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Properties

Label 13T2
Degree $13$
Order $26$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $D_{13}$

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Show commands: Magma

magma: G := TransitiveGroup(13, 2);
 

Group action invariants

Degree $n$:  $13$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $2$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{13}$
CHM label:  $D(13)=13:2$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,3,4,5,6,7,8,9,10,11,12,13), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

26T2

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 1 $ $13$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
$ 13 $ $2$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)$
$ 13 $ $2$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)$
$ 13 $ $2$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)$
$ 13 $ $2$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)$
$ 13 $ $2$ $13$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)$
$ 13 $ $2$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $26=2 \cdot 13$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  26.1
magma: IdentifyGroup(G);
 
Character table:    
     2  1  1   .   .   .   .   .   .
    13  1  .   1   1   1   1   1   1

       1a 2a 13a 13b 13c 13d 13e 13f
    2P 1a 1a 13b 13d 13f 13e 13c 13a
    3P 1a 2a 13c 13f 13d 13a 13b 13e
    5P 1a 2a 13e 13c 13b 13f 13a 13d
    7P 1a 2a 13f 13a 13e 13b 13d 13c
   11P 1a 2a 13b 13d 13f 13e 13c 13a
   13P 1a 2a  1a  1a  1a  1a  1a  1a

X.1     1  1   1   1   1   1   1   1
X.2     1 -1   1   1   1   1   1   1
X.3     2  .   A   C   B   D   F   E
X.4     2  .   B   E   D   A   C   F
X.5     2  .   C   D   E   F   B   A
X.6     2  .   D   F   A   B   E   C
X.7     2  .   E   A   F   C   D   B
X.8     2  .   F   B   C   E   A   D

A = E(13)^3+E(13)^10
B = E(13)^4+E(13)^9
C = E(13)^6+E(13)^7
D = E(13)+E(13)^12
E = E(13)^5+E(13)^8
F = E(13)^2+E(13)^11

magma: CharacterTable(G);