Properties

Label 26T39
26T39 1 2 1->2 4 1->4 3 2->3 6 3->6 23 3->23 5 4->5 24 4->24 5->6 8 5->8 7 6->7 10 7->10 21 7->21 9 8->9 22 8->22 11 9->11 9->11 12 10->12 10->12 13 11->13 14 12->14 16 13->16 19 13->19 15 14->15 20 14->20 17 15->17 18 16->18 17->18 17->19 18->20 19->22 20->21 21->24 22->23 26 23->26 25 24->25 25->1 25->26 26->2
Degree $26$
Order $5616$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $\PSL(3,3)$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(26, 39);
 
Copy content sage:G = TransitiveGroup(26, 39)
 
Copy content oscar:G = transitive_group(26, 39)
 
Copy content gap:G := TransitiveGroup(26, 39);
 

Group invariants

Abstract group:  $\PSL(3,3)$
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:  $5616=2^{4} \cdot 3^{3} \cdot 13$
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 $n$:  $26$
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 $t$:  $39$
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(26).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(26), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(26), G));
 
Generators:  $(1,2)(3,23)(4,24)(5,6)(7,21)(8,22)(9,11)(10,12)(13,19)(14,20)(17,18)(25,26)$, $(1,4,5,8,9,11,13,16,18,20,21,24,25)(2,3,6,7,10,12,14,15,17,19,22,23,26)$
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 13: $\PSL(3,3)$

Low degree siblings

13T7 x 2, 26T39, 39T43 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{26}$ $1$ $1$ $0$ $()$
2A $2^{12},1^{2}$ $117$ $2$ $12$ $( 1, 6)( 2, 5)( 3,15)( 4,16)( 7,14)( 8,13)( 9,10)(11,21)(12,22)(17,18)(19,20)(23,24)$
3A $3^{6},1^{8}$ $104$ $3$ $12$ $( 1,22,23)( 2,21,24)( 5,16,20)( 6,15,19)( 7, 9,26)( 8,10,25)$
3B $3^{8},1^{2}$ $624$ $3$ $16$ $( 1,15,25)( 2,16,26)( 3,18,11)( 4,17,12)( 5, 9,24)( 6,10,23)( 7,21,20)( 8,22,19)$
4A $4^{6},1^{2}$ $702$ $4$ $18$ $( 1, 4, 6,16)( 2, 3, 5,15)( 7,21,14,11)( 8,22,13,12)( 9,20,10,19)(17,23,18,24)$
6A $6^{3},2^{3},1^{2}$ $936$ $6$ $18$ $( 1,26,22, 7,23, 9)( 2,25,21, 8,24,10)( 5,19,16, 6,20,15)(11,18)(12,17)(13,14)$
8A1 $8^{3},2$ $702$ $8$ $22$ $( 1,12, 4, 8, 6,22,16,13)( 2,11, 3, 7, 5,21,15,14)( 9,17,20,23,10,18,19,24)(25,26)$
8A-1 $8^{3},2$ $702$ $8$ $22$ $( 1,13,16,22, 6, 8, 4,12)( 2,14,15,21, 5, 7, 3,11)( 9,24,19,18,10,23,20,17)(25,26)$
13A1 $13^{2}$ $432$ $13$ $24$ $( 1, 3,14,19,18, 6,10,25,21, 7,12,15,24)( 2, 4,13,20,17, 5, 9,26,22, 8,11,16,23)$
13A-1 $13^{2}$ $432$ $13$ $24$ $( 1,24,15,12, 7,21,25,10, 6,18,19,14, 3)( 2,23,16,11, 8,22,26, 9, 5,17,20,13, 4)$
13A2 $13^{2}$ $432$ $13$ $24$ $( 1,14,18,10,21,12,24, 3,19, 6,25, 7,15)( 2,13,17, 9,22,11,23, 4,20, 5,26, 8,16)$
13A-2 $13^{2}$ $432$ $13$ $24$ $( 1,15, 7,25, 6,19, 3,24,12,21,10,18,14)( 2,16, 8,26, 5,20, 4,23,11,22, 9,17,13)$

Malle's constant $a(G)$:     $1/12$

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 3B 4A 6A 8A1 8A-1 13A1 13A-1 13A2 13A-2
Size 1 117 104 624 702 936 702 702 432 432 432 432
2 P 1A 1A 3A 3B 2A 3A 4A 4A 13A2 13A-2 13A-1 13A1
3 P 1A 2A 1A 1A 4A 2A 8A1 8A-1 13A1 13A-1 13A2 13A-2
13 P 1A 2A 3A 3B 4A 6A 8A-1 8A1 1A 1A 1A 1A
Type
5616.a.1a R 1 1 1 1 1 1 1 1 1 1 1 1
5616.a.12a R 12 4 3 0 0 1 0 0 −1 −1 −1 −1
5616.a.13a R 13 −3 4 1 1 0 −1 −1 0 0 0 0
5616.a.16a1 C 16 0 −2 1 0 0 0 0 ζ13−6+ζ13−5+ζ13−2 ζ132+ζ135+ζ136 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134
5616.a.16a2 C 16 0 −2 1 0 0 0 0 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133
5616.a.16a3 C 16 0 −2 1 0 0 0 0 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133 ζ13−6+ζ13−5+ζ13−2 ζ132+ζ135+ζ136
5616.a.16a4 C 16 0 −2 1 0 0 0 0 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2
5616.a.26a R 26 2 −1 −1 2 −1 0 0 0 0 0 0
5616.a.26b1 C 26 −2 −1 −1 0 1 −ζ8−ζ83 ζ8+ζ83 0 0 0 0
5616.a.26b2 C 26 −2 −1 −1 0 1 ζ8+ζ83 −ζ8−ζ83 0 0 0 0
5616.a.27a R 27 3 0 0 −1 0 −1 −1 1 1 1 1
5616.a.39a R 39 −1 3 0 −1 −1 1 1 0 0 0 0

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