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Properties

Label 26T9
Degree $26$
Order $156$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{26}:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(26, 9);
 

Group action invariants

Degree $n$:  $26$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $9$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{26}:C_6$
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:  (1,6,20,4,26,12)(2,5,19,3,25,11)(7,14,21,23,17,10)(8,13,22,24,18,9)(15,16), (1,13,3,7,21,5)(2,14,4,8,22,6)(9,16,24,26,20,11)(10,15,23,25,19,12)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$12$:  $C_6\times C_2$
$78$:  $C_{13}:C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: $C_{13}:C_6$

Low degree siblings

26T9

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 $2^{13}$ $1$ $2$ $13$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)$
2B $2^{12},1^{2}$ $13$ $2$ $12$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,26)(10,25)(11,24)(12,23)(13,21)(14,22)(15,19)(16,20)$
2C $2^{13}$ $13$ $2$ $13$ $( 1,23)( 2,24)( 3,22)( 4,21)( 5,19)( 6,20)( 7,18)( 8,17)( 9,15)(10,16)(11,14)(12,13)(25,26)$
3A1 $3^{8},1^{2}$ $13$ $3$ $16$ $( 1, 3,21)( 2, 4,22)( 5,13, 7)( 6,14, 8)( 9,24,20)(10,23,19)(11,16,26)(12,15,25)$
3A-1 $3^{8},1^{2}$ $13$ $3$ $16$ $( 1,21, 3)( 2,22, 4)( 5, 7,13)( 6, 8,14)( 9,20,24)(10,19,23)(11,26,16)(12,25,15)$
6A1 $6^{4},2$ $13$ $6$ $21$ $( 1,25,20, 2,26,19)( 3, 6,11, 4, 5,12)( 7,18,21, 8,17,22)( 9,23,13,10,24,14)(15,16)$
6A-1 $6^{4},1^{2}$ $13$ $6$ $20$ $( 1, 5,21, 7, 3,13)( 2, 6,22, 8, 4,14)( 9,11,20,26,24,16)(10,12,19,25,23,15)$
6B1 $6^{4},2$ $13$ $6$ $21$ $( 1,10,11, 6,24,22)( 2, 9,12, 5,23,21)( 3, 4)( 7,18,13,25,16,19)( 8,17,14,26,15,20)$
6B-1 $6^{4},2$ $13$ $6$ $21$ $( 1,19,13,15,24, 4)( 2,20,14,16,23, 3)( 5,10,26,12, 7,18)( 6, 9,25,11, 8,17)(21,22)$
6C1 $6^{4},1^{2}$ $13$ $6$ $20$ $( 1,13, 3, 7,21, 5)( 2,14, 4, 8,22, 6)( 9,16,24,26,20,11)(10,15,23,25,19,12)$
6C-1 $6^{4},2$ $13$ $6$ $21$ $( 1,15,11, 2,16,12)( 3, 8,17, 4, 7,18)( 5,25,24, 6,26,23)( 9,10)(13,19,21,14,20,22)$
13A1 $13^{2}$ $6$ $13$ $24$ $( 1,21,16, 9, 3,24,17,11, 5,26,20,13, 7)( 2,22,15,10, 4,23,18,12, 6,25,19,14, 8)$
13A2 $13^{2}$ $6$ $13$ $24$ $( 1,16, 3,17, 5,20, 7,21, 9,24,11,26,13)( 2,15, 4,18, 6,19, 8,22,10,23,12,25,14)$
26A1 $26$ $6$ $26$ $25$ $( 1,12,21, 6,16,25, 9,19, 3,14,24, 8,17, 2,11,22, 5,15,26,10,20, 4,13,23, 7,18)$
26A5 $26$ $6$ $26$ $25$ $( 1,25,24,22,20,18,16,14,11,10, 7, 6, 3, 2,26,23,21,19,17,15,13,12, 9, 8, 5, 4)$

Malle's constant $a(G)$:     $1/12$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $156=2^{2} \cdot 3 \cdot 13$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  156.8
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 13A1 13A2 26A1 26A5
Size 1 1 13 13 13 13 13 13 13 13 13 13 6 6 6 6
2 P 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 3A1 3A-1 3A1 3A-1 13A2 13A1 13A1 13A2
3 P 1A 2A 2B 2C 1A 1A 2A 2B 2C 2C 2B 2A 13A1 13A2 26A1 26A5
13 P 1A 2A 2B 2C 3A1 3A-1 6C1 6A-1 6B-1 6B1 6A1 6C-1 1A 1A 2A 2A
Type
156.8.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
156.8.1b R 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1
156.8.1c R 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1
156.8.1d R 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1 1 1 1 1
156.8.1e1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1
156.8.1e2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1
156.8.1f1 C 1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 −1 −1
156.8.1f2 C 1 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 −1 −1
156.8.1g1 C 1 −1 1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 1 −1 −1
156.8.1g2 C 1 −1 1 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 1 −1 −1
156.8.1h1 C 1 1 −1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 1 1 1 1
156.8.1h2 C 1 1 −1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 1 1 1 1
156.8.6a1 R 6 6 0 0 0 0 0 0 0 0 0 0 ζ13−6+ζ13−5+ζ13−2+ζ132+ζ135+ζ136 −ζ13−6−ζ13−5−ζ13−2−1−ζ132−ζ135−ζ136 −ζ13−6−ζ13−5−ζ13−2−1−ζ132−ζ135−ζ136 ζ13−6+ζ13−5+ζ13−2+ζ132+ζ135+ζ136
156.8.6a2 R 6 6 0 0 0 0 0 0 0 0 0 0 −ζ13−6−ζ13−5−ζ13−2−1−ζ132−ζ135−ζ136 ζ13−6+ζ13−5+ζ13−2+ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2+ζ132+ζ135+ζ136 −ζ13−6−ζ13−5−ζ13−2−1−ζ132−ζ135−ζ136
156.8.6b1 R 6 −6 0 0 0 0 0 0 0 0 0 0 ζ13−6+ζ13−5+ζ13−2+ζ132+ζ135+ζ136 −ζ13−6−ζ13−5−ζ13−2−1−ζ132−ζ135−ζ136 ζ13−6+ζ13−5+ζ13−2+1+ζ132+ζ135+ζ136 −ζ13−6−ζ13−5−ζ13−2−ζ132−ζ135−ζ136
156.8.6b2 R 6 −6 0 0 0 0 0 0 0 0 0 0 −ζ13−6−ζ13−5−ζ13−2−1−ζ132−ζ135−ζ136 ζ13−6+ζ13−5+ζ13−2+ζ132+ζ135+ζ136 −ζ13−6−ζ13−5−ζ13−2−ζ132−ζ135−ζ136 ζ13−6+ζ13−5+ζ13−2+1+ζ132+ζ135+ζ136

magma: CharacterTable(G);