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Properties

Label 26T38
Degree $26$
Order $5408$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{13}^2.D_4$

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Show commands: Magma

magma: G := TransitiveGroup(26, 38);
 

Group action invariants

Degree $n$:  $26$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $38$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{13}^2.D_4$
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:  (1,21,3,18,5,15,7,25,9,22,11,19,13,16,2,26,4,23,6,20,8,17,10,14,12,24), (1,13,12,11,10,9,8,7,6,5,4,3,2)(14,24,26,16)(15,19,25,21)(17,22,23,18)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$32$:  $C_4\wr C_2$

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

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 $ $16$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$
$ 13, 13 $ $16$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
$ 13, 13 $ $16$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $13$ $(14,19,24,16,21,26,18,23,15,20,25,17,22)$
$ 13, 13 $ $32$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$
$ 13, 13 $ $32$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
$ 13, 13 $ $32$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $13$ $(14,24,21,18,15,25,22,19,16,26,23,20,17)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $13$ $(14,21,15,22,16,23,17,24,18,25,19,26,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $169$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
$ 26 $ $208$ $26$ $( 1,21, 3,18, 5,15, 7,25, 9,22,11,19,13,16, 2,26, 4,23, 6,20, 8,17,10,14,12,24 )$
$ 26 $ $208$ $26$ $( 1,18, 7,22,13,26, 6,17,12,21, 5,25,11,16, 4,20,10,24, 3,15, 9,19, 2,23, 8,14 )$
$ 26 $ $208$ $26$ $( 1,22, 6,21,11,20, 3,19, 8,18,13,17, 5,16,10,15, 2,14, 7,26,12,25, 4,24, 9,23 )$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $52$ $2$ $( 1,16)( 2,21)( 3,26)( 4,18)( 5,23)( 6,15)( 7,20)( 8,25)( 9,17)(10,22)(11,14) (12,19)(13,24)$
$ 4, 4, 4, 4, 4, 4, 1, 1 $ $338$ $4$ $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,19,26,22)(16,24,25,17)(18,21,23,20)$
$ 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 $ $104$ $26$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,24)(15,23)(16,22)(17,21)(18,20) (25,26)$
$ 13, 2, 2, 2, 2, 2, 2, 1 $ $104$ $26$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,21)(15,20)(16,19)(17,18)(22,26) (23,25)$
$ 13, 2, 2, 2, 2, 2, 2, 1 $ $104$ $26$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,15)(16,26)(17,25)(18,24)(19,23) (20,22)$
$ 4, 4, 4, 4, 4, 4, 2 $ $676$ $4$ $( 1,21, 8,17)( 2,26, 7,25)( 3,18, 6,20)( 4,23, 5,15)( 9,22,13,16)(10,14,12,24) (11,19)$
$ 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 $ $169$ $4$ $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20)$
$ 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $26$ $4$ $(15,22,26,19)(16,17,25,24)(18,20,23,21)$
$ 13, 4, 4, 4, 1 $ $104$ $52$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,24,26,16)(15,19,25,21) (17,22,23,18)$
$ 13, 4, 4, 4, 1 $ $104$ $52$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,21,25,18)(15,16,24,23) (17,19,22,20)$
$ 13, 4, 4, 4, 1 $ $104$ $52$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,15,23,22)(16,18,21,19) (17,26,20,24)$
$ 4, 4, 4, 2, 2, 2, 2, 2, 2, 1, 1 $ $338$ $4$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,19,26,22)(16,24,25,17) (18,21,23,20)$
$ 8, 8, 8, 2 $ $676$ $8$ $( 1,21, 5,15,11,19, 7,25)( 2,26,13,16,10,14,12,24)( 3,18, 8,17, 9,22, 4,23) ( 6,20)$
$ 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $26$ $4$ $(15,19,26,22)(16,24,25,17)(18,21,23,20)$
$ 13, 4, 4, 4, 1 $ $104$ $52$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,24,22,25)(15,16,21,20) (17,26,19,23)$
$ 13, 4, 4, 4, 1 $ $104$ $52$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,21,17,23)(15,26,16,18) (19,20,25,24)$
$ 13, 4, 4, 4, 1 $ $104$ $52$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,15,20,19)(16,25,18,22) (21,24,26,23)$
$ 4, 4, 4, 2, 2, 2, 2, 2, 2, 1, 1 $ $338$ $4$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,22,26,19)(16,17,25,24) (18,20,23,21)$
$ 8, 8, 8, 2 $ $676$ $8$ $( 1,21, 6,20, 5,15,13,16)( 2,26,11,19, 4,23, 8,17)( 3,18)( 7,25,10,14,12,24, 9,22)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $5408=2^{5} \cdot 13^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  5408.l
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);