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Properties

Label 14T30
14T30 1 2 1->2 4 1->4 12 1->12 3 2->3 6 2->6 8 2->8 3->4 3->4 3->12 4->3 5 4->5 5->6 7 5->7 6->7 11 6->11 7->2 7->8 7->11 8->6 9 8->9 10 9->10 9->10 9->10 10->1 10->11 11->5 11->12 12->9 14 12->14 13 13->14 14->1
Degree $14$
Order $1092$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\PSL(2,13)$

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Show commands: Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(14, 30);
 
Copy content sage:G = TransitiveGroup(14, 30)
 
Copy content oscar:G = transitive_group(14, 30)
 

Group invariants

Abstract group:  $\PSL(2,13)$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
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)
 
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)
 
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)
 
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)
 
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
 

Group action invariants

Degree $n$:  $14$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Transitive number $t$:  $30$
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]
 
CHM label:   $L(14)=PSL(2,13)$
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)
 
Primitive:  yes
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)
 
$\card{\Aut(F/K)}$:  $1$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(14).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(14), G)[1])
 
Generators:  $(1,12)(2,6)(3,4)(7,11)(9,10)(13,14)$, $(1,4,3,12,9,10)(2,8,6,11,5,7)$, $(1,2,3,4,5,6,7,8,9,10,11,12,14)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: None

Low degree siblings

28T120, 42T176

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{14}$ $1$ $1$ $0$ $()$
2A $2^{6},1^{2}$ $91$ $2$ $6$ $( 1, 6)( 2,10)( 3,14)( 4,11)( 5,12)( 8,13)$
3A $3^{4},1^{2}$ $182$ $3$ $8$ $( 1, 8, 2)( 3,11,12)( 4, 5,14)( 6,13,10)$
6A $6^{2},1^{2}$ $182$ $6$ $10$ $( 1,10, 8, 6, 2,13)( 3, 5,11,14,12, 4)$
7A1 $7^{2}$ $156$ $7$ $12$ $( 1,13, 4, 5, 8,10,14)( 2, 7,11, 3,12, 9, 6)$
7A2 $7^{2}$ $156$ $7$ $12$ $( 1, 4, 8,14,13, 5,10)( 2,11,12, 6, 7, 3, 9)$
7A3 $7^{2}$ $156$ $7$ $12$ $( 1, 5,14, 4,10,13, 8)( 2, 3, 6,11, 9, 7,12)$
13A1 $13,1$ $84$ $13$ $12$ $( 1,13, 2, 5, 6,14,12, 7, 9, 4, 3,10,11)$
13A2 $13,1$ $84$ $13$ $12$ $( 1, 2, 6,12, 9, 3,11,13, 5,14, 7, 4,10)$

Malle's constant $a(G)$:     $1/6$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(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)
 

Regular extensions

$f_{ 1 } =$ $\left(x^{3}-x^{2} + 35 x - 27\right)^{4} \left(x^{2} + 36\right) - 4 \left(x^{2} + 39\right)^{6} \left(7 x^{2} - 2 x + 247\right)/\left(27 \left(39 t^{2}+1\right)\right)$ Copy content Toggle raw display