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Properties

Label 26T43
Degree $26$
Order $8112$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{13}\wr C_2:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(26, 43);
 

Group action invariants

Degree $n$:  $26$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $43$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{13}\wr C_2:C_6$
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:  (1,9)(2,8)(3,7)(4,6)(10,13)(11,12)(14,18,22,26,17,21,25,16,20,24,15,19,23), (1,12,13,6,3,11,7,9,8,2,5,10)(14,15,22,19,24,20,18,17,23,26,21,25), (1,25,5,22,9,19,13,16,4,26,8,23,12,20,3,17,7,14,11,24,2,21,6,18,10,15)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$3$:  $C_3$
$4$:  $C_2^2$ x 7
$6$:  $C_6$ x 7
$8$:  $C_2^3$
$12$:  $C_6\times C_2$ x 7
$16$:  $Q_8:C_2$
$24$:  24T3
$48$:  24T17

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T43 x 2

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 $ $1$ $1$ $()$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $24$ $13$ $(14,22,17,25,20,15,23,18,26,21,16,24,19)$
$ 13, 13 $ $24$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $48$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$
$ 13, 13 $ $48$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$
$ 13, 13 $ $24$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $26$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
$ 13, 2, 2, 2, 2, 2, 2, 1 $ $312$ $26$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(14,22,17,25,20,15,23,18,26,21,16, 24,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $169$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
$ 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, 3, 3, 3, 3, 1, 1 $ $338$ $6$ $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,23,17)(16,19,20)(18,24,26)(21,25,22)$
$ 6, 6, 6, 6, 1, 1 $ $169$ $6$ $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,18,17,26,23,24)(16,22,20,25,19,21)$
$ 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, 3, 3, 3, 3, 1, 1 $ $338$ $6$ $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,17,23)(16,20,19)(18,26,24)(21,22,25)$
$ 6, 6, 6, 6, 1, 1 $ $169$ $6$ $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,24,23,26,17,18)(16,21,19,25,20,22)$
$ 12, 12, 1, 1 $ $338$ $12$ $( 2, 7,11, 9,10, 3,13, 8, 4, 6, 5,12)(15,21,24,19,23,25,26,20,17,22,18,16)$
$ 12, 12, 1, 1 $ $169$ $12$ $( 2, 8,11, 6,10,12,13, 7, 4, 9, 5, 3)(15,21,24,19,23,25,26,20,17,22,18,16)$
$ 12, 12, 1, 1 $ $169$ $12$ $( 2, 7,11, 9,10, 3,13, 8, 4, 6, 5,12)(15,20,24,22,23,16,26,21,17,19,18,25)$
$ 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 $ $169$ $12$ $( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(15,25,18,19,17,21,26,16,23,22,24,20)$
$ 12, 12, 1, 1 $ $169$ $12$ $( 2, 3, 5, 9, 4, 7,13,12,10, 6,11, 8)(15,16,18,22,17,20,26,25,23,19,24,21)$
$ 4, 4, 4, 4, 4, 4, 1, 1 $ $338$ $4$ $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,22,26,19)(16,17,25,24)(18,20,23,21)$
$ 4, 4, 4, 4, 4, 4, 1, 1 $ $169$ $4$ $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,22,26,19)(16,17,25,24)(18,20,23,21)$
$ 4, 4, 4, 4, 4, 4, 1, 1 $ $169$ $4$ $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20)$
$ 26 $ $312$ $26$ $( 1,25, 5,22, 9,19,13,16, 4,26, 8,23,12,20, 3,17, 7,14,11,24, 2,21, 6,18,10,15 )$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $26$ $2$ $( 1,15)( 2,24)( 3,20)( 4,16)( 5,25)( 6,21)( 7,17)( 8,26)( 9,22)(10,18)(11,14) (12,23)(13,19)$
$ 4, 4, 4, 4, 4, 4, 2 $ $338$ $4$ $( 1,25,10,15)( 2,21, 9,19)( 3,17, 8,23)( 4,26, 7,14)( 5,22, 6,18)(11,24,13,16) (12,20)$
$ 6, 6, 6, 6, 2 $ $338$ $6$ $( 1,22, 8,17, 3,15)( 2,25,11,26,12,16)( 4,18)( 5,21, 7,14,13,19) ( 6,24,10,23, 9,20)$
$ 12, 12, 2 $ $338$ $12$ $( 1,22, 7,14, 2,25, 4,18,11,26, 3,15)( 5,21, 8,17,12,16,13,19,10,23, 6,24) ( 9,20)$
$ 6, 6, 6, 6, 2 $ $338$ $6$ $( 1,21, 3,23, 8,15)( 2,22,12,19,11,18)( 4,24)( 5,25,13,20, 7,14) ( 6,26, 9,16,10,17)$
$ 12, 12, 2 $ $338$ $12$ $( 1,21,12,19, 4,24,11,18,13,20, 8,15)( 2,22, 3,23, 7,14,10,17, 9,16, 5,25) ( 6,26)$
$ 6, 6, 6, 6, 2 $ $338$ $6$ $( 1,26, 4,20,13,15)( 2,24, 7,14, 9,23)( 3,22,10,21, 5,18)( 6,16) ( 8,25,12,17,11,19)$
$ 12, 12, 2 $ $338$ $12$ $( 1,26,11,19, 7,14, 6,16, 9,23,13,15)( 2,24, 8,25, 3,22, 5,18,12,17, 4,20) (10,21)$
$ 6, 6, 6, 6, 2 $ $338$ $6$ $( 1,18, 6,19,12,15)( 2,26)( 3,21,11,20, 5,24)( 4,16, 7,14, 8,22) ( 9,17,13,23,10,25)$
$ 12, 12, 2 $ $338$ $12$ $( 1,18, 9,17, 2,26,13,23, 5,24,12,15)( 3,21, 4,16, 8,22,11,20,10,25, 6,19) ( 7,14)$
$ 26 $ $312$ $26$ $( 1,24, 6,20,11,16, 3,25, 8,21,13,17, 5,26,10,22, 2,18, 7,14,12,23, 4,19, 9,15 )$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $26$ $2$ $( 1,15)( 2,22)( 3,16)( 4,23)( 5,17)( 6,24)( 7,18)( 8,25)( 9,19)(10,26)(11,20) (12,14)(13,21)$
$ 4, 4, 4, 4, 4, 4, 2 $ $338$ $4$ $( 1,24, 9,15)( 2,18, 8,21)( 3,25, 7,14)( 4,19, 6,20)( 5,26)(10,22,13,17) (11,16,12,23)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $8112=2^{4} \cdot 3 \cdot 13^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  8112.bd
magma: IdentifyGroup(G);
 
Character table:    38 x 38 character table

magma: CharacterTable(G);