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Properties

Label 26T26
Degree $26$
Order $2704$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{13}^2:\OD_{16}$

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Show commands: Magma

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

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$16$:  $C_8:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

There are no siblings 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 $ $8$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)$
$ 13, 13 $ $16$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $16$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $16$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $16$ $13$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $16$ $13$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $16$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $16$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $16$ $13$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $16$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $26$ $2$ $(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$
$ 13, 2, 2, 2, 2, 2, 2, 1 $ $104$ $26$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
$ 13, 2, 2, 2, 2, 2, 2, 1 $ $104$ $26$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
$ 13, 2, 2, 2, 2, 2, 2, 1 $ $104$ $26$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
$ 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 $ $169$ $4$ $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20)$
$ 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 $ $169$ $4$ $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,22,26,19)(16,17,25,24)(18,20,23,21)$
$ 8, 8, 8, 2 $ $338$ $8$ $( 1,26, 3,18,13,17,11,25)( 2,22, 8,24,12,21, 6,19)( 4,14, 5,23,10,16, 9,20) ( 7,15)$
$ 8, 8, 8, 2 $ $338$ $8$ $( 1,15, 7,26, 3,23,10,25)( 2,19)( 4,14, 5,18,13,24,12,20)( 6,22, 8,17,11,16, 9,21)$
$ 8, 8, 8, 2 $ $338$ $8$ $( 1,22,10,24, 4,14, 8,25)( 2,15, 5,20, 3,21,13,16)( 6,26,11,17,12,23, 7,19) ( 9,18)$
$ 8, 8, 8, 2 $ $338$ $8$ $( 1,19, 6,15, 5,21,13,25)( 2,26,11,24, 4,14, 8,16)( 3,20)( 7,22,10,17,12,18, 9,23)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $2704=2^{4} \cdot 13^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  2704.s
magma: IdentifyGroup(G);
 
Character table:

Size
2 P
13 P
Type

magma: CharacterTable(G);