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Properties

Label 26T3
Degree $26$
Order $52$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{26}$

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Show commands: Magma

magma: G := TransitiveGroup(26, 3);
 

Group action invariants

Degree $n$:  $26$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $3$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{26}$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,22,15,10,3,23,18,12,5,26,20,13,7,2,21,16,9,4,24,17,11,6,25,19,14,8), (1,21)(2,22)(3,20)(4,19)(5,18)(6,17)(7,15)(8,16)(9,14)(10,13)(23,26)(24,25)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$26$:  $D_{13}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: $D_{13}$

Low degree siblings

26T3

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $52=2^{2} \cdot 13$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  52.4
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 13A1 13A2 13A3 13A4 13A5 13A6 26A1 26A3 26A5 26A7 26A9 26A11
Size 1 1 13 13 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 13A4 13A1 13A3 13A6 13A5 13A2 13A2 13A5 13A6 13A3 13A1 13A4
13 P 1A 2A 2B 2C 13A6 13A5 13A2 13A4 13A1 13A3 26A7 26A11 26A5 26A9 26A3 26A1
Type
52.4.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
52.4.1b R 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
52.4.1c R 1 −1 1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
52.4.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1
52.4.2a1 R 2 2 0 0 ζ13−6+ζ136 ζ13−1+ζ13 ζ13−5+ζ135 ζ13−2+ζ132 ζ13−4+ζ134 ζ13−3+ζ133 ζ13−3+ζ133 ζ13−4+ζ134 ζ13−2+ζ132 ζ13−5+ζ135 ζ13−1+ζ13 ζ13−6+ζ136
52.4.2a2 R 2 2 0 0 ζ13−5+ζ135 ζ13−3+ζ133 ζ13−2+ζ132 ζ13−6+ζ136 ζ13−1+ζ13 ζ13−4+ζ134 ζ13−4+ζ134 ζ13−1+ζ13 ζ13−6+ζ136 ζ13−2+ζ132 ζ13−3+ζ133 ζ13−5+ζ135
52.4.2a3 R 2 2 0 0 ζ13−4+ζ134 ζ13−5+ζ135 ζ13−1+ζ13 ζ13−3+ζ133 ζ13−6+ζ136 ζ13−2+ζ132 ζ13−2+ζ132 ζ13−6+ζ136 ζ13−3+ζ133 ζ13−1+ζ13 ζ13−5+ζ135 ζ13−4+ζ134
52.4.2a4 R 2 2 0 0 ζ13−3+ζ133 ζ13−6+ζ136 ζ13−4+ζ134 ζ13−1+ζ13 ζ13−2+ζ132 ζ13−5+ζ135 ζ13−5+ζ135 ζ13−2+ζ132 ζ13−1+ζ13 ζ13−4+ζ134 ζ13−6+ζ136 ζ13−3+ζ133
52.4.2a5 R 2 2 0 0 ζ13−2+ζ132 ζ13−4+ζ134 ζ13−6+ζ136 ζ13−5+ζ135 ζ13−3+ζ133 ζ13−1+ζ13 ζ13−1+ζ13 ζ13−3+ζ133 ζ13−5+ζ135 ζ13−6+ζ136 ζ13−4+ζ134 ζ13−2+ζ132
52.4.2a6 R 2 2 0 0 ζ13−1+ζ13 ζ13−2+ζ132 ζ13−3+ζ133 ζ13−4+ζ134 ζ13−5+ζ135 ζ13−6+ζ136 ζ13−6+ζ136 ζ13−5+ζ135 ζ13−4+ζ134 ζ13−3+ζ133 ζ13−2+ζ132 ζ13−1+ζ13
52.4.2b1 R 2 −2 0 0 ζ13−6+ζ136 ζ13−1+ζ13 ζ13−5+ζ135 ζ13−2+ζ132 ζ13−4+ζ134 ζ13−3+ζ133 −ζ13−3−ζ133 −ζ13−4−ζ134 −ζ13−2−ζ132 −ζ13−5−ζ135 −ζ13−1−ζ13 −ζ13−6−ζ136
52.4.2b2 R 2 −2 0 0 ζ13−5+ζ135 ζ13−3+ζ133 ζ13−2+ζ132 ζ13−6+ζ136 ζ13−1+ζ13 ζ13−4+ζ134 −ζ13−4−ζ134 −ζ13−1−ζ13 −ζ13−6−ζ136 −ζ13−2−ζ132 −ζ13−3−ζ133 −ζ13−5−ζ135
52.4.2b3 R 2 −2 0 0 ζ13−4+ζ134 ζ13−5+ζ135 ζ13−1+ζ13 ζ13−3+ζ133 ζ13−6+ζ136 ζ13−2+ζ132 −ζ13−2−ζ132 −ζ13−6−ζ136 −ζ13−3−ζ133 −ζ13−1−ζ13 −ζ13−5−ζ135 −ζ13−4−ζ134
52.4.2b4 R 2 −2 0 0 ζ13−3+ζ133 ζ13−6+ζ136 ζ13−4+ζ134 ζ13−1+ζ13 ζ13−2+ζ132 ζ13−5+ζ135 −ζ13−5−ζ135 −ζ13−2−ζ132 −ζ13−1−ζ13 −ζ13−4−ζ134 −ζ13−6−ζ136 −ζ13−3−ζ133
52.4.2b5 R 2 −2 0 0 ζ13−2+ζ132 ζ13−4+ζ134 ζ13−6+ζ136 ζ13−5+ζ135 ζ13−3+ζ133 ζ13−1+ζ13 −ζ13−1−ζ13 −ζ13−3−ζ133 −ζ13−5−ζ135 −ζ13−6−ζ136 −ζ13−4−ζ134 −ζ13−2−ζ132
52.4.2b6 R 2 −2 0 0 ζ13−1+ζ13 ζ13−2+ζ132 ζ13−3+ζ133 ζ13−4+ζ134 ζ13−5+ζ135 ζ13−6+ζ136 −ζ13−6−ζ136 −ζ13−5−ζ135 −ζ13−4−ζ134 −ζ13−3−ζ133 −ζ13−2−ζ132 −ζ13−1−ζ13

magma: CharacterTable(G);