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Properties

Label 26T5
Degree $26$
Order $78$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{13}:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(26, 5);
 

Group action invariants

Degree $n$:  $26$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $5$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{13}:C_6$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
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)
magma: Generators(G);
 

Low degree resolvents

|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 TypeSizeOrderRepresentative
$1^{26}$ $1$ $1$ $()$
$3^{8},1^{2}$ $13$ $3$ $( 3, 8,19)( 4, 7,20)( 5,14,11)( 6,13,12)( 9,26,21)(10,25,22)(15,18,23) (16,17,24)$
$3^{8},1^{2}$ $13$ $3$ $( 3,19, 8)( 4,20, 7)( 5,11,14)( 6,12,13)( 9,21,26)(10,22,25)(15,23,18) (16,24,17)$
$2^{13}$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)$
$6^{4},2$ $13$ $6$ $( 1, 2)( 3, 7,19, 4, 8,20)( 5,13,11, 6,14,12)( 9,25,21,10,26,22) (15,17,23,16,18,24)$
$6^{4},2$ $13$ $6$ $( 1, 2)( 3,20, 8, 4,19, 7)( 5,12,14, 6,11,13)( 9,22,26,10,21,25) (15,24,18,16,23,17)$
$26$ $3$ $26$ $( 1, 3, 6, 8,10,11,13,16,18,19,22,24,25, 2, 4, 5, 7, 9,12,14,15,17,20,21,23,26 )$
$13^{2}$ $3$ $13$ $( 1, 4, 6, 7,10,12,13,15,18,20,22,23,25)( 2, 3, 5, 8, 9,11,14,16,17,19,21,24, 26)$
$26$ $3$ $26$ $( 1, 5,10,14,18,21,25, 3, 7,11,15,19,23, 2, 6, 9,13,17,22,26, 4, 8,12,16,20,24 )$
$13^{2}$ $3$ $13$ $( 1, 6,10,13,18,22,25, 4, 7,12,15,20,23)( 2, 5, 9,14,17,21,26, 3, 8,11,16,19, 24)$
$26$ $3$ $26$ $( 1, 9,18,26, 7,16,23, 5,13,21, 4,11,20, 2,10,17,25, 8,15,24, 6,14,22, 3,12,19 )$
$13^{2}$ $3$ $13$ $( 1,10,18,25, 7,15,23, 6,13,22, 4,12,20)( 2, 9,17,26, 8,16,24, 5,14,21, 3,11, 19)$
$13^{2}$ $3$ $13$ $( 1,15, 4,18, 6,20, 7,22,10,23,12,25,13)( 2,16, 3,17, 5,19, 8,21, 9,24,11,26, 14)$
$26$ $3$ $26$ $( 1,16, 4,17, 6,19, 7,21,10,24,12,26,13, 2,15, 3,18, 5,20, 8,22, 9,23,11,25,14 )$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $78=2 \cdot 3 \cdot 13$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  78.2
magma: IdentifyGroup(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 13A1 13A-2 13A2 13A-1 13A-1 13A-2 13A2 13A1
3 P 1A 2A 1A 1A 2A 2A 13A-2 13A-1 13A1 13A2 26A-1 26A-5 26A5 26A1
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 ζ132+ζ135+ζ136 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133 ζ13−6+ζ13−5+ζ13−2 ζ132+ζ135+ζ136
78.2.3a2 C 3 3 0 0 0 0 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2
78.2.3a3 C 3 3 0 0 0 0 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133 ζ13−6+ζ13−5+ζ13−2 ζ132+ζ135+ζ136 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133
78.2.3a4 C 3 3 0 0 0 0 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2 ζ13−6+ζ13−5+ζ13−2 ζ132+ζ135+ζ136 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134
78.2.3b1 C 3 −3 0 0 0 0 ζ13−6+ζ13−5+ζ13−2 ζ132+ζ135+ζ136 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134 −ζ13−3−ζ13−1−ζ134 −ζ13−4−ζ13−ζ133 −ζ13−6−ζ13−5−ζ13−2 −ζ132−ζ135−ζ136
78.2.3b2 C 3 −3 0 0 0 0 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133 −ζ13−4−ζ13−ζ133 −ζ13−3−ζ13−1−ζ134 −ζ132−ζ135−ζ136 −ζ13−6−ζ13−5−ζ13−2
78.2.3b3 C 3 −3 0 0 0 0 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133 ζ13−6+ζ13−5+ζ13−2 ζ132+ζ135+ζ136 −ζ132−ζ135−ζ136 −ζ13−6−ζ13−5−ζ13−2 −ζ13−3−ζ13−1−ζ134 −ζ13−4−ζ13−ζ133
78.2.3b4 C 3 −3 0 0 0 0 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2 −ζ13−6−ζ13−5−ζ13−2 −ζ132−ζ135−ζ136 −ζ13−4−ζ13−ζ133 −ζ13−3−ζ13−1−ζ134

magma: CharacterTable(G);