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Properties

Label 26T37
Degree $26$
Order $4056$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{13}^2:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(26, 37);
 

Group action invariants

Degree $n$:  $26$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $37$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{13}^2:C_6$
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:  (1,18,12,14,6,15)(2,20)(3,22,5,26,11,25)(4,24,8,19,7,17)(9,21,10,23,13,16), (1,16,6,24,12,18)(2,15)(3,14,11,19,5,25)(4,26,7,23,8,22)(9,21,13,17,10,20)
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
$8$:  $D_{4}$
$12$:  $C_6\times C_2$
$24$:  $D_4 \times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T32 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 $ $24$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
$ 13, 13 $ $24$ $13$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $24$ $13$ $(14,23,19,15,24,20,16,25,21,17,26,22,18)$
$ 13, 13 $ $24$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$
$ 13, 13 $ $12$ $13$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$
$ 13, 13 $ $24$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $12$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$
$ 13, 13 $ $12$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$
$ 13, 13 $ $12$ $13$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $169$ $3$ $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$
$ 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 $ $338$ $6$ $( 1,18,12,14, 6,15)( 2,20)( 3,22, 5,26,11,25)( 4,24, 8,19, 7,17) ( 9,21,10,23,13,16)$
$ 26 $ $156$ $26$ $( 1,26, 5,24, 9,22,13,20, 4,18, 8,16,12,14, 3,25, 7,23,11,21, 2,19, 6,17,10,15 )$
$ 26 $ $156$ $26$ $( 1,14,12,15,10,16, 8,17, 6,18, 4,19, 2,20,13,21,11,22, 9,23, 7,24, 5,25, 3,26 )$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $26$ $2$ $( 1,18)( 2,24)( 3,17)( 4,23)( 5,16)( 6,22)( 7,15)( 8,21)( 9,14)(10,20)(11,26) (12,19)(13,25)$
$ 6, 6, 6, 6, 2 $ $338$ $6$ $( 1,24,12,14, 7,15)( 2,16, 8,20,10,17)( 3,21, 4,26,13,19)( 5,18, 9,25, 6,23) (11,22)$
$ 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, 1, 1 $ $169$ $6$ $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,18,17,26,23,24)(16,22,20,25,19,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $169$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
$ 4, 4, 4, 4, 4, 4, 1, 1 $ $338$ $4$ $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,22,26,19)(16,17,25,24)(18,20,23,21)$
$ 12, 12, 1, 1 $ $338$ $12$ $( 2, 3, 5, 9, 4, 7,13,12,10, 6,11, 8)(15,25,18,19,17,21,26,16,23,22,24,20)$
$ 12, 12, 1, 1 $ $338$ $12$ $( 2, 7,11, 9,10, 3,13, 8, 4, 6, 5,12)(15,21,24,19,23,25,26,20,17,22,18,16)$
$ 6, 6, 6, 6, 2 $ $338$ $6$ $( 1,18, 2,15, 5,19)( 3,25, 8,23,10,17)( 4,22,11,14, 6,16)( 7,26) ( 9,20,13,21,12,24)$
$ 26 $ $156$ $26$ $( 1,26, 6,20,11,14, 3,21, 8,15,13,22, 5,16,10,23, 2,17, 7,24,12,18, 4,25, 9,19 )$
$ 26 $ $156$ $26$ $( 1,15, 5,18, 9,21,13,24, 4,14, 8,17,12,20, 3,23, 7,26,11,16, 2,19, 6,22,10,25 )$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $26$ $2$ $( 1,23)( 2,14)( 3,18)( 4,22)( 5,26)( 6,17)( 7,21)( 8,25)( 9,16)(10,20)(11,24) (12,15)(13,19)$
$ 6, 6, 6, 6, 2 $ $338$ $6$ $( 1,24, 8,17, 6,19)( 2,23, 4,21, 9,16)( 3,22,13,25,12,26)( 5,20) ( 7,18,10,15,11,14)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $4056=2^{3} \cdot 3 \cdot 13^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  4056.bd
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);