Properties

Label 26T25
Degree $26$
Order $2704$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{13}^2.C_2^2$

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Show commands: Magma

magma: G := TransitiveGroup(26, 25);
 

Group invariants

Abstract group:  $D_{13}^2.C_2^2$
magma: IdentifyGroup(G);
 
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
magma: NilpotencyClass(G);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $C_2^3$
$16$:  $Q_8:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T25 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{26}$ $1$ $1$ $0$ $()$
2A $2^{6},1^{14}$ $26$ $2$ $6$ $( 1, 4)( 2, 3)( 5,13)( 6,12)( 7,11)( 8,10)$
2B $2^{13}$ $26$ $2$ $13$ $( 1,23)( 2,14)( 3,18)( 4,22)( 5,26)( 6,17)( 7,21)( 8,25)( 9,16)(10,20)(11,24)(12,15)(13,19)$
2C $2^{13}$ $26$ $2$ $13$ $( 1,23)( 2,17)( 3,24)( 4,18)( 5,25)( 6,19)( 7,26)( 8,20)( 9,14)(10,21)(11,15)(12,22)(13,16)$
2D $2^{12},1^{2}$ $169$ $2$ $12$ $( 1, 5)( 2, 4)( 6,13)( 7,12)( 8,11)( 9,10)(14,21)(15,20)(16,19)(17,18)(22,26)(23,25)$
4A1 $4^{6},1^{2}$ $169$ $4$ $18$ $( 1,13, 5, 6)( 2, 8, 4,11)( 7, 9,12,10)(14,22,21,26)(15,17,20,18)(16,25,19,23)$
4A-1 $4^{6},1^{2}$ $169$ $4$ $18$ $( 1, 6, 5,13)( 2,11, 4, 8)( 7,10,12, 9)(14,26,21,22)(15,18,20,17)(16,23,19,25)$
4B $4^{6},2$ $338$ $4$ $19$ $( 1,19, 5,16)( 2,15, 4,20)( 3,24)( 6,25,13,23)( 7,21,12,14)( 8,17,11,18)( 9,26,10,22)$
4C $4^{6},2$ $338$ $4$ $19$ $( 1,18, 3,17)( 2,24)( 4,23,13,25)( 5,16,12,19)( 6,22,11,26)( 7,15,10,20)( 8,21, 9,14)$
4D $4^{6},1^{2}$ $338$ $4$ $18$ $( 1, 8,12, 5)( 2, 3,11,10)( 4, 6, 9, 7)(14,17,19,16)(15,22,18,24)(20,21,26,25)$
13A1 $13,1^{13}$ $8$ $13$ $12$ $(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13A2 $13,1^{13}$ $8$ $13$ $12$ $(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13A4 $13^{2}$ $8$ $13$ $24$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
13B1 $13,1^{13}$ $8$ $13$ $12$ $(14,26,25,24,23,22,21,20,19,18,17,16,15)$
13B2 $13^{2}$ $8$ $13$ $24$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13B4 $13^{2}$ $8$ $13$ $24$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13C1 $13^{2}$ $8$ $13$ $24$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,22,17,25,20,15,23,18,26,21,16,24,19)$
13C2 $13^{2}$ $8$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13C4 $13^{2}$ $8$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,21,15,22,16,23,17,24,18,25,19,26,20)$
13D1 $13^{2}$ $16$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
13D2 $13^{2}$ $16$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13D4 $13^{2}$ $16$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13E1 $13^{2}$ $16$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,22,17,25,20,15,23,18,26,21,16,24,19)$
13E2 $13^{2}$ $16$ $13$ $24$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13E4 $13^{2}$ $16$ $13$ $24$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
26A1 $26$ $104$ $26$ $25$ $( 1,25, 5,14, 9,16,13,18, 4,20, 8,22,12,24, 3,26, 7,15,11,17, 2,19, 6,21,10,23)$
26A3 $26$ $104$ $26$ $25$ $( 1,17, 2,24, 3,18, 4,25, 5,19, 6,26, 7,20, 8,14, 9,21,10,15,11,22,12,16,13,23)$
26A7 $26$ $104$ $26$ $25$ $( 1,24, 3,25, 5,26, 7,14, 9,15,11,16,13,17, 2,18, 4,19, 6,20, 8,21,10,22,12,23)$
26B1 $13,2^{6},1$ $104$ $26$ $18$ $( 1, 4)( 2, 3)( 5,13)( 6,12)( 7,11)( 8,10)(14,17,20,23,26,16,19,22,25,15,18,21,24)$
26B3 $13,2^{6},1$ $104$ $26$ $18$ $( 1, 4)( 2, 3)( 5,13)( 6,12)( 7,11)( 8,10)(14,20,26,19,25,18,24,17,23,16,22,15,21)$
26B7 $13,2^{6},1$ $104$ $26$ $18$ $( 1, 4)( 2, 3)( 5,13)( 6,12)( 7,11)( 8,10)(14,19,24,16,21,26,18,23,15,20,25,17,22)$
26C1 $26$ $104$ $26$ $25$ $( 1,19,13,15,12,24,11,20,10,16, 9,25, 8,21, 7,17, 6,26, 5,22, 4,18, 3,14, 2,23)$
26C3 $26$ $104$ $26$ $25$ $( 1,24,11,25, 8,26, 5,14, 2,15,12,16, 9,17, 6,18, 3,19,13,20,10,21, 7,22, 4,23)$
26C7 $26$ $104$ $26$ $25$ $( 1,20,10,17, 6,14, 2,24,11,21, 7,18, 3,15,12,25, 8,22, 4,19,13,16, 9,26, 5,23)$

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(G);
 

Character table

34 x 34 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed