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Properties

Label 26T33
26T33 1 16 1->16 23 1->23 2 21 2->21 22 2->22 3 3->21 26 3->26 4 18 4->18 20 4->20 5 19 5->19 5->23 6 15 6->15 6->18 7 17 7->17 7->20 8 8->16 25 8->25 9 9->15 9->17 10 14 10->14 10->22 11 11->14 11->26 12 12->19 12->25 13 24 13->24 13->24 14->1 14->9 15->5 15->8 16->1 16->2 17->10 18->3 19->2 19->10 20->4 20->11 21->7 21->11 22->3 22->5 23->12 23->12 24->6 24->8 25->4 25->13 26->7 26->13
Degree $26$
Order $4056$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{13}:F_{13}$

Related objects

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Show commands: Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(26, 33);
 
Copy content sage:G = TransitiveGroup(26, 33)
 
Copy content oscar:G = transitive_group(26, 33)
 

Group invariants

Abstract group:  $D_{13}:F_{13}$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Order:  $4056=2^{3} \cdot 3 \cdot 13^{2}$
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:  yes
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 $n$:  $26$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Transitive number $t$:  $33$
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]
 
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:  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)
 
$\card{\Aut(F/K)}$:  $1$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(26).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(26), G)[1])
 
Generators:  $(1,23,12,25,4,20,11,26,13,24,8,16)(2,22,3,21,7,17,10,14,9,15,5,19)(6,18)$, $(1,16,2,21,11,14)(3,26,7,20,4,18)(5,23,12,19,10,22)(6,15,8,25,13,24)(9,17)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $C_6$ x 3
$8$:  $C_4\times C_2$
$12$:  $C_{12}$ x 2, $C_6\times C_2$
$24$:  24T2
$156$:  $F_{13}$ x 2
$312$:  26T10 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T33 x 5

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{26}$ $1$ $1$ $0$ $()$
2A $2^{13}$ $13$ $2$ $13$ $( 1,19)( 2,26)( 3,20)( 4,14)( 5,21)( 6,15)( 7,22)( 8,16)( 9,23)(10,17)(11,24)(12,18)(13,25)$
2B $2^{13}$ $13$ $2$ $13$ $( 1,15)( 2,21)( 3,14)( 4,20)( 5,26)( 6,19)( 7,25)( 8,18)( 9,24)(10,17)(11,23)(12,16)(13,22)$
2C $2^{12},1^{2}$ $169$ $2$ $12$ $( 1, 3)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$
3A1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1,12, 6)( 3, 5,11)( 4, 8, 7)( 9,10,13)(15,17,23)(16,20,19)(18,26,24)(21,22,25)$
3A-1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1, 6,12)( 3,11, 5)( 4, 7, 8)( 9,13,10)(15,23,17)(16,19,20)(18,24,26)(21,25,22)$
4A1 $4^{6},1^{2}$ $169$ $4$ $18$ $( 1,10, 3, 7)( 4,12,13, 5)( 6, 9,11, 8)(15,19,26,22)(16,24,25,17)(18,21,23,20)$
4A-1 $4^{6},1^{2}$ $169$ $4$ $18$ $( 1, 7, 3,10)( 4, 5,13,12)( 6, 8,11, 9)(15,22,26,19)(16,17,25,24)(18,20,23,21)$
4B1 $4^{6},2$ $169$ $4$ $19$ $( 1,19, 4,20)( 2,15, 3,24)( 5,16,13,23)( 6,25,12,14)( 7,21,11,18)( 8,17,10,22)( 9,26)$
4B-1 $4^{6},2$ $169$ $4$ $19$ $( 1,20, 4,19)( 2,24, 3,15)( 5,23,13,16)( 6,14,12,25)( 7,18,11,21)( 8,22,10,17)( 9,26)$
6A1 $6^{4},1^{2}$ $169$ $6$ $20$ $( 1,11,12, 3, 6, 5)( 4,10, 8,13, 7, 9)(15,18,17,26,23,24)(16,22,20,25,19,21)$
6A-1 $6^{4},1^{2}$ $169$ $6$ $20$ $( 1, 5, 6, 3,12,11)( 4, 9, 7,13, 8,10)(15,24,23,26,17,18)(16,21,19,25,20,22)$
6B1 $6^{4},2$ $169$ $6$ $21$ $( 1,17,11,19,10,24)( 2,25, 7,26,13,22)( 3,20)( 4,15,12,14, 6,18)( 5,23, 8,21, 9,16)$
6B-1 $6^{4},2$ $169$ $6$ $21$ $( 1,24,10,19,11,17)( 2,22,13,26, 7,25)( 3,20)( 4,18, 6,14,12,15)( 5,16, 9,21, 8,23)$
6C1 $6^{4},2$ $169$ $6$ $21$ $( 1,23)( 2,15,10,16, 4,25)( 3,20, 6,22, 7,14)( 5,17,11,21,13,18)( 8,19,12,26, 9,24)$
6C-1 $6^{4},2$ $169$ $6$ $21$ $( 1,23)( 2,25, 4,16,10,15)( 3,14, 7,22, 6,20)( 5,18,13,21,11,17)( 8,24, 9,26,12,19)$
12A1 $12^{2},1^{2}$ $169$ $12$ $22$ $( 1, 4,11,10,12, 8, 3,13, 6, 7, 5, 9)(15,25,18,19,17,21,26,16,23,22,24,20)$
12A-1 $12^{2},1^{2}$ $169$ $12$ $22$ $( 1, 9, 5, 7, 6,13, 3, 8,12,10,11, 4)(15,20,24,22,23,16,26,21,17,19,18,25)$
12A5 $12^{2},1^{2}$ $169$ $12$ $22$ $( 1, 8, 5,10, 6, 4, 3, 9,12, 7,11,13)(15,21,24,19,23,25,26,20,17,22,18,16)$
12A-5 $12^{2},1^{2}$ $169$ $12$ $22$ $( 1,13,11, 7,12, 9, 3, 4, 6,10, 5, 8)(15,16,18,22,17,20,26,25,23,19,24,21)$
12B1 $12^{2},2$ $169$ $12$ $23$ $( 1,24, 7,19, 2,21, 4,15,11,20, 3,18)( 5,25, 8,16,12,17,13,14,10,23, 6,22)( 9,26)$
12B-1 $12^{2},2$ $169$ $12$ $23$ $( 1,18, 3,20,11,15, 4,21, 2,19, 7,24)( 5,22, 6,23,10,14,13,17,12,16, 8,25)( 9,26)$
12B5 $12^{2},2$ $169$ $12$ $23$ $( 1,21, 3,19,11,24, 4,18, 2,20, 7,15)( 5,17, 6,16,10,25,13,22,12,23, 8,14)( 9,26)$
12B-5 $12^{2},2$ $169$ $12$ $23$ $( 1,15, 7,20, 2,18, 4,24,11,19, 3,21)( 5,14, 8,23,12,22,13,25,10,16, 6,17)( 9,26)$
13A $13^{2}$ $12$ $13$ $24$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13B $13^{2}$ $12$ $13$ $24$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13C $13,1^{13}$ $24$ $13$ $12$ $(14,26,25,24,23,22,21,20,19,18,17,16,15)$
13D $13^{2}$ $24$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,19,24,16,21,26,18,23,15,20,25,17,22)$
13E $13^{2}$ $24$ $13$ $24$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,19,24,16,21,26,18,23,15,20,25,17,22)$
13F $13^{2}$ $24$ $13$ $24$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13G $13^{2}$ $24$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13H $13^{2}$ $24$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,18,22,26,17,21,25,16,20,24,15,19,23)$
26A $26$ $156$ $26$ $25$ $( 1,25, 8,22, 2,19, 9,16, 3,26,10,23, 4,20,11,17, 5,14,12,24, 6,21,13,18, 7,15)$
26B $26$ $156$ $26$ $25$ $( 1,14, 5,25, 9,23,13,21, 4,19, 8,17,12,15, 3,26, 7,24,11,22, 2,20, 6,18,10,16)$

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)
 

Character table

34 x 34 character table

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 

Regular extensions

Data not computed