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Properties

Label 26T10
Degree $26$
Order $312$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times F_{13}$

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Show commands: Magma

magma: G := TransitiveGroup(26, 10);
 

Group action invariants

Degree $n$:  $26$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $10$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times F_{13}$
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:  (1,8,6,23,17,19)(2,7,5,24,18,20)(3,16,11,22,9,13)(4,15,12,21,10,14)(25,26), (1,11,6,20,21,26,7,24,3,15,14,9)(2,12,5,19,22,25,8,23,4,16,13,10)
magma: Generators(G);
 

Low degree resolvents

|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
$156$:  $F_{13}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: $F_{13}$

Low degree siblings

26T10

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $312=2^{3} \cdot 3 \cdot 13$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  312.45
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 4A1 4A-1 4B1 4B-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 12A1 12A-1 12A5 12A-5 12B1 12B-1 12B5 12B-5 13A 26A
Size 1 1 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 12 12
2 P 1A 1A 1A 1A 3A-1 3A1 2B 2B 2B 2B 3A-1 3A-1 3A1 3A1 3A-1 3A1 6A-1 6A-1 6A1 6A1 6A-1 6A-1 6A1 6A1 13A 13A
3 P 1A 2A 2B 2C 1A 1A 4B1 4B-1 4A1 4A-1 2B 2C 2A 2C 2A 2B 4B1 4A1 4B1 4A1 4A-1 4B-1 4A-1 4B-1 13A 26A
13 P 1A 2A 2B 2C 3A1 3A-1 4B-1 4B1 4A-1 4A1 6A-1 6C-1 6B-1 6C1 6B1 6A1 12B5 12A5 12B1 12A1 12A-1 12B-1 12A-5 12B-5 1A 2A
Type
312.45.1a R 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
312.45.1b R 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
312.45.1c R 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
312.45.1d R 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
312.45.1e1 C 1 1 1 1 ζ3−1 ζ3 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1
312.45.1e2 C 1 1 1 1 ζ3 ζ3−1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1
312.45.1f1 C 1 −1 −1 1 1 1 −i i −i i −1 −1 −1 −1 1 1 i −i i −i i −i i −i 1 −1
312.45.1f2 C 1 −1 −1 1 1 1 i −i i −i −1 −1 −1 −1 1 1 −i i −i i −i i −i i 1 −1
312.45.1g1 C 1 1 −1 −1 1 1 −i i i −i −1 −1 1 1 −1 −1 i −i i −i −i i −i i 1 1
312.45.1g2 C 1 1 −1 −1 1 1 i −i −i i −1 −1 1 1 −1 −1 −i i −i i i −i i −i 1 1
312.45.1h1 C 1 −1 1 −1 ζ3−1 ζ3 −1 −1 1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 −1
312.45.1h2 C 1 −1 1 −1 ζ3 ζ3−1 −1 −1 1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 −1
312.45.1i1 C 1 −1 1 −1 ζ3−1 ζ3 1 1 −1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 −1
312.45.1i2 C 1 −1 1 −1 ζ3 ζ3−1 1 1 −1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 −1
312.45.1j1 C 1 1 1 1 ζ3−1 ζ3 −1 −1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 1
312.45.1j2 C 1 1 1 1 ζ3 ζ3−1 −1 −1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 1
312.45.1k1 C 1 −1 −1 1 −ζ122 ζ124 −ζ123 ζ123 −ζ123 ζ123 −ζ124 ζ122 ζ122 −ζ124 ζ124 −ζ122 −ζ125 ζ12 −ζ12 ζ125 −ζ125 ζ12 −ζ12 ζ125 1 −1
312.45.1k2 C 1 −1 −1 1 ζ124 −ζ122 ζ123 −ζ123 ζ123 −ζ123 ζ122 −ζ124 −ζ124 ζ122 −ζ122 ζ124 ζ12 −ζ125 ζ125 −ζ12 ζ12 −ζ125 ζ125 −ζ12 1 −1
312.45.1k3 C 1 −1 −1 1 −ζ122 ζ124 ζ123 −ζ123 ζ123 −ζ123 −ζ124 ζ122 ζ122 −ζ124 ζ124 −ζ122 ζ125 −ζ12 ζ12 −ζ125 ζ125 −ζ12 ζ12 −ζ125 1 −1
312.45.1k4 C 1 −1 −1 1 ζ124 −ζ122 −ζ123 ζ123 −ζ123 ζ123 ζ122 −ζ124 −ζ124 ζ122 −ζ122 ζ124 −ζ12 ζ125 −ζ125 ζ12 −ζ12 ζ125 −ζ125 ζ12 1 −1
312.45.1l1 C 1 1 −1 −1 −ζ122 ζ124 −ζ123 ζ123 ζ123 −ζ123 −ζ124 ζ122 −ζ122 ζ124 −ζ124 ζ122 −ζ125 ζ12 −ζ12 ζ125 ζ125 −ζ12 ζ12 −ζ125 1 1
312.45.1l2 C 1 1 −1 −1 ζ124 −ζ122 ζ123 −ζ123 −ζ123 ζ123 ζ122 −ζ124 ζ124 −ζ122 ζ122 −ζ124 ζ12 −ζ125 ζ125 −ζ12 −ζ12 ζ125 −ζ125 ζ12 1 1
312.45.1l3 C 1 1 −1 −1 −ζ122 ζ124 ζ123 −ζ123 −ζ123 ζ123 −ζ124 ζ122 −ζ122 ζ124 −ζ124 ζ122 ζ125 −ζ12 ζ12 −ζ125 −ζ125 ζ12 −ζ12 ζ125 1 1
312.45.1l4 C 1 1 −1 −1 ζ124 −ζ122 −ζ123 ζ123 ζ123 −ζ123 ζ122 −ζ124 ζ124 −ζ122 ζ122 −ζ124 −ζ12 ζ125 −ζ125 ζ12 ζ12 −ζ125 ζ125 −ζ12 1 1
312.45.12a R 12 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1
312.45.12b R 12 −12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 1

magma: CharacterTable(G);