  $\PSL(2,7)$
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:  $168=2^{3} \cdot 3 \cdot 7$
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 $n$:  $42$
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 $t$:  $37$
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);
 
$\card{\Aut(F/K)}$:  $2$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(42).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(42), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(42), G));
 
Generators:  $(1,10,30,22,14,40,31)(2,9,29,21,13,39,32)(3,11,25,19,16,41,36)(4,12,26,20,15,42,35)(5,7,28,23,17,37,33)(6,8,27,24,18,38,34)$, $(1,28,33)(2,27,34)(3,30,35)(4,29,36)(5,25,31)(6,26,32)(7,10,11)(8,9,12)(13,41,21)(14,42,22)(15,40,19)(16,39,20)(17,38,24)(18,37,23)$
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

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 6: None

Degree 7: $\GL(3,2)$ x 2

Degree 14: None

Degree 21: $\PSL(2,7)$

Low degree siblings

7T5 x 2, 8T37, 14T10 x 2, 21T14, 24T284, 28T32, 42T38 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{42}$ $1$ $1$ $0$ $()$
2A $2^{20},1^{2}$ $21$ $2$ $20$ $( 1, 8)( 2, 7)( 3,11)( 4,12)( 5,10)( 6, 9)(13,34)(14,33)(15,31)(16,32)(17,35)(18,36)(19,20)(21,22)(25,27)(26,28)(29,30)(37,39)(38,40)(41,42)$
3A $3^{14}$ $56$ $3$ $28$ $( 1,20, 7)( 2,19, 8)( 3,24,11)( 4,23,12)( 5,22, 9)( 6,21,10)(13,30,33)(14,29,34)(15,25,36)(16,26,35)(17,28,32)(18,27,31)(37,42,40)(38,41,39)$
4A $4^{10},1^{2}$ $42$ $4$ $30$ $( 1,17, 8,35)( 2,18, 7,36)( 3,15,11,31)( 4,16,12,32)( 5,14,10,33)( 6,13, 9,34)(19,22,20,21)(25,37,27,39)(26,38,28,40)(29,42,30,41)$
7A1 $7^{6}$ $24$ $7$ $36$ $( 1,26,16,37,23,36,12)( 2,25,15,38,24,35,11)( 3,29,18,39,19,31,10)( 4,30,17,40,20,32, 9)( 5,27,13,42,21,34, 7)( 6,28,14,41,22,33, 8)$
7A-1 $7^{6}$ $24$ $7$ $36$ $( 1,12,36,23,37,16,26)( 2,11,35,24,38,15,25)( 3,10,31,19,39,18,29)( 4, 9,32,20,40,17,30)( 5, 7,34,21,42,13,27)( 6, 8,33,22,41,14,28)$

Malle's constant $a(G)$:     $1/20$

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 3A 4A 7A1 7A-1
Size 1 21 56 42 24 24
2 P 1A 1A 3A 2A 7A1 7A-1
3 P 1A 2A 1A 4A 7A-1 7A1
7 P 1A 2A 3A 4A 1A 1A
Type
168.42.1a R 1 1 1 1 1 1
168.42.3a1 C 3 −1 0 1 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
168.42.3a2 C 3 −1 0 1 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
168.42.6a R 6 2 0 0 −1 −1
168.42.7a R 7 −1 1 −1 0 0
168.42.8a R 8 0 −1 0 1 1

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