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Properties

Label 26T5
26T5 1 2 1->2 10 1->10 9 2->9 3 3->2 7 3->7 4 4->1 8 4->8 5 13 5->13 19 5->19 6 14 6->14 20 6->20 12 7->12 7->19 11 8->11 8->20 9->3 25 9->25 10->4 26 10->26 11->6 21 11->21 12->5 22 12->22 13->11 14->12 15 15->6 17 15->17 16 16->5 18 16->18 23 17->23 24 17->24 18->23 18->24 19->4 19->16 20->3 20->15 21->8 21->10 22->7 22->9 23->16 23->25 24->15 24->26 25->18 25->21 26->17 26->22
Degree $26$
Order $78$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{13}:C_6$

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Copy content magma:G := TransitiveGroup(26, 5);
 

Group invariants

Abstract group:  $C_{13}:C_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $78=2 \cdot 3 \cdot 13$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $26$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $5$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,10,4)(2,9,3)(5,19,16)(6,20,15)(7,12,22)(8,11,21)(17,24,26)(18,23,25)$, $(1,2)(3,7,19,4,8,20)(5,13,11,6,14,12)(9,25,21,10,26,22)(15,17,23,16,18,24)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$39$:  $C_{13}:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: $C_{13}:C_3$

Low degree siblings

There are no siblings 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 $2^{13}$ $1$ $2$ $13$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)$
3A1 $3^{8},1^{2}$ $13$ $3$ $16$ $( 1,18,13)( 2,17,14)( 3,24, 5)( 4,23, 6)( 7,10,15)( 8, 9,16)(11,21,26)(12,22,25)$
3A-1 $3^{8},1^{2}$ $13$ $3$ $16$ $( 1,13,18)( 2,14,17)( 3, 5,24)( 4, 6,23)( 7,15,10)( 8,16, 9)(11,26,21)(12,25,22)$
6A1 $6^{4},2$ $13$ $6$ $21$ $( 1,14,18, 2,13,17)( 3, 6,24, 4, 5,23)( 7,16,10, 8,15, 9)(11,25,21,12,26,22)(19,20)$
6A-1 $6^{4},2$ $13$ $6$ $21$ $( 1,17,13, 2,18,14)( 3,23, 5, 4,24, 6)( 7, 9,15, 8,10,16)(11,22,26,12,21,25)(19,20)$
13A1 $13^{2}$ $3$ $13$ $24$ $( 1,25,23,22,20,18,15,13,12,10, 7, 6, 4)( 2,26,24,21,19,17,16,14,11, 9, 8, 5, 3)$
13A-1 $13^{2}$ $3$ $13$ $24$ $( 1, 4, 6, 7,10,12,13,15,18,20,22,23,25)( 2, 3, 5, 8, 9,11,14,16,17,19,21,24,26)$
13A2 $13^{2}$ $3$ $13$ $24$ $( 1,23,20,15,12, 7, 4,25,22,18,13,10, 6)( 2,24,19,16,11, 8, 3,26,21,17,14, 9, 5)$
13A-2 $13^{2}$ $3$ $13$ $24$ $( 1, 6,10,13,18,22,25, 4, 7,12,15,20,23)( 2, 5, 9,14,17,21,26, 3, 8,11,16,19,24)$
26A1 $26$ $3$ $26$ $25$ $( 1,14,25,11,23, 9,22, 8,20, 5,18, 3,15, 2,13,26,12,24,10,21, 7,19, 6,17, 4,16)$
26A-1 $26$ $3$ $26$ $25$ $( 1,24,20,16,12, 8, 4,26,22,17,13, 9, 6, 2,23,19,15,11, 7, 3,25,21,18,14,10, 5)$
26A5 $26$ $3$ $26$ $25$ $( 1, 9,18,26, 7,16,23, 5,13,21, 4,11,20, 2,10,17,25, 8,15,24, 6,14,22, 3,12,19)$
26A-5 $26$ $3$ $26$ $25$ $( 1, 8,13,19,25, 5,12,17,23, 3,10,16,22, 2, 7,14,20,26, 6,11,18,24, 4, 9,15,21)$

Malle's constant $a(G)$:     $1/13$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 6A1 6A-1 13A1 13A-1 13A2 13A-2 26A1 26A-1 26A5 26A-5
Size 1 1 13 13 13 13 3 3 3 3 3 3 3 3
2 P 1A 1A 3A-1 3A1 3A1 3A-1 13A2 13A-2 13A-1 13A1 13A1 13A-1 13A2 13A-2
3 P 1A 2A 1A 1A 2A 2A 13A1 13A-1 13A2 13A-2 26A1 26A-1 26A5 26A-5
13 P 1A 2A 3A1 3A-1 6A1 6A-1 1A 1A 1A 1A 2A 2A 2A 2A
Type
78.2.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
78.2.1b R 1 −1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
78.2.1c1 C 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1 1 1 1 1
78.2.1c2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1 1 1 1 1
78.2.1d1 C 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 1 −1 −1 −1 −1
78.2.1d2 C 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 1 −1 −1 −1 −1
78.2.3a1 C 3 3 0 0 0 0 ζ13−6+ζ13−5+ζ13−2 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133 ζ132+ζ135+ζ136
78.2.3a2 C 3 3 0 0 0 0 ζ132+ζ135+ζ136 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133 ζ13−6+ζ13−5+ζ13−2 ζ132+ζ135+ζ136 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134 ζ13−6+ζ13−5+ζ13−2
78.2.3a3 C 3 3 0 0 0 0 ζ13−3+ζ13−1+ζ134 ζ13−6+ζ13−5+ζ13−2 ζ132+ζ135+ζ136 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2 ζ13−4+ζ13+ζ133
78.2.3a4 C 3 3 0 0 0 0 ζ13−4+ζ13+ζ133 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133 ζ13−6+ζ13−5+ζ13−2 ζ132+ζ135+ζ136 ζ13−3+ζ13−1+ζ134
78.2.3b1 C 3 −3 0 0 0 0 ζ13−6+ζ13−5+ζ13−2 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134 ζ132+ζ135+ζ136 −ζ13−6−ζ13−5−ζ13−2 −ζ13−3−ζ13−1−ζ134 −ζ13−4−ζ13−ζ133 −ζ132−ζ135−ζ136
78.2.3b2 C 3 −3 0 0 0 0 ζ132+ζ135+ζ136 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133 ζ13−6+ζ13−5+ζ13−2 −ζ132−ζ135−ζ136 −ζ13−4−ζ13−ζ133 −ζ13−3−ζ13−1−ζ134 −ζ13−6−ζ13−5−ζ13−2
78.2.3b3 C 3 −3 0 0 0 0 ζ13−3+ζ13−1+ζ134 ζ13−6+ζ13−5+ζ13−2 ζ132+ζ135+ζ136 ζ13−4+ζ13+ζ133 −ζ13−3−ζ13−1−ζ134 −ζ132−ζ135−ζ136 −ζ13−6−ζ13−5−ζ13−2 −ζ13−4−ζ13−ζ133
78.2.3b4 C 3 −3 0 0 0 0 ζ13−4+ζ13+ζ133 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2 ζ13−3+ζ13−1+ζ134 −ζ13−4−ζ13−ζ133 −ζ13−6−ζ13−5−ζ13−2 −ζ132−ζ135−ζ136 −ζ13−3−ζ13−1−ζ134

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed