Properties

Label 26T45
Degree $26$
Order $8112$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{13}^2.C_{12}$

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Show commands: Magma

magma: G := TransitiveGroup(26, 45);
 

Group invariants

Abstract group:  $D_{13}^2.C_{12}$
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 $t$:  $45$
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:  $(1,19,13,22,11,15,7,14,12,25,9,21,3,26,4,23,6,17,10,18,5,20,8,24)(2,16)$, $(1,21,3,25,7,20,2,23,5,16,11,15,10,26,8,22,4,14,9,24,6,18,13,19)(12,17)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $C_6$ x 3
$8$:  $C_4\times C_2$
$12$:  $C_{12}$ x 2, $C_6\times C_2$
$16$:  $C_8:C_2$
$24$:  24T2
$48$:  24T16

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

There are no siblings 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 $2^{6},1^{14}$ $26$ $2$ $6$ $(14,18)(15,17)(19,26)(20,25)(21,24)(22,23)$
2B $2^{12},1^{2}$ $169$ $2$ $12$ $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$
3A1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1, 9, 3)( 2, 5, 6)( 4,10,12)( 7,11, 8)(15,23,17)(16,19,20)(18,24,26)(21,25,22)$
3A-1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1, 3, 9)( 2, 6, 5)( 4,12,10)( 7, 8,11)(15,17,23)(16,20,19)(18,26,24)(21,22,25)$
4A1 $4^{6},1^{2}$ $169$ $4$ $18$ $( 1, 4, 6, 3)( 2, 9, 5,11)( 7, 8,13,12)(14,18,25,21)(15,23,24,16)(17,20,22,19)$
4A-1 $4^{6},1^{2}$ $169$ $4$ $18$ $( 1, 7, 3,10)( 4, 5,13,12)( 6, 8,11, 9)(14,18,24,20)(15,26,23,25)(16,21,22,17)$
4B $4^{6},1^{2}$ $338$ $4$ $18$ $( 1, 8,12, 5)( 2, 3,11,10)( 4, 6, 9, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20)$
6A1 $6^{4},1^{2}$ $169$ $6$ $20$ $( 1,10, 9,12, 3, 4)( 2, 7, 5,11, 6, 8)(15,24,23,26,17,18)(16,21,19,25,20,22)$
6A-1 $6^{4},1^{2}$ $169$ $6$ $20$ $( 1, 4, 3,12, 9,10)( 2, 8, 6,11, 5, 7)(15,18,17,26,23,24)(16,22,20,25,19,21)$
6B1 $6^{2},3^{4},1^{2}$ $338$ $6$ $18$ $( 1, 2, 5)( 3, 8,10)( 4,11, 6)( 9,13,12)(14,18,19,16,25,24)(17,22,20,26,21,23)$
6B-1 $6^{2},3^{4},1^{2}$ $338$ $6$ $18$ $( 1, 8, 6)( 2, 4, 9)( 3,13,12)( 7,10,11)(14,18,21,20,16,26)(15,22,24,19,25,23)$
8A1 $8^{3},2$ $338$ $8$ $22$ $( 1,19, 9,15,10,21, 2,25)( 3,18, 6,23, 8,22, 5,17)( 4,24,11,14, 7,16,13,26)(12,20)$
8A-1 $8^{3},2$ $338$ $8$ $22$ $( 1,20,12,25, 9,26,11,21)( 2,24, 7,18, 8,22, 3,15)( 4,19,10,17, 6,14,13,16)( 5,23)$
8B1 $8^{3},2$ $338$ $8$ $22$ $( 1,23)( 2,17, 9,14,13,16, 6,19)( 3,24, 4,18,12,22,11,15)( 5,25, 7,26,10,21, 8,20)$
8B-1 $8^{3},2$ $338$ $8$ $22$ $( 1,17)( 2,21, 6,24,13,26, 9,23)( 3,25,11,18,12,22, 4,16)( 5,20, 8,19,10,14, 7,15)$
12A1 $12^{2},1^{2}$ $169$ $12$ $22$ $( 2, 3, 5, 9, 4, 7,13,12,10, 6,11, 8)(14,18,26,16,22,21,19,15,20,17,24,25)$
12A-1 $12^{2},1^{2}$ $169$ $12$ $22$ $( 1, 6, 2,13,12, 5, 8, 3, 7, 9,10, 4)(14,18,20,21,15,25,17,26,24,23,16,19)$
12A5 $12^{2},1^{2}$ $169$ $12$ $22$ $( 1, 5, 3, 4,10, 7, 2,11,13,12, 6, 9)(14,18,16,17,23,20,15,24,26,25,19,22)$
12A-5 $12^{2},1^{2}$ $169$ $12$ $22$ $( 1,10, 5, 2, 8, 9, 7,11, 3, 6,13,12)(14,18,23,26,20,19,21,17,25,22,15,16)$
12B1 $12^{2},1^{2}$ $338$ $12$ $22$ $( 1, 7,10, 5, 9,11,12, 6, 3, 8, 4, 2)(15,20,24,22,23,16,26,21,17,19,18,25)$
12B-1 $12^{2},1^{2}$ $338$ $12$ $22$ $( 1,11, 4, 5, 3, 7,12, 2, 9, 8,10, 6)(15,16,18,22,17,20,26,25,23,19,24,21)$
13A $13,1^{13}$ $24$ $13$ $12$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)$
13B $13^{2}$ $48$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,21,15,22,16,23,17,24,18,25,19,26,20)$
13C $13^{2}$ $48$ $13$ $24$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13D $13^{2}$ $48$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
24A1 $24,2$ $338$ $24$ $24$ $( 1,26, 8,18, 7,21, 9,15, 5,14,13,16,10,25, 3,20, 4,17, 2,23, 6,24,11,22)(12,19)$
24A-1 $24,2$ $338$ $24$ $24$ $( 1,24,12,21,11,26, 4,22, 7,20, 2,19, 6,25, 8,15, 9,23, 3,14,13,16, 5,17)(10,18)$
24A5 $24,2$ $338$ $24$ $24$ $( 1,18, 4,20, 9,19,13,26,11,16,12,21, 5,25, 2,23,10,24, 6,17, 8,14, 7,22)( 3,15)$
24A-5 $24,2$ $338$ $24$ $24$ $( 1,25, 6,20,10,16, 8,18, 9,17, 2,24,12,14, 7,19, 3,23, 5,21, 4,22,11,15)(13,26)$
24B1 $24,2$ $338$ $24$ $24$ $( 1,15, 6,23, 2,14,13,16,12,17, 5,24, 8,21, 3,26, 7,22, 9,20,10,19, 4,25)(11,18)$
24B-1 $24,2$ $338$ $24$ $24$ $( 1,23)( 2,20, 3,17, 5,24, 9,25, 4,14, 7,18,13,26,12,16,10,22, 6,21,11,19, 8,15)$
24B5 $24,2$ $338$ $24$ $24$ $( 1,15, 8,16, 9,18,11,22, 2,17,10,20,13,26, 6,25, 5,23, 3,19,12,24, 4,21)( 7,14)$
24B-5 $24,2$ $338$ $24$ $24$ $( 1,14,13,16, 2,25,11,20, 6,17, 3,23, 9,24,10,22, 8,26,12,18, 4,21, 7,15)( 5,19)$
26A $13,2^{6},1$ $312$ $26$ $18$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,18)(15,17)(19,26)(20,25)(21,24)(22,23)$

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(G);
 

Character table

35 x 35 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed