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Properties

Label 26T40
Degree $26$
Order $6084$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{13}^2:(C_6\times S_3)$

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Show commands: Magma

magma: G := TransitiveGroup(26, 40);
 

Group action invariants

Degree $n$:  $26$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $40$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{13}^2:(C_6\times S_3)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,19,5,14,9,22,13,17,4,25,8,20,12,15,3,23,7,18,11,26,2,21,6,16,10,24), (1,19,11,25,10,14)(2,17,7,20,13,21)(3,15)(4,26,12,23,6,22)(5,24,8,18,9,16)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$36$:  $C_6\times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

39T46 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Cycle 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 $ $1$ $1$ $()$
$ 13, 13 $ $18$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $18$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $13$ $(14,26,25,24,23,22,21,20,19,18,17,16,15)$
$ 13, 13 $ $36$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $36$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $18$ $13$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$
$ 13, 13 $ $18$ $13$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $13$ $(14,25,23,21,19,17,15,26,24,22,20,18,16)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $338$ $3$ $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$
$ 26 $ $234$ $26$ $( 1,19, 5,14, 9,22,13,17, 4,25, 8,20,12,15, 3,23, 7,18,11,26, 2,21, 6,16,10,24 )$
$ 26 $ $234$ $26$ $( 1,22, 8,23, 2,24, 9,25, 3,26,10,14, 4,15,11,16, 5,17,12,18, 6,19,13,20, 7,21 )$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $39$ $2$ $( 1,15)( 2,17)( 3,19)( 4,21)( 5,23)( 6,25)( 7,14)( 8,16)( 9,18)(10,20)(11,22) (12,24)(13,26)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $26$ $3$ $(15,17,23)(16,20,19)(18,26,24)(21,22,25)$
$ 13, 3, 3, 3, 3, 1 $ $156$ $39$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,22,20)(15,25,16)(17,18,21) (19,24,26)$
$ 13, 3, 3, 3, 3, 1 $ $156$ $39$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,17,26)(15,20,22)(16,23,18) (21,25,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $169$ $3$ $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$
$ 6, 6, 6, 6, 2 $ $507$ $6$ $( 1,19,10,24,11,26)( 2,21,13,17, 7,18)( 3,23)( 4,25, 6,16,12,15) ( 5,14, 9,22, 8,20)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $26$ $3$ $(15,23,17)(16,19,20)(18,24,26)(21,25,22)$
$ 13, 3, 3, 3, 3, 1 $ $156$ $39$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,22,16)(15,18,19)(17,23,25) (20,24,21)$
$ 13, 3, 3, 3, 3, 1 $ $156$ $39$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,17,18)(15,26,21)(16,22,24) (19,23,20)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $169$ $3$ $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$
$ 6, 6, 6, 6, 2 $ $507$ $6$ $( 1,19,12,15, 7,18)( 2,21, 8,20,10,24)( 3,23, 4,25,13,17)( 5,14, 9,22, 6,16) (11,26)$
$ 6, 6, 2, 2, 2, 2, 2, 2, 1, 1 $ $338$ $6$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,18,17,26,23,24)(16,22,20,25,19, 21)$
$ 6, 6, 6, 6, 1, 1 $ $169$ $6$ $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,24,23,26,17,18)(16,21,19,25,20,22)$
$ 6, 6, 6, 6, 2 $ $507$ $6$ $( 1,19, 6,22,12,23)( 2,17)( 3,15,11,25, 5,24)( 4,26, 7,20, 8,18) ( 9,16,13,21,10,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $169$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
$ 6, 6, 6, 6, 1, 1 $ $338$ $6$ $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,24,23,26,17,18)(16,21,19,25,20,22)$
$ 26 $ $234$ $26$ $( 1,19,13,21,12,23,11,25,10,14, 9,16, 8,18, 7,20, 6,22, 5,24, 4,26, 3,15, 2,17 )$
$ 26 $ $234$ $26$ $( 1,18, 7,19,13,20, 6,21,12,22, 5,23,11,24, 4,25,10,26, 3,14, 9,15, 2,16, 8,17 )$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $39$ $2$ $( 1,17)( 2,15)( 3,26)( 4,24)( 5,22)( 6,20)( 7,18)( 8,16)( 9,14)(10,25)(11,23) (12,21)(13,19)$
$ 6, 6, 2, 2, 2, 2, 2, 2, 1, 1 $ $338$ $6$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,24,23,26,17,18)(16,21,19,25,20, 22)$
$ 6, 6, 6, 6, 1, 1 $ $169$ $6$ $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,18,17,26,23,24)(16,22,20,25,19,21)$
$ 6, 6, 6, 6, 2 $ $507$ $6$ $( 1,19, 8,18, 3,15)( 2,17,11,25,12,23)( 4,26)( 5,24, 7,20,13,21) ( 6,22,10,14, 9,16)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $6084=2^{2} \cdot 3^{2} \cdot 13^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  6084.i
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);