Properties

Label 42T176
42T176 1 7 1->7 21 1->21 2 8 2->8 20 2->20 3 9 3->9 19 3->19 4 16 4->16 17 4->17 5 5->16 18 5->18 6 6->17 6->18 12 7->12 32 7->32 10 8->10 31 8->31 11 9->11 33 9->33 10->5 13 10->13 11->4 15 11->15 12->6 14 12->14 13->21 29 13->29 14->19 28 14->28 15->20 30 15->30 22 16->22 42 16->42 24 17->24 40 17->40 23 18->23 41 18->41 37 19->37 39 19->39 20->37 38 20->38 21->38 21->39 22->6 22->8 23->4 23->7 24->5 24->9 25 25->2 25->24 26 26->3 26->23 27 27->1 27->22 28->26 28->29 29->27 29->30 30->25 30->28 31->32 31->42 32->33 32->41 33->31 33->40 34 34->14 34->27 35 35->15 35->26 36 36->13 36->25 37->1 37->35 38->3 38->36 39->2 39->34 40->12 40->35 41->10 41->34 42->11 42->36
Degree $42$
Order $1092$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $\PSL(2,13)$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(42, 176);
 
Copy content sage:G = TransitiveGroup(42, 176)
 
Copy content oscar:G = transitive_group(42, 176)
 
Copy content gap:G := TransitiveGroup(42, 176);
 

Group invariants

Abstract group:  $\PSL(2,13)$
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:  $1092=2^{2} \cdot 3 \cdot 7 \cdot 13$
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 $t$:  $176$
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);
 
$\card{\Aut(F/K)}$:  $3$
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:  $(1,21,38,36,13,29,27)(2,20,37,35,15,30,25)(3,19,39,34,14,28,26)(4,16,22,8,31,42,11)(5,18,23,7,32,41,10)(6,17,24,9,33,40,12)$, $(1,7,12,14,19,37)(2,8,10,13,21,39)(3,9,11,15,20,38)(4,17,40,35,26,23)(5,16,42,36,25,24)(6,18,41,34,27,22)(28,29,30)(31,32,33)$
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

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 6: None

Degree 7: None

Degree 14: $\PSL(2,13)$

Degree 21: None

Low degree siblings

14T30, 28T120

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 $2^{18},1^{6}$ $91$ $2$ $18$ $( 4,21)( 5,19)( 6,20)( 7,27)( 8,25)( 9,26)(10,30)(11,28)(12,29)(13,23)(14,24)(15,22)(16,31)(17,33)(18,32)(37,40)(38,42)(39,41)$
3A $3^{14}$ $182$ $3$ $28$ $( 1, 2, 3)( 4,38,32)( 5,39,33)( 6,37,31)( 7,23,11)( 8,22,12)( 9,24,10)(13,28,27)(14,30,26)(15,29,25)(16,20,40)(17,19,41)(18,21,42)(34,35,36)$
6A $6^{6},3^{2}$ $182$ $6$ $34$ $( 1, 3, 2)( 4,18,38,21,32,42)( 5,17,39,19,33,41)( 6,16,37,20,31,40)( 7,28,23,27,11,13)( 8,29,22,25,12,15)( 9,30,24,26,10,14)(34,36,35)$
7A1 $7^{6}$ $156$ $7$ $36$ $( 1,27,29,13,36,38,21)( 2,25,30,15,35,37,20)( 3,26,28,14,34,39,19)( 4,11,42,31, 8,22,16)( 5,10,41,32, 7,23,18)( 6,12,40,33, 9,24,17)$
7A2 $7^{6}$ $156$ $7$ $36$ $( 1,29,36,21,27,13,38)( 2,30,35,20,25,15,37)( 3,28,34,19,26,14,39)( 4,42, 8,16,11,31,22)( 5,41, 7,18,10,32,23)( 6,40, 9,17,12,33,24)$
7A3 $7^{6}$ $156$ $7$ $36$ $( 1,13,21,29,38,27,36)( 2,15,20,30,37,25,35)( 3,14,19,28,39,26,34)( 4,31,16,42,22,11, 8)( 5,32,18,41,23,10, 7)( 6,33,17,40,24,12, 9)$
13A1 $13^{3},1^{3}$ $84$ $13$ $36$ $( 1,36, 7,30, 6,26,41,32,19,24,37,10,13)( 2,35, 8,28, 5,27,42,31,21,23,39,11,15)( 3,34, 9,29, 4,25,40,33,20,22,38,12,14)$
13A2 $13^{3},1^{3}$ $84$ $13$ $36$ $( 1, 7, 6,41,19,37,13,36,30,26,32,24,10)( 2, 8, 5,42,21,39,15,35,28,27,31,23,11)( 3, 9, 4,40,20,38,14,34,29,25,33,22,12)$

Malle's constant $a(G)$:     $1/18$

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 6A 7A1 7A2 7A3 13A1 13A2
Size 1 91 182 182 156 156 156 84 84
2 P 1A 1A 3A 3A 7A2 7A3 7A1 13A2 13A1
3 P 1A 2A 1A 2A 7A3 7A1 7A2 13A1 13A2
7 P 1A 2A 3A 6A 1A 1A 1A 13A2 13A1
13 P 1A 2A 3A 6A 7A1 7A2 7A3 1A 1A
Type
1092.25.1a R 1 1 1 1 1 1 1 1 1
1092.25.7a1 R 7 −1 1 −1 0 0 0 ζ13−6+ζ13−5+ζ13−2+1+ζ132+ζ135+ζ136 −ζ13−6−ζ13−5−ζ13−2−ζ132−ζ135−ζ136
1092.25.7a2 R 7 −1 1 −1 0 0 0 −ζ13−6−ζ13−5−ζ13−2−ζ132−ζ135−ζ136 ζ13−6+ζ13−5+ζ13−2+1+ζ132+ζ135+ζ136
1092.25.12a1 R 12 0 0 0 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73 −1 −1
1092.25.12a2 R 12 0 0 0 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7 −1 −1
1092.25.12a3 R 12 0 0 0 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72 −1 −1
1092.25.13a R 13 1 1 1 −1 −1 −1 0 0
1092.25.14a R 14 2 −1 −1 0 0 0 1 1
1092.25.14b R 14 −2 −1 1 0 0 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