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Properties

Label 26T43
Degree $26$
Order $8112$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{13}\wr C_2:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(26, 43);
 

Group invariants

Abstract group:  $D_{13}\wr C_2:C_6$
magma: IdentifyGroup(G);
 
Order:  $8112=2^{4} \cdot 3 \cdot 13^{2}$
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 $t$:  $43$
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:  $(1,9)(2,8)(3,7)(4,6)(10,13)(11,12)(14,18,22,26,17,21,25,16,20,24,15,19,23)$, $(1,12,13,6,3,11,7,9,8,2,5,10)(14,15,22,19,24,20,18,17,23,26,21,25)$, $(1,25,5,22,9,19,13,16,4,26,8,23,12,20,3,17,7,14,11,24,2,21,6,18,10,15)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$3$:  $C_3$
$4$:  $C_2^2$ x 7
$6$:  $C_6$ x 7
$8$:  $C_2^3$
$12$:  $C_6\times C_2$ x 7
$16$:  $Q_8:C_2$
$24$:  24T3
$48$:  24T17

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T43 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 $2^{13}$ $26$ $2$ $13$ $( 1,23)( 2,19)( 3,15)( 4,24)( 5,20)( 6,16)( 7,25)( 8,21)( 9,17)(10,26)(11,22)(12,18)(13,14)$
2C $2^{13}$ $26$ $2$ $13$ $( 1,22)( 2,16)( 3,23)( 4,17)( 5,24)( 6,18)( 7,25)( 8,19)( 9,26)(10,20)(11,14)(12,21)(13,15)$
2D $2^{12},1^{2}$ $169$ $2$ $12$ $( 1,13)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(14,20)(15,19)(16,18)(21,26)(22,25)(23,24)$
3A1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1, 2,11)( 3, 7, 4)( 5,12,10)( 6, 8,13)(14,16,21)(15,25,24)(18,26,20)(19,22,23)$
3A-1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1,11, 2)( 3, 4, 7)( 5,10,12)( 6,13, 8)(14,21,16)(15,24,25)(18,20,26)(19,23,22)$
4A1 $4^{6},1^{2}$ $169$ $4$ $18$ $( 1, 3,13,11)( 2, 8,12, 6)( 4, 5,10, 9)(14,15,20,19)(16,25,18,22)(21,24,26,23)$
4A-1 $4^{6},1^{2}$ $169$ $4$ $18$ $( 1,11,13, 3)( 2, 6,12, 8)( 4, 9,10, 5)(14,19,20,15)(16,22,18,25)(21,23,26,24)$
4B $4^{6},2$ $338$ $4$ $19$ $( 1,14, 7,16)( 2,23, 6,20)( 3,19, 5,24)( 4,15)( 8,25,13,18)( 9,21,12,22)(10,17,11,26)$
4C $4^{6},1^{2}$ $338$ $4$ $18$ $( 1, 4, 6, 3)( 2, 9, 5,11)( 7, 8,13,12)(14,22,21,26)(15,17,20,18)(16,25,19,23)$
4D $4^{6},2$ $338$ $4$ $19$ $( 1,24, 7,21)( 2,17, 6,15)( 3,23, 5,22)( 4,16)( 8,14,13,18)( 9,20,12,25)(10,26,11,19)$
6A1 $6^{4},1^{2}$ $169$ $6$ $20$ $( 1,12, 5,13, 2, 9)( 3, 6,10,11, 8, 4)(14,26,16,20,21,18)(15,23,25,19,24,22)$
6A-1 $6^{4},1^{2}$ $169$ $6$ $20$ $( 1, 9, 2,13, 5,12)( 3, 4, 8,11,10, 6)(14,18,21,20,16,26)(15,22,24,19,25,23)$
6B1 $6^{2},3^{4},1^{2}$ $338$ $6$ $18$ $( 1, 7, 2, 4,11, 3)( 5, 8,12,13,10, 6)(14,21,16)(15,24,25)(18,20,26)(19,23,22)$
6B-1 $6^{2},3^{4},1^{2}$ $338$ $6$ $18$ $( 1, 3,11, 4, 2, 7)( 5, 6,10,13,12, 8)(14,16,21)(15,25,24)(18,26,20)(19,22,23)$
6C1 $6^{4},2$ $338$ $6$ $21$ $( 1,21, 6,23, 8,16)( 2,24, 9,19, 4,17)( 3,14,12,15,13,18)( 5,20)( 7,26,11,25,10,22)$
6C-1 $6^{4},2$ $338$ $6$ $21$ $( 1,16, 8,23, 6,21)( 2,17, 4,19, 9,24)( 3,18,13,15,12,14)( 5,20)( 7,22,10,25,11,26)$
6D1 $6^{4},2$ $338$ $6$ $21$ $( 1,18, 8,22, 6,19)( 2,26, 4,16, 9,17)( 3,21,13,23,12,15)( 5,24)( 7,14,10,25,11,20)$
6D-1 $6^{4},2$ $338$ $6$ $21$ $( 1,19, 6,22, 8,18)( 2,17, 9,16, 4,26)( 3,15,12,23,13,21)( 5,24)( 7,20,11,25,10,14)$
12A1 $12^{2},1^{2}$ $169$ $12$ $22$ $( 1, 4,12, 3, 5, 6,13,10, 2,11, 9, 8)(14,22,26,15,16,23,20,25,21,19,18,24)$
12A-1 $12^{2},1^{2}$ $169$ $12$ $22$ $( 1, 8, 9,11, 2,10,13, 6, 5, 3,12, 4)(14,24,18,19,21,25,20,23,16,15,26,22)$
12A5 $12^{2},1^{2}$ $169$ $12$ $22$ $( 1, 6, 9, 3, 2, 4,13, 8, 5,11,12,10)(14,23,18,15,21,22,20,24,16,19,26,25)$
12A-5 $12^{2},1^{2}$ $169$ $12$ $22$ $( 1,10,12,11, 5, 8,13, 4, 2, 3, 9, 6)(14,25,26,19,16,24,20,22,21,15,18,23)$
12B1 $12^{2},2$ $338$ $12$ $23$ $( 1,18, 5,14, 8,24, 7,25, 3,16,13,19)( 2,17, 9,23,11,21, 6,26,12,20,10,22)( 4,15)$
12B-1 $12^{2},2$ $338$ $12$ $23$ $( 1,19,13,16, 3,25, 7,24, 8,14, 5,18)( 2,22,10,20,12,26, 6,21,11,23, 9,17)( 4,15)$
12C1 $12^{2},1^{2}$ $338$ $12$ $22$ $( 1, 2,13, 4, 9,12, 6, 5, 7, 3,11, 8)(14,17,23,22,20,16,21,18,25,26,15,19)$
12C-1 $12^{2},1^{2}$ $338$ $12$ $22$ $( 1, 8,11, 3, 7, 5, 6,12, 9, 4,13, 2)(14,19,15,26,25,18,21,16,20,22,23,17)$
12D1 $12^{2},2$ $338$ $12$ $23$ $( 1,22,13,24, 3,18, 7,23, 8,21, 5,14)( 2,20,10,17,12,26, 6,25,11,15, 9,19)( 4,16)$
12D-1 $12^{2},2$ $338$ $12$ $23$ $( 1,14, 5,21, 8,23, 7,18, 3,24,13,22)( 2,19, 9,15,11,25, 6,26,12,17,10,20)( 4,16)$
13A $13,1^{13}$ $24$ $13$ $12$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)$
13B $13^{2}$ $24$ $13$ $24$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13C $13^{2}$ $24$ $13$ $24$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,15,16,17,18,19,20,21,22,23,24,25,26)$
13D $13^{2}$ $48$ $13$ $24$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13E $13^{2}$ $48$ $13$ $24$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,15,16,17,18,19,20,21,22,23,24,25,26)$
26A $13,2^{6},1$ $312$ $26$ $18$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,21)(15,20)(16,19)(17,18)(22,26)(23,25)$
26B $26$ $312$ $26$ $25$ $( 1,25, 5,22, 9,19,13,16, 4,26, 8,23,12,20, 3,17, 7,14,11,24, 2,21, 6,18,10,15)$
26C $26$ $312$ $26$ $25$ $( 1,23, 3,24, 5,25, 7,26, 9,14,11,15,13,16, 2,17, 4,18, 6,19, 8,20,10,21,12,22)$

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(G);
 

Character table

38 x 38 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed