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Properties

Label 26T15
Degree $26$
Order $1014$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{13}^2:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(26, 15);
 

Group invariants

Abstract group:  $C_{13}^2:C_6$
magma: IdentifyGroup(G);
 
Order:  $1014=2 \cdot 3 \cdot 13^{2}$
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 $t$:  $15$
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:  $(1,23,3,25,9,18)(2,24,6,15,5,14)(4,26,12,21,10,19)(7,16,8,17,11,20)(13,22)$, $(1,24,2,14,3,17,4,20,5,23,6,26,7,16,8,19,9,22,10,25,11,15,12,18,13,21)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$39$:  $C_{13}:C_3$
$78$:  $C_{13}:C_6$, 26T5

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T15 x 5

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 $2^{13}$ $13$ $2$ $13$ $( 1,17)( 2,20)( 3,23)( 4,26)( 5,16)( 6,19)( 7,22)( 8,25)( 9,15)(10,18)(11,21)(12,24)(13,14)$
3A1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1, 2, 5)( 3, 8,10)( 4,11, 6)( 9,13,12)(14,24,15)(16,17,20)(18,23,25)(19,26,21)$
3A-1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1, 5, 2)( 3,10, 8)( 4, 6,11)( 9,12,13)(14,15,24)(16,20,17)(18,25,23)(19,21,26)$
6A1 $6^{4},2$ $169$ $6$ $21$ $( 1,16, 2,17, 5,20)( 3,18, 8,23,10,25)( 4,19,11,26, 6,21)( 7,22)( 9,24,13,15,12,14)$
6A-1 $6^{4},2$ $169$ $6$ $21$ $( 1,20, 5,17, 2,16)( 3,25,10,23, 8,18)( 4,21, 6,26,11,19)( 7,22)( 9,14,12,15,13,24)$
13A1 $13^{2}$ $3$ $13$ $24$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
13A-1 $13^{2}$ $3$ $13$ $24$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,15,16,17,18,19,20,21,22,23,24,25,26)$
13A2 $13^{2}$ $3$ $13$ $24$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13A-2 $13^{2}$ $3$ $13$ $24$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,16,18,20,22,24,26,15,17,19,21,23,25)$
13B1 $13^{2}$ $6$ $13$ $24$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
13B2 $13^{2}$ $6$ $13$ $24$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13C1 $13^{2}$ $6$ $13$ $24$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,20,26,19,25,18,24,17,23,16,22,15,21)$
13C-1 $13^{2}$ $6$ $13$ $24$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13C2 $13^{2}$ $6$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
13C-2 $13^{2}$ $6$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,18,22,26,17,21,25,16,20,24,15,19,23)$
13D1 $13^{2}$ $6$ $13$ $24$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,19,24,16,21,26,18,23,15,20,25,17,22)$
13D-1 $13^{2}$ $6$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13D2 $13^{2}$ $6$ $13$ $24$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13D-2 $13^{2}$ $6$ $13$ $24$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13E1 $13,1^{13}$ $6$ $13$ $12$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)$
13E-1 $13,1^{13}$ $6$ $13$ $12$ $(14,21,15,22,16,23,17,24,18,25,19,26,20)$
13E2 $13,1^{13}$ $6$ $13$ $12$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)$
13E-2 $13,1^{13}$ $6$ $13$ $12$ $(14,15,16,17,18,19,20,21,22,23,24,25,26)$
13F1 $13^{2}$ $6$ $13$ $24$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,18,22,26,17,21,25,16,20,24,15,19,23)$
13F-1 $13^{2}$ $6$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26)$
13F2 $13^{2}$ $6$ $13$ $24$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,21,15,22,16,23,17,24,18,25,19,26,20)$
13F-2 $13^{2}$ $6$ $13$ $24$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,16,18,20,22,24,26,15,17,19,21,23,25)$
13G1 $13^{2}$ $6$ $13$ $24$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,21,15,22,16,23,17,24,18,25,19,26,20)$
13G-1 $13^{2}$ $6$ $13$ $24$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
13G2 $13^{2}$ $6$ $13$ $24$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13G-2 $13^{2}$ $6$ $13$ $24$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,22,17,25,20,15,23,18,26,21,16,24,19)$
13H1 $13^{2}$ $6$ $13$ $24$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13H-1 $13^{2}$ $6$ $13$ $24$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,17,20,23,26,16,19,22,25,15,18,21,24)$
13H2 $13^{2}$ $6$ $13$ $24$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13H-2 $13^{2}$ $6$ $13$ $24$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,20,26,19,25,18,24,17,23,16,22,15,21)$
26A1 $26$ $39$ $26$ $25$ $( 1,16, 5,15, 9,14,13,26, 4,25, 8,24,12,23, 3,22, 7,21,11,20, 2,19, 6,18,10,17)$
26A-1 $26$ $39$ $26$ $25$ $( 1,17,10,18, 6,19, 2,20,11,21, 7,22, 3,23,12,24, 8,25, 4,26,13,14, 9,15, 5,16)$
26A5 $26$ $39$ $26$ $25$ $( 1,14, 8,22, 2,17, 9,25, 3,20,10,15, 4,23,11,18, 5,26,12,21, 6,16,13,24, 7,19)$
26A-5 $26$ $39$ $26$ $25$ $( 1,19, 7,24,13,16, 6,21,12,26, 5,18,11,23, 4,15,10,20, 3,25, 9,17, 2,22, 8,14)$

Malle's constant $a(G)$:     $1/12$

magma: ConjugacyClasses(G);
 

Character table

40 x 40 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed