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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

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

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);